['Johnstone-Gleason covers for partially ordered sets'] author [{'given': 'Vakhtang', 'family': 'Abashidze'}] publication date 2023-01-09 volume 39 issue 01 page range ('1', '20') url http://www.tac.mta.ca/tac/volumes/39/1/39-01abs.html abstract In 1958, Andrew Gleason proved that for every compact Hausdorff space X there exists an extremally disconnected compact Hausdorff space X̃ and a continuous surjection p:X̃→ X with the property that every other continuous surjection from an extremally disconnected compact Hausdorff space onto X factors via surjection through p. Later, several authors have extended this construction to wider contexts, including the Gleason cover for an elementary topos introduced by Johnstone in 1980. We investigate properties of the Gleason cover for not necessarily sober T_0 Alexandroff spaces, i. e. spaces determined by partially ordered sets. First, we introduce the notion of co-local homeomorphism for such spaces, and prove that for every finite T_0 topological space X there exists a unique irreducible co-local homeomorphism p:X̃→ X from finite extremally disconnected space X̃ onto X. Next, we extend this approach to arbitrary Alexandroff topological spaces. We finish with several characterizations of Alexandroff spaces with Alexandroff Gleason covers. keywords Gleason cover, Alexandroff space, co-local homeomorphism ams class 06F30, 18F60, 18F70, 54B30, 54G05, 54D80 dois [] DOI citations: V. Abashidze, Absolute for finite $T_0$ spaces. Master thesis, Saint Andrew the First-Called Georgian University of the Patriarchate of Georgia, 2016 None B Banaschewski. Projective covers in categories of topological spaces and topological algebras. General topology and its relations to modern analysis and algebra, pages 63–91, 1971. None A Błaszczyk. Extremally disconnected resolutions of T0 -spaces. Colloquium Mathematicum, 1(32):57–68, 1974. ['Extremally disconnected resolutions of $T_0$-spaces'] 10.4064/cm-32-1-57-68 Jürgen Flachmeyer z. Z. Topologische projektivräume. Mathematische Nachrichten, 26(1-4):57–66, 1963. ['Topologische Projektivräume'] 10.1002/mana.19630260106 Andrew M Gleason et al.Projective topological spaces.Illinois Journal of Mathematics, 2(4A):482–489, 1958. ['Projective topological spaces'] 10.1215/ijm/1255454110 Stavros D Iliadis. Absolutes of hausdorff spaces. Doklady Akademii Nauk, 149(1):22–25, 1963. None L. Rudolf J. Mioduszewski. H-closed and extremally disconnected Hausdorff spaces. Dissertationes Mathematicae, 66, 1969. None Peter T Johnstone. The gleason cover of a topos, i. Journal of Pure and Applied Algebra, 19:171–192, 1980. ['The gleason cover of a topos, I'] 10.1016/0022-4049(80)90100-0 Peter T Johnstone. The gleason cover of a topos, ii. Journal of Pure and Applied Algebra, 22(3):229–247, 1981. ['The gleason cover of a topos, II'] 10.1016/0022-4049(81)90100-6 Peter T. Johnstone. Sketches of an Elephant: A Topos Theory Compendium: Volumes 1 and 2. Num- ber 43 in Oxford Logic Guides. Oxford Science Publications, 2002. None Jorge Picado and Aleš Pultr. Frames and Locales: topology without points. Springer Science \& Business Media, 2011. ['Frames and Locales'] 10.1007/978-3-0348-0154-6 Jack R Porter and R Grant Woods. Extensions and absolutes of Hausdorff spaces. Springer Science \& Business Media, 1988. ['Extensions and Absolutes of Hausdorff Spaces'] 10.1007/978-1-4612-3712-9 Marcel Erné; Kurt Stege. Counting finite posets and topologies. Order, 8, 1991. None V.M.Ul’janov. On compactifications satisfying the first axiom of countability and absolutes. Math- ematics of the USSR Sbornik”, 27(N 2):199–226, 1975. None S. Willard. General Topology. Addison Wesley series in mathematics/Lynn H.Loomis. Addison- Wesley Publishing Company, 1970. None L. B. Šapiro. The absolutes of topological spaces and of continuous mappings. Dokl. Akad. Nauk SSSR, 226(3):523–526, 1976. \endrefs \end{document} None -------------- ['A Hofmann-Mislove theorem for approach spaces'] author [{'given': 'Junche', 'family': 'Yu'}, {'given': 'Dexue', 'family': 'Zhang'}] publication date 2023-01-12 volume 39 issue 02 page range ('21', '50') url http://www.tac.mta.ca/tac/volumes/39/2/39-02abs.html abstract The Hofmann-Mislove theorem says that the ordered set of open filters of the open-set lattice of a sober topological space is isomorphic to the ordered set of compact saturated sets (ordered by reverse inclusion) of that space. This paper concerns a metric analogy of this result. To this end, the notion of compact functions of approach spaces is introduced. Such functions are an analog of compact subsets in the enriched context. It is shown that for a sober approach space X, the metric space of proper open [0,∞]-filters of the metric space of upper regular functions of X is isomorphic to the opposite of the metric space of inhabited and saturated compact functions of X, establishing a Hofmann-Mislove theorem for approach spaces. keywords Approach space, compact function, sober approach space, metric space, open [0,∞]-filter of a metric space ams class 18B35, 18F60, 54A05, 54B30 dois [] DOI citations: S. Antoniuk, P. Waszkiewicz, A duality of generalized metric spaces, Topology and its Applications 158 (2011) 2371-2381. ['A duality of generalized metric spaces'] 10.1016/j.topol.2011.04.013 B. Banaschewski, R. Lowen, C. Van Olmen, Sober approach spaces, Topology and its Applications 153 (2006) 3059-3070. ['Sober approach spaces'] 10.1016/j.topol.2006.05.003 M.M. Bonsangue, F. van Breugel, J.J.M.M. Rutten, Generalized metric space: completion, topology, and powerdomains via the Yoneda embedding, Theoretical Computer Science 193 (1998) 1-51. ['Generalized metric spaces: Completion, topology, and powerdomains via the Yoneda embedding'] 10.1016/s0304-3975(97)00042-x F. Borceux, {\em Handbook of Categorical Algebra, Vol. 2}, Cambridge University Press, 1994. ['Handbook of Categorical Algebra'] 10.1017/cbo9780511525865 M.M. Clementino, D. Hofmann, W. Tholen, One setting for all: metric, topology, uniformity, approach structure, Applied Categorical Structures 12 (2004) 127-154. ['One Setting for All: Metric, Topology, Uniformity, Approach Structure'] 10.1023/b:apcs.0000018144.87456.10 R.C. Flagg, P. S\"{u}nderhauf, The essence of ideal completion in quantitative form, Theoretical Computer Science 278 (2002) 141-158. ['The essence of ideal completion in quantitative form'] 10.1016/s0304-3975(00)00334-0 R.C. Flagg, P. S\"{u}nderhauf, K.R. Wagner, A logical approach to quantitative domain theory, Topology Atlas Preprint No. 23, 1996. http://at.yorku.ca/e/a/p/p/23.htm None G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, D.S. Scott, \emph{Continuous Lattices and Domains}, %Encyclopedia of Mathematics and its Applications, Vol. 93, Cambridge University Press, Cambridge, 2003. ['Continuous Lattices and Domains'] 10.1017/cbo9780511542725 J. Goubault-Larrecq, \emph{Non-Hausdorff Topology and Domain Theory}, Cambridge University Press, Cambridge, 2013. ['Non-Hausdorff Topology and Domain Theory'] 10.1017/cbo9781139524438 G. Gutierres, D. Hofmann, Approaching metric domains, Applied Categorical Structures 21 (2013) 617-650. ['Approaching Metric Domains'] 10.1007/s10485-011-9274-z D. Hofmann, Topological theories and closed objects, Advances in Mathematics 215 (2007) 789-824. ['Topological theories and closed objects'] 10.1016/j.aim.2007.04.013 D. Hofmann, Injective spaces via adjunction, Journal of Pure and Applied Algebra 215 (2011) 283-302. ['Injective spaces via adjunction'] 10.1016/j.jpaa.2010.04.021 D. Hofmann, G. J. Seal, W. Tholen (eds.), {\em Monoidal Topology: A Categorical Approach to Order, Metric, and Topology}, %volume 153 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2014. ['Monoidal Topology'] 10.1017/cbo9781107517288 D. Hofmann, P. Waszkiewicz, A duality of quantale-enriched categories, Journal of Pure and Applied Algebra 216 (2012) 1866-1878. ['A duality of quantale-enriched categories'] 10.1016/j.jpaa.2012.02.024 K.H. Hofmann, M.W. Mislove, Local compactness and continuous lattices, in: \emph{Continuous Lattices}, Lecture Notes in Mathematics, volume 871, Springer, 1981, pp. 209-248. ['Local compactness and continuous lattices'] 10.1007/bfb0089908 H. Lai, D. Zhang, Completely distributive enriched categories are not always continuous, Theory and Applications of Categories 35 (2020) 64-88. None H. Lai, D. Zhang, G. Zhang, A comparative study of ideals in fuzzy orders, Fuzzy Sets and Systems 382 (2020) 1-28. ['A comparative study of ideals in fuzzy orders'] 10.1016/j.fss.2018.11.019 F.W. Lawvere, Metric spaces, generalized logic, and closed categories, Rendiconti del Seminario Mat\'{e}matico e Fisico di Milano 43 (1973) 135-166. ['Metric spaces, generalized logic, and closed categories'] 10.1007/bf02924844 W. Li, D. Zhang, Sober metric approach spaces, Topology and its Applications 233 (2018) 67-88. ['Sober metric approach spaces'] 10.1016/j.topol.2017.10.019 W. Li, D. Zhang, Scott approach distance on metric spaces, Applied Categorical Structures 26 (2018) 1067-1093. ['Scott Approach Distance on Metric Spaces'] 10.1007/s10485-018-9527-1 R. Lowen, Kuratowski's measure of noncompactness revisited, The Quarterly Journal of Mathematics 39 (1988) 235-254. ["KURATOWSKI'S MEASURE OF NON-COMPACTNESS REVISITED"] 10.1093/qmath/39.2.235 R. Lowen, Approach spaces: a common supercategory of {\sf TOP} and {\sf MET}, Mathematische Nachrichten 141 (1989) 183-226. ['Approach Spaces A Common Supercategory of TOP and MET'] 10.1002/mana.19891410120 R. Lowen, \emph{Approach Spaces: the Missing Link in the Topology-Uniformity-Metric Triad}, Oxford University Press, 1997. ['Approach Spaces'] 10.1093/oso/9780198500308.001.0001 R. Lowen, \emph{Index Analysis, Approach Theory at Work}, Springer, 2015. ['Index Analysis'] 10.1007/978-1-4471-6485-2 D.S. Scott, Continuous lattices, in: %F.W. Lawvere (ed.), \emph{Toposes, Algebraic Geometry and Logic}, Lecture Notes in Mathematics, vol. 274, pp. 97-136. Springer, Berlin, 1972. None M. B. Smyth, Quasi-uniformities: Reconciling domains with metric spaces, in: \emph{Mathematical Foundations of Programming Language Semantics}, Lecture Notes in Computer Science, vol 298, pp. 236-253, Springer, Berlin, 1988. ['Quasi-uniformities: Reconciling domains with metric spaces'] 10.1007/3-540-19020-1_12 I. Stubbe, Categorical structures enriched in a quantaloid: categories, distributors and functors, Theory and Applications of Categories 14 (2005) 1-45. None I. Stubbe, Categorical structures enriched in a quantaloid: tensored and cotensored categories, Theory and Applications of Categories 16 (2006) 283-306. None S. Vickers, Localic completion of generalized metric spaces, Theory and Application of Categories 14 (2005) 328-356. None K.R. Wagner, Liminf convergence in $\Omega$-categories, Theoretical Computer Science 184 (1997) 61-104. ['Liminf convergence in Ω-categories'] 10.1016/s0304-3975(96)00223-x B. Windels, The Scott approach structure: an extension of the Scott topology for quantitative domain theory, Acta Mathematica Hungarica 88 (2000) 35-44. \end{thebibliography} \end{document} ['The Scott Approach Structure: An Extension of the Scott Topology for Quantitative Domain Theory'] 10.1023/a:1006792209118 -------------- ['Finite symmetries of quantum character stacks'] author [{'given': 'Corina', 'family': 'Keller'}, {'given': 'Lukas', 'family': 'Müller'}] publication date 2023-01-17 volume 39 issue 03 page range ('51', '97') url http://www.tac.mta.ca/tac/volumes/39/3/39-03abs.html abstract For a finite group D, we study categorical factorisation homology on oriented surfaces equipped with principal D-bundles, which `integrates' a (linear) balanced braided category 𝒜 with D-action over those surfaces. For surfaces with at least one boundary component, we identify the value of factorisation homology with the category of modules over an explicit algebra in 𝒜, extending the work of Ben-Zvi, Brochier and Jordan to surfaces with D-bundles. Furthermore, we show that the value of factorisation homology on annuli, boundary conditions, and point defects can be described in terms of equivariant representation theory. Our main example comes from an action of Dynkin diagram automorphisms on representation categories of quantum groups. We show that in this case factorisation homology gives rise to a quantisation of the moduli space of flat twisted bundles. keywords Factorisation homology, character varities, quantum groups, ribbon categories ams class 17B37, 18M15, 18M60, 57K16 dois [] DOI citations: S.{\,}Axelrod, S.{\,}Della Pietra, E.{\,}Witten. \emph{Geometric quantization of Chern--Simons gauge theory.} Journal of Differential Geometry (\textbf{1991}). 33(3):787--902. ['Geometric quantization of Chern-Simons gauge theory'] 10.4310/jdg/1214446565 D.{\,}Ayala, J.{\,}Francis. \emph{A factorization homology primer.} arXiv:1903.10961 (\textbf{2019}). ['A factorization homology primer'] 10.1201/9781351251624-2 D.{\,}Ayala, J.{\,}Francis. \emph{Factorization homology of topological manifolds.} Journal of Topology (\textbf{2015}). 8(4):1045--1084. ['Factorization homology of topological manifolds'] 10.1112/jtopol/jtv028 D.{\,}Ayala, J.{\,}Francis, H.{\,}L.{\,}Tanaka. \emph{Factorization homology of stratified spaces.} Selecta Mathematica (\textbf{2017}). 23(1):293--362. ['Factorization homology of stratified spaces'] 10.1007/s00029-016-0242-1 J.{\,}C.{\,}Baez, J.{\,}Dolan. \emph{Higher dimensional algebra and topological quantum field theory.} Journal of Mathematical Physics (\textbf{1995}). 36:6073--6105. ['Higher-dimensional algebra and topological quantum field theory'] 10.1063/1.531236 A.{\,}Brochier, D.{\,}Jordan, N.{\,}Snyder. \emph{On dualizability of braided tensor categories.} Compositio Mathematica (\textbf{2021}). 157(3):435--483. ['On dualizability of braided tensor categories'] 10.1112/s0010437x20007630 S.~Bunk, L.~Müller, and R.{\,}J.~Szabo. \emph{Smooth 2-group extensions and symmetries of bundle gerbes.} Communications in Mathematical Physics (\textbf{2021}). 384:1829--1911. ['Smooth 2-Group Extensions and Symmetries of Bundle Gerbes'] 10.1007/s00220-021-04099-7 A.{\,}Brochier. \emph{A Kohno--Drinfeld theorem for the monodromy of cyclotomic KZ connections.} Communications in Mathematical Physics (\textbf{2012}). 311:55--96. ['A Kohno–Drinfeld Theorem for the Monodromy of Cyclotomic KZ Connections'] 10.1007/s00220-012-1424-0 A.{\,}Brochier. \emph{Cyclotomic associators and finite type invariants for tangles in the solid torus.} Algebraic \& Geometric Topology (\textbf{2013}). 13:3365--3409. ['Cyclotomic associators and finite type invariants for tangles in the solid torus'] 10.2140/agt.2013.13.3365 D.{\,}Ben-Zvi, A.{\,}Brochier, D.{\,}Jordan. \emph{Integrating quantum groups over surfaces.} Journal of Topology (\textbf{2018}). 11(4):874--917. None D.{\,}Ben-Zvi, A.{\,}Brochier, D.{\,}Jordan. \emph{Quantum character varieties and braided monoidal categories.} Selecta Mathematica (\textbf{2018}). 24(5):4711--4748. ['Quantum character varieties and braided module categories'] 10.1007/s00029-018-0426-y D.{\,}Ben-Zvi, D.{\,}Nadler. \emph{Loop spaces and representations.} Duke Mathematical Journal (\textbf{2013}). 162(9):1587--1619. ['Loop spaces and representations'] 10.1215/00127094-2266130 D.{\,}Ben-Zvi, D.{\,}Nadler. \emph{Betti geometric Langlands.} arXiv:1606.08523 (\textbf{2016}). ['Betti Geometric Langlands'] 10.1090/pspum/097.2/01 V.{\,}Chari, A.{\,}N.{\,}Pressley. \emph{A guide to quantum groups.} Cambridge University Press, Cambridge, (\textbf{1995}). None D.{\,}Calaque, M.{\,}Gonzalez. \emph{A moperadic approach to cyclotomic associators.} arXiv:2004.00572 (\textbf{2020}). None J.{\,}Donin, A.{\,}Mudrov. \emph{Reflection equation, twist, and equivariant quantization.} Israel Journal of Mathematics (\textbf{2003}). 136:11--28. ['Reflection equation, twist, and equivariant quantization'] 10.1007/bf02807191 C.{\,}L.{\,}Douglas, C.{\,}Schommer-Pries, N.{\,}Snyder. \emph{Dualizable tensor categories.} Memoirs of the American Mathematical Society (\textbf{2020}). 268(1308). %arXiv:1312.7188 (\textbf{2013}). ['Dualizable tensor categories'] 10.1090/memo/1308 B.{\,}Enriquez. \emph{Quasi-reflection algebras and cyclotomic associators.} Selecta Mathematica (\textbf{2008}). 13:391--463. ['Quasi-reflection algebras and cyclotomic associators'] 10.1007/s00029-007-0048-2 J.{\,}Fuchs, J.{\,}Priel, C.{\,}Schweigert, A.{\,}Valentino. \emph{On the Brauer groups of symmetries of abelian Dijkgraaf--Witten theories}. Communications in Mathematical Physics (\textbf{2015}). 339(2):385--405. ['On the Brauer Groups of Symmetries of Abelian Dijkgraaf–Witten Theories'] 10.1007/s00220-015-2420-y V.{\,}V.{\,}Fock, A.{\,}A.{\,}Rosly. \emph{Poisson structure on moduli of flat connections on Riemann surfaces and $r$-matrix}. arXiv:9802054 (\textbf{1998}). ['Poisson structure on moduli of flat connections on Riemann surfaces and the 𝑟-matrix'] 10.1090/trans2/191/03 B.{\,}Fresse. \emph{Homotopy of operads and Grothendieck--Teichm\"uller groups. Part 1: The algebraic theory and its topological background.} Mathematical Surveys and Monographs 217, American Mathematical Society, Providence, RI (\textbf{2017}). None J.{\,}Fuchs, G.{\,}Schaumann, C.{\,}Schweigert. \emph{A trace for bimodule categories}. Applied Categorical Structures (\textbf{2017}). 25:227–-268. ['A Trace for Bimodule Categories'] 10.1007/s10485-016-9425-3 J.{\,}Fuchs, C.{\,}Schweigert, A.{\,}Valentino. \emph{Bicategories for boundary conditions and for surface defects in 3-d TFT}. Communications in Mathematical Physics (\textbf{2015}). 321:543--575. ['Bicategories for Boundary Conditions and for Surface Defects in 3-d TFT'] 10.1007/s00220-013-1723-0 C.{\,}Galindo. \emph{Coherence for monoidal $G$-categories and braided $G$-crossed categories}. Journal of Algebra (\textbf{2017}). 487:118--137. ['Coherence for monoidal G-categories and braided G-crossed categories'] 10.1016/j.jalgebra.2017.05.027 I.{\,}Ganev. \emph{The wonderful compactification for quantum groups}. Journal of the London Mathematical Society (\textbf{2018}). 99(2):778--806. ['The wonderful compactification for quantum groups'] 10.1112/jlms.12193 G.{\,}Ginot. \emph{Notes on factorization algebras, factorization homology and applications.} In: D.{\,}Calaque, T.{\,}Strobl (eds.). \emph{Mathematical Aspects of Quantum Field Theories, Mathematical Physics Studies}, 429--552. Springer International Publishing (\textbf{2015}). ['Mathematical Aspects of Quantum Field Theories'] 10.1007/978-3-319-09949-1 S.{\,}Gelaki, D.{\,}Naidu, D.{\,}Nikshych. \emph{Centers of graded fusion categories}. Algebra \& Number Theory (\textbf{2009}). 3(8):959--990. ['Centers of graded fusion categories'] 10.2140/ant.2009.3.959 P.{\,}Safronov. \emph{A categorical approach to quantum moment maps}. Theory and Applications of Categories (\textbf{2021}). 37(24):818--862. None O.{\,}Gwilliam, C.{\,}Scheimbauer. \emph{Duals and adjoints in the factorization higher Morita category.} arxiv:1804.10924 (\textbf{2018}). None S. Galatius, G. Szűcsr. \emph{The equivariant cobordism category.} Journal of Topology (\textbf{2021}). 14:215--257. ['The equivariant cobordism category'] 10.1112/topo.12181 R.{\,}Haugseng. \emph{The higher Morita category of $E_n$-algebras.} Geometry and Topology (\textbf{2017}). 21(3):1631--1730. ['The higher Morita category of\n𝔼n–algebras'] 10.2140/gt.2017.21.1631 N.{\,}J.{\,}Hitchin. \emph{Flat connections and geometric quantization.} Communications in Mathematical Physics (\textbf{1990}). 131(2):347--380. ['Flat connections and geometric quantization'] 10.1007/bf02161419 A. Henriques, D. Penneys, J. Tener. \emph{Categorified trace for module tensor categories over braided tensor categories.} Documenta Mathematica (\textbf{2016}). 21:1089--1149. ['Categorified trace for module tensor categories over braided tensor categories'] 10.4171/dm/553 J.{\,}E.{\,}Humphreys. \emph{Reflection groups and Coxeter groups.} Cambridge University Press (\textbf{1990}). ['Reflection Groups and Coxeter Groups'] 10.1017/cbo9780511623646 N.{\,}Idrissi. \emph{Swiss-cheese operad and Drinfeld center.} Israel Journal of Mathematics (\textbf{2017}). 221(2):941--972. ['Swiss-Cheese operad and Drinfeld center'] 10.1007/s11856-017-1579-7 T.{\,}Johnson-Freyd, C. Scheimbauer. \emph{(Op)lax natural transformations, twisted quantum field theories, and ``even higher" Morita categories.} Advances in Mathematics (\textbf{2017}). 307:147--223. ['(Op)lax natural transformations, twisted quantum field theories, and “even higher” Morita categories'] 10.1016/j.aim.2016.11.014 A. Kapustin, E. Witten. \emph{Electric-magnetic duality and the geometric Langlands program.} Communications in Number Theory and Physics (\textbf{2005}). 1:1--236. ['Electric-magnetic duality and the geometric Langlands program'] 10.4310/cntp.2007.v1.n1.a1 J.{\,}Lurie. \emph{Higher algebra.} Preprint available at: \url{https://www.math.ias.edu/~lurie/} None J.{\,}Lurie. \emph{On the classification of topological field theories.} Current Developments in Mathematics (\textbf{2009}). 129--280. ['On the Classification of Topological Field Theories'] 10.4310/cdm.2008.v2008.n1.a3 V.{\,}Lyubashenko. \emph{Modular transformations for tensor categories.} Journal of Pure and Applied Algebra (\textbf{1995}). 98(3):279--327. ['Modular transformations for tensor categories'] 10.1016/0022-4049(94)00045-k E.{\,}Meinrenken. {\em Convexity for twisted conjugation}. Mathematical Research Letters (\textbf{2017}). 24:1797--1818. ['Convexity for twisted conjugation'] 10.4310/mrl.2017.v24.n6.a12 L.{\,}Müller, R.{\,}J.{\,}Szabo. {\em {'t Hooft anomalies of discrete gauge theories and non-abelian group cohomology}}. Communications in Mathematical Physics (\textbf{2020}). 375:1581--1627. ['’t Hooft Anomalies of Discrete Gauge Theories and Non-abelian Group Cohomology'] 10.1007/s00220-019-03546-w L.{\,}Müller, L.{\,}Szegedy, R.{\,}J.{\,}Szabo. \emph{Symmetry defects and orbifolds of two-dimensional Yang--Mills theory.} Letters in Mathematical Physics (\textbf{2022}). 112(2). ['Symmetry defects and orbifolds of two-dimensional Yang–Mills theory'] 10.1007/s11005-021-01476-0 L.{\,}Müller, L.{\,}Woike. \emph{Equivariant higher Hochschild homology and topological field theories.} Homology, Homotopy and Applications (\textbf{2020}). 22(1):27--54. ['Equivariant higher Hochschild homology and topological field theories'] 10.4310/hha.2020.v22.n1.a3 L.{\,}Müller, L.{\,}Woike. \emph{The little bundles operad.} Algebraic \& Geometric Topology (\textbf{2020}). 20(4):2029--2070. ['The little bundles operad'] 10.2140/agt.2020.20.2029 L.{\,}Müller, L.{\,}Woike. \emph{Cyclic framed little disks algebras, Grothendieck--Verdier duality and handlebody group representations}. The Quarterly Journal of Mathematics (\textbf{2022}). ['Cyclic Framed Little Disks Algebras, Grothendieck–Verdier Duality And Handlebody Group Representations'] 10.1093/qmath/haac015 V.{\,}Ostrik. \emph{Module categories, weak Hopf algebras and modular invariants}. Transformation Groups (\textbf{2003}). 8:177--206. ['Module categories, weak Hopf algebras and modular invariants'] 10.1007/s00031-003-0515-6 P.{\,}Salvatore, N.{\,}Wahl. \emph{Framed discs operads and Batalin--Vilkovisky algebras.} The Quarterly Journal of Mathematics (\textbf{2003}). 54(2):213--231. ['Framed Discs Operads and Batalin-Vilkovisky Algebras'] 10.1093/qjmath/54.2.213 C.{\,}Scheimbauer. \emph{Factorization homology as a fully extended topological field theory.} Ph.D. thesis, ETH Zurich \textbf{(2014)}. None M.{\,}A.{\,}Semenov-Tian-Shansky. \emph{Poisson--Lie groups, quantum duality principle, and the quantum double.} Contemporary Mathematics (\textbf{1994}). 175:219--248. ['Poisson Lie groups, quantum duality principle, and the quantum double'] 10.1090/conm/175/01845 V.{\,}Turaev. \emph{Homotopy field theory in dimension 3 and crossed group-categories.} arXiv:0005291 (\textbf{2000}). None V.{\,}Turaev. \emph{Homotopy quantum field theory}. With appendices by M. Müger and A. Virelizier. European Mathematical Society (\textbf{2010}). ['Homotopy Quantum Field Theory'] 10.4171/086 A.{\,}A.{\,}Voronov. \emph{The Swiss-cheese operad.} Homotopy invariant algebraic structures, 239:365--373, in Contemporary Mathematics, American Mathematical Society, Providence, RI (\textbf{1999}). ['The Swiss-cheese operad'] 10.1090/conm/239/03610 T.{\,}A.{\,}Wasserman. \emph{The Drinfeld centre of a symmetric fusion category is 2-fold monoidal.} Advances in Mathematics (\textbf{2020}). 366. ['The Drinfeld centre of a symmetric fusion category is 2-fold monoidal'] 10.1016/j.aim.2020.107090 T.{\,}A.{\,}N. Weelinck. \emph{Equivariant factorization homology of global quotient orbifolds}. Advances in Mathematics (\textbf{2020}). 366. ['Equivariant factorization homology of global quotient orbifolds'] 10.1016/j.aim.2020.107072 T.{\,}Willwacher. \emph{The homotopy braces formality morphism.} Duke Mathematical Journal (\textbf{2016}). 165(10):1815--1964. ['The homotopy braces formality morphism'] 10.1215/00127094-3450644 E.{\,}Witten, \emph{On quantum gauge theories in two dimensions}. Communications in Mathematical Physics (\textbf{1991}). 141:153--209. ['On quantum gauge theories in two dimensions'] 10.1007/bf02100009 L.{\,}Woike. \emph{Higher categorical and operadic concepts for orbifold constructions - A study at the interface of topology and representation theory.} Ph.D. thesis available at: \url{https://ediss.sub.uni-hamburg.de/handle/ediss/8444} (\textbf{2020}). None J.{\,}C.{\,}Wang, X.{\,}G.{\,}Wen, E.{\,}Witten. \emph{Symmetric gapped interfaces of SPT and SET states: Systematic constructions}. Physical Review X \textbf{(2018)}. 8(3):031048. ['Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions'] 10.1103/physrevx.8.031048 A.{\,}J.{\,}Zerouali. \emph{Twisted moduli spaces and Duistermaat--Heckman measures.} Journal of Geometry and Physics \textbf{(2021)}. 161. \endrefs \end{document} ['Twisted moduli spaces and Duistermaat–Heckman measures'] 10.1016/j.geomphys.2020.104042 -------------- ['Enhanced twisted arrow categories'] author [{'given': 'Fernando Abellán', 'family': 'García'}, {'given': 'Walker H.', 'family': 'Stern'}] publication date 2023-01-26 volume 39 issue 04 page range ('98', '149') url http://www.tac.mta.ca/tac/volumes/39/4/39-04abs.html abstract Given an ∞-bicategory with underlying ∞-category , we construct a Cartesian fibration ()→×^, which we call the enhanced twisted arrow ∞-category, classifying the restricted mapping category functor Map_:^×→^×→_∞. With the aid of this new construction, we provide a description of the ∞-category of natural transformations Nat(F,G) as an end for any functors F and G from an ∞-category to an ∞-bicategory. As an application of our results, we demonstrate that the definition of weighted colimits studied by Gepner-Haugseng-Nikolaus satisfies the expected 2-dimensional universal property. keywords (∞,2)-category, twisted arrow category, natural transformation, weighted colimit ams class 18N10, 18N60, 18N65 dois [] DOI citations: Abell\'an Garc\'ia, F., Dyckerhoff, T., and Stern, W. H. \enquote{A relative 2-nerve}. {\em Algebr. Geom. Topol.} 20-6 (2020) pp. 3147--3182 ['A relative 2–nerve'] 10.2140/agt.2020.20.3147 Abell\'an Garc\'ia, F. and Stern, W. H. \emph{Theorem A for marked 2-categories.} J. Pure \& Applied Algebra, Vol. 226, Issue 9, 2022 ['Theorem A for marked 2-categories'] 10.1016/j.jpaa.2022.107040 Abell\'an Garc\'ia, F. and Stern, W. H. \enquote{2-Cartesian fibrations II: Higher cofinality}. 2022. arXiv:\href{https://arxiv.org/abs/2201.09589}{2201.09589} None Abell\'an Garc\'ia, F. \emph{Marked colimits and higher cofinality.} Homotopy Relat. Struct. 17, 1–22 (2022). DOI: \href{https://doi.org/10.1007/s40062-021-00296-2}{s40062-021-00296-2} None Barwick, C., Glasman, S. , and Nardin, D. \enquote{Dualizing cartesian and cocartesian fibrations}. {\em Theory and Applications of Categories}. Vol. 33 No. 4 (2018), pp. 67-94. None Dyckerhoff, T. and Kapranov, M. \enquote{Higher Segal Spaces}. Springer Lecture Notes in Mathematics 2244. {\em Springer}, 2019. ['Higher Segal Spaces'] 10.1007/978-3-030-27124-4 Barwick, C. \enquote{Spectral Mackey Functors and equivariant Algebraic K-Theory (I).} {\em Advances in Mathematics} Vol. 304 (Jan. 2017), pp. 646-727. ['Spectral Mackey functors and equivariant algebraic K-theory (I)'] 10.1016/j.aim.2016.08.043 Cisinski, D. \enquote{Higher Categories and Homotopical Algebra.} Cambridge University Press, 2019. ['Higher Categories and Homotopical Algebra'] 10.1017/9781108588737 Dugger, D. and Spivak, David I. \enquote{Rigidification of quasi-categories}. {\em Algebraic \& Geometric Topology} 11.1 (jan. 2011), pp. 225--261. \textsc{doi}: \href{https://doi.org/10.2140/agt.2011.11.225}{10.2140/agt.2011.11.225}. ['Rigidification of quasi-categories'] 10.2140/agt.2011.11.225 Dugger, D. \enquote{A primer on homotopy colimits}. University of Oregon, 2008. None Gagna, A., Harpaz, Y., Lanari, E. \enquote{On the equivalence of all models for $(\infty,2)$-categories.} {\em J. London Math. Soc.} 2022. None Gagna, A., Harpaz, Y., Lanari, E. \enquote{Fibrations and lax limits of $(\infty,2)$-categories.} 2020. arXiv:\href{https://arxiv.org/abs/2012.04537}{2012.04537} None Gepner, D., Haugseng R. and Nikolaus, T. \enquote{Lax colimits and free fibrations in $\infty$-categories}. {\em Documenta Mathematica } 1.15 (jan. 2015), vol 22. pp. 1225-1266 None Lurie, J. \enquote{Derived Algebraic Geometry {X}: Formal Moduli Problems} 2011 available at the author's webpage: \href{people.math.harvard.edu/~lurie/papers/DAG-X.pdf}{DAG X} None Lurie, J. \enquote{$(\infty, 2)$-categories and the Goodwillie Calculus}. 2009. arXiv: \href{http://arxiv.org/abs/0905.0462}{0905.0462} None Lurie, J. \enquote{Higher Topos Theory}. Princeton University Press, 2009. None Lurie, J. \enquote {Higher Algebra}. 2017 {\em Available at} \href{http://people.math.harvard.edu/~lurie/papers/HA.pdf}{\em the author's webpage.} None Lurie, J. \enquote{Kerodon}. 2022 \href{https://kerodon.net}{https://kerodon.net} \end{thebibliography} \end{document} None -------------- ['A Gray-categorical pasting theorem'] author [{'given': 'Nicola Di', 'family': 'Vittorio'}] publication date 2023-02-01 volume 39 issue 05 page range ('150', '171') url http://www.tac.mta.ca/tac/volumes/39/5/39-05abs.html abstract The notion of -category, a semi-strict 3-category in which the middle four interchange is weakened to an isomorphism, is central in the study of three-dimensional category theory. In this context it is common practice to use 2-dimensional pasting diagrams to express composites of 2-cells, however there is no thorough treatment in the literature justifying this procedure. We fill this gap by providing a formal approach to pasting in -categories and by proving that such composites are uniquely defined up to a contractible groupoid of choices. keywords Gray-categories, pasting diagrams ams class 18N20, 16S15, 03E20 dois [] DOI citations: Franz Baader and Tobias Nipkov, ``Term rewriting and all that'', Cambridge university press, 1999. None Nicola Di Vittorio, ``2-derivators'', \url{https://doi.org/10.25949/19817653.v1}, Master of Research Thesis, Macquarie University, 2020. None Simon Forest, ``Computational descriptions of higher categories'', PhD thesis, Institut Polytechnique de Paris, 2021. None Simon Forest and Samuel Mimram, ``Coherence of Gray categories via rewriting'', \emph{3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)}, Vol. 108, Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, 2018. None Robert Gordon, John Power and Ross Street, ``Coherence for tricategories'', Vol. 558, American Mathematical Soc., 1995. ['Coherence for tricategories'] 10.1090/memo/0558 Philip Hackney et al., ``An $(\infty, 2)$-categorical pasting theorem'', arXiv: 2106.03660, 2021. None G{\'e}rard Huet, ``Confluent reductions: Abstract properties and applications to term rewriting systems'', \emph{Journal of the ACM (JACM)}, 27.4 (1980), pp. 797-821. ['Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems'] 10.1145/322217.322230 Michael Johnson, ``Pasting diagrams in $n$-categories with applications to coherence theorems and categories of paths'', PhD thesis, University of Sydney, 1987. None Niles Johnson and Donald Yau, ``2-dimensional categories'', Oxford University Press, USA, 2021. ['2-Dimensional Categories'] 10.1093/oso/9780198871378.001.0001 G. M. Kelly, ``Basic concepts of enriched category theory'', \emph{Reprints in Theory and Applications of Categories} Vol. 10, 2005. None John Power, ``A 2-categorical pasting theorem'', \emph{Journal of Algebra}, 129.2 (1990), pp. 439-445. ['A 2-categorical pasting theorem'] 10.1016/0021-8693(90)90229-h Emily Riehl and Dominic Verity, ``Elements of $\infty$-Category Theory'', Cambridge University Press vol. 194, 2022. ['Elements of ∞-Category Theory'] 10.1017/9781108936880 Terese, ``Term rewriting systems'', Cambridge Tracts in Theoretical Computer Science 55. Cambridge University Press, 2003. None Dominic Verity, ``Enriched categories, internal categories and change of base'', \emph{Reprints in Theory and Applications of Categories}, 20 (2011), pp. 1--266. \endrefs None -------------- ['Pointed semibiproducts of monoids'] author [{'given': 'Nelson', 'family': 'Martins-Ferreira'}] publication date 2023-02-17 volume 39 issue 06 page range ('172', '185') url http://www.tac.mta.ca/tac/volumes/39/6/39-06abs.html abstract A new notion of a (pointed) semibiproduct is introduced, which, in the case of groups amounts to an extension equipped with a set-theoretical section. When the section is a group homomorphism then a pointed semibiproduct is the same as a group split extension. The main result of the paper is a characterization of pointed semibiproducts of monoids using a structure that is a generalization of the action that is used in the definition of a semidirect product of groups. keywords Semibiproduct, biproduct, semidirect product of groups and monoids, pointed semibiproduct, semibiproduct extension, pointed monoid action system, Schreier extension ams class 18G50, 20M10, 20M32 dois [] DOI citations: L. Lamport, Latex User's Guide \& %Reference Manual. Addison-Wesley (fifth edition), 1986. \begin{thebibliography}{10} None F. Borceux, D. Bourn, \emph{Mal'cev, protomodular, homological and semi-abelian categories}, Mathematics and Its Applications 566, Kluwer 2004. ['Mal’cev, Protomodular, Homological and Semi-Abelian Categories'] 10.1007/978-1-4020-1962-3 D. Bourn, \emph{From Groups to Categorial Algebra : Introduction to Protomodular and Mal'tsev Categories}, Compact Textbooks in Mathematics, Birkhäuser 2017. ['From Groups to Categorial Algebra'] 10.1007/978-3-319-57219-2 D. Bourn and G. Janelidze , \emph{Protomodularity, descent, and semidirect products}, Theory and Applications of Categories, \textbf{4}(2) (1998) 37--46. None D. Bourn and Z. Janelidze, \emph{Subtractive Categories and Extended Subtractions}, Appl. Categor. Struct. \textbf{17} (2009) 317--343. %https://doi.org/10.1007/s10485-008-9182-z ['Subtractive Categories and Extended Subtractions'] 10.1007/s10485-008-9182-z D. Bourn and Z. Janelidze, \emph{Categorical (binary) difference terms and protomodularity}, Algebra Universalis \textbf{66} (2011) 277--316. ['Categorical (binary) difference terms and protomodularity'] 10.1007/s00012-011-0156-x D. Bourn, N. Martins-Ferreira, A. Montoli and M. Sobral, \emph{Schreier split epimorphisms in monoids and in semirings}, Textos de Matemática Série B \textbf{45} (Departamento de Matemática, Universidade de Coimbra) 2013. %(ISBN 978-972-8564-49-0 None D. Bourn, N. Martins-Ferreira, A. Montoli, M. Sobral, \emph{Schreier split epimorphisms between monoids}, Semigroup Forum \textbf{88} (2014) 739--752. ['Schreier split epimorphisms between monoids'] 10.1007/s00233-014-9571-6 D. Bourn, N. Martins-Ferreira, A. Montoli and M. Sobral, \emph{Monoids and pointed S-protomodular categories}, Homology, Homotopy and Applications \textbf{48}(1) (2016) 151--172. ['Monoids and pointed $S$-protomodular categories'] 10.4310/hha.2016.v18.n1.a9 R. Brown, \emph{Possible connections between wiskered categories and groupoids, Leibniz algebras, automorphism structures and local-to-global questions}, Journal of Homotopy and Related Structures \textbf{1}(1) (2010) 1--13. None M. M. Clementino, N. Martins-ferreira and A. Montoli, \emph{On the categorical behaviour of preordered groups}, J. Pure Appl. Algebra \textbf{223} (2019) 4226--4245. ['On the categorical behaviour of preordered groups'] 10.1016/j.jpaa.2019.01.006 P. Faul, \emph{A characterization of weakly Schreier extensions of monoids}, J. Pure Appl. Algebra \textbf{225}(2) (2021) article no. 106489. ['A characterization of weakly Schreier extensions of monoids'] 10.1016/j.jpaa.2020.106489 I. Fleischer, \emph{Monoid extension theory}, J. Pure Appl. Algebra \textbf{21} (1981) 15l--159. ['Monoid extension theory'] 10.1016/0022-4049(81)90004-9 J. J. Ganci, \emph{Schreier extensions of topological semigroups}, PhD, Louisiana State University (1975). ['Schreier Extensions of Topological Semigroups.'] 10.31390/gradschool_disstheses.2789 M. Gran, G. Janelidze and M. Sobral, \emph{Split extensions and semidirect products of unitary magmas}, Comment. Math. Univ. Carolin. \textbf{60}(4) (2019) 509--527. %arxiv: 1906.02310v2 (math.CT). None G. Janelidze, L. Márki, W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra \textbf{168} (2002) 367--386. ['Semi-abelian categories'] 10.1016/s0022-4049(01)00103-7 J. Leech, \emph{Extending groups by monoids}, Journal of Algebra \textbf{74} (1982) 1--19. ['Extending groups by monoids'] 10.1016/0021-8693(82)90002-3 S. MacLane, \emph{Duality for groups}, Bull. Amer. Math. Soc. \textbf{56}(6) (1950) 485--516. ['Duality for groups'] 10.1090/s0002-9904-1950-09427-0 S. Mac Lane, \emph{Homology}, Springer-Verlag, Berlin, 1963. ['Homology'] 10.1007/978-3-642-62029-4 S. Mac Lane, \emph{Categories for the Working Mathematician}, 2ed, Graduate Texts in Mathematics 5, Springer, 1998. ['Categories for the Working Mathematician'] 10.1007/978-1-4612-9839-7 S. Mac Lane, \emph{Cohomology Theory in Abstract Groups. III} Annals of Mathematics, Second Series, \textbf{50}(3) (1949) 736--761. None N. Martins-Ferreira, \emph{On the notion of pseudocategory internal to a category with a 2-cell structure}, Tbil. Math. J. \textbf{8} (1) (2015) 107--141. ['On the notion of pseudocategory internal to a category with a 2-cell structure'] 10.1515/tmj-2015-0007 N. Martins-Ferreira, \emph{Semi-biproducts in monoids}, arXiv: 2002.05985v1 (2020). None N. Martins-Ferreira, \emph{Semi-biproducts of monoids}, arXiv: arXiv:2109.06278 (math.CT). None N. Martins-Ferreira, \emph{On semibiproducts of magmas and semigroups}, arXiv:2208.12704 (2022). ['On semibiproducts of magmas and semigroups'] 10.1007/s00233-024-10483-1 N. Martins-Ferreira and M. Sobral, \emph{Schreier split extensions of preordered monoids}, arXiv: 2004.14895v1 (math.CT). ['Schreier split extensions of preordered monoids'] 10.1016/j.jlamp.2021.100643 N. Martins-Ferreira and M. Sobral, \emph{Schreier split extensions of preordered monoids}, Journal of Logical and Algebraic Methods in Programming \textbf{120} (2021). ['Schreier split extensions of preordered monoids'] 10.1016/j.jlamp.2021.100643 N. Martins-Ferreira, A. Montoli, M. Sobral and A. Patchkoria, \emph{On the classification of Schreier extensions of monoids with non-abelian kernel}, Forum Math. \textbf{32} (3) (2020) 607--623. ['On the classification of Schreier extensions of monoids with non-abelian kernel'] 10.1515/forum-2019-0164 A. Montoli, D. Rodelo and T. Van der Linden, \emph{Intrinsic Schreier split extensions}, Appl. Categ. Structures \textbf{28} (2020) 517–538. ['Intrinsic Schreier Split Extensions'] 10.1007/s10485-019-09588-4 D. G. Northcott, \emph{An introduction to Homological Algebra}, Cambridge University Press, 1960. None C. Wells, \emph{Extension theories for monoids}, Semigroup Forum \textbf{16} (1978) 13--35. \end{thebibliography} %\endrefs ['Extension theories for monoids'] 10.1007/bf02194611 -------------- ['Corrigendum to “On the normally ordered tensor product and duality for Tate objects”'] author [{'given': 'Oliver', 'family': 'Braunling'}, {'given': 'Michael', 'family': 'Groechenig'}, {'given': 'Aron', 'family': 'Heleodoro'}, {'given': 'Jesse', 'family': 'Wolfson'}] publication date 2023-02-27 volume 39 issue 07 page range ('186', '188') url http://www.tac.mta.ca/tac/volumes/39/7/39-07abs.html abstract The definition of the shuffle product on ∞-Tate objects in the published article is erroneous. There is no problem with the shuffle product for n-Tate objects. keywords Tate vector space, Tate object, normally ordered product, higher adeles, higher local fields ams class 14A22, 18B30 dois [] DOI citations: Braunling, Groechenig, Heleodoro, Wolfson \emph{On the normally ordered tensor product and duality for Tate objects} Theory Appl. Categ., 33 (2018), 296-349 \end{thebibliography} \end{document} None -------------- ['On the construction of Noetherian forms for algebraic structures'] author [{'given': 'Francois Koch van', 'family': 'Niekerk'}] publication date 2023-03-01 volume 39 issue 08 page range ('189', '206') url http://www.tac.mta.ca/tac/volumes/39/8/39-08abs.html abstract A Noetherian form is a self-dual axiomatic context in which the Noether isomorphism theorems and other homomorphism theorems can be established. These theorems for group-like algebraic structures (for example groups, rings without unity and vector spaces) can be obtained by choosing a Noetherian form based on lattices of subalgebras. In this paper we show that by replacing lattices of subalgebras with some other lattices, it becomes possible to move beyond group-like structures and encompass all types of algebraic structures (including sets, monoids, lattices). Moreover, we show that in a suitable sense, existence of a Noetherian form for a give type of mathematical structure is intimately linked with algebraicity of structures. The isomorphism theorems resulting from applying these Noetherian forms recover the isomorphism theorems known for general algebraic structures in the literature. keywords Isomorphism Theorems, Monads, Noetherian Forms, Varieties ams class 08C05, 08A30, 18C15, 18D99 dois [] DOI citations: -------------- ['Fréchet modules and descent'] author [{'given': 'Oren', 'family': 'Ben-Bassat'}, {'given': 'Kobi', 'family': 'Kremnizer'}] publication date 2023-03-08 volume 39 issue 09 page range ('207', '266') url http://www.tac.mta.ca/tac/volumes/39/9/39-09abs.html abstract Motivated by classical functional analysis results over the complex numbers and results in the bornological setting over the complex numbers of R. Meyer, we study several aspects of the study of Ind-Banach modules over Banach rings. This allows for a synthesis of some aspects of homological algebra and functional analysis. This includes a study of nuclear modules and of modules which are flat with respect to the projective tensor product. We also study metrizable and Fréchet Ind-Banach modules. We give explicit descriptions of projective limits of Banach rings as ind-objects. We study exactness properties of the projective tensor product with respect to kernels and countable products. As applications, we describe a theory of quasi-coherent modules in Banach algebraic geometry. We prove descent theorems for quasi-coherent modules in various analytic and arithmetic contexts and relate them to well known complexes of modules coming from covers. keywords Banach, Fréchet, derived algebraic geometry, derived analytic geometry, descent, rings and modules, Banach algebras ams class 46J05, 46J10, 46J15, 46M15, 46M18, 46M40, 46M05, 46M10, 18F20, 26E30, 46S10, 32P05 dois [] DOI citations: Ad\'{a}mek, J., Lawvere, F. W. and Rosicky, J., {How algebraic is algebra?,} Theory and Applications of Categories, Vol. 8, No. 9, 253-283, 2001. None Ad\'{a}mek, J., Rosicky, J., {Locally Presentable and Accessible Categories,} London Mathematical Society, Lecture Note Series 189, Cambridge University Press, 1994. None Ardakov, K., Wadsley, S., {On irreducible representations of compact $p$-adic analytic groups,} Annals of Mathematics 178, 453-557. ['On irreducible representations of compact p-adic analytic groups'] 10.4007/annals.2013.178.2.3 Ardakov, K., Ben-Bassat, O., {Bounded linear endomorphisms of rigid analytic functions,} Proceedings of the London Mathematical Society 117 (5), 881-900, 2018. ['Bounded linear endomorphisms of rigid analytic functions'] 10.1112/plms.12142 Aristov, O., Pirkovskii, A. Yu., {Open embeddings and pseudoflat epimorphisms,} {\tt arXiv:1908.02117} ['Open embeddings and pseudoflat epimorphisms'] 10.1016/j.jmaa.2019.123817 Bambozzi, F., {On a generalization of affinoid varieties,} PhD thesis, University of Padova, 2013. None Bambozzi, F., Ben-Bassat, O., {Dagger geometry as Banach algebraic geometry,} Journal of Number Theory 162, 391-462, 2016. ['Dagger geometry as Banach algebraic geometry'] 10.1016/j.jnt.2015.10.023 Bambozzi, F., Ben-Bassat, O., Kremnizer, K., {Stein domains in Banach algebraic geometry,} Journal of Functional Analysis, Volume 274, Issue 7, 2018. ['Stein domains in Banach algebraic geometry'] 10.1016/j.jfa.2018.01.003 Bambozzi, F., Ben-Bassat, O., Kremnizer, K., {Analytic geometry over F1 and the Fargues-Fontaine curve,} Advances in Mathematics 356, 2019. None Banica, C., Stanasila, O., {Algebraic methods in the %global theory of Complex Spaces,} Editura Academiei, 1976. None Bousfield, A.K., Friedlander, E.M., Lecture Notes in Mathematics, Vol. 658, Springer Berlin, 80--130, 1978. None Ben-Bassat, O., Kremnizer, K., Non-Archimedean analytic geometry as relative algebraic geometry, Annales de la Facult\'{e} des Sciences de Toulouse S\'{e}r. 6, 26, 1, 2017. ['Non-Archimedean analytic geometry as relative algebraic geometry'] 10.5802/afst.1526 Ben-Bassat, O., Kelly, J., Kremnizer, K., {A perspective on derived algebraic geometry,} preprint. None Ben-Bassat, O., Temkin, M., {Berkovich Spaces and Tubular Descent,} Advances in Mathematics 234, 217-238, 2013. ['Berkovich spaces and tubular descent'] 10.1016/j.aim.2012.10.016 Behrend, K., Dhillon, A., {Connected components %of moduli stacks of torsors via Tamagawa numbers,} %Canad. J. Math. 61, 3-28, 2009. None Bosch, S., G\"{u}ntzer, U., Remmert, R., {Non-archimedean analysis. A systematic approach to rigid analytic geometry,} Springer, 1984. ['Non-Archimedean Analysis'] 10.1007/978-3-642-52229-1 Berkovich, V., {Spectral Theory and Analytic Geometry Over Non-archimedean Fields,} American Mathematical Society, 1990. ['Spectral theory'] 10.1090/surv/033/08 Berkovich, V., {Non-archimedean Analytic Spaces,} Advanced School on p-adic Analysis and Applications, ICTP, Trieste, 2009. None Bourbaki, N., Topological Vector Spaces, Volume 1, Springer, 1987. ['Topological Vector Spaces'] 10.1007/978-3-642-61715-7 Block, J., {Mayer-Vietoris sequences in cyclic homology of topological algebras,} MSRI 01208-88, available from author's homepage, 1987. None Cuntz, J. and Deninger, C., {An alternative to Witt vectors} {\tt arXiv:1311.2774v2} None Chaung, J. and Lazarev, A., {Homological Epimorphims and Homotopy Epimorphisms,} {\tt arXiv:1908.11283v1} None Cigler, J., Losert, V., Michor, P., {Banach Modules and Functors on Categories of Banach Spaces,} Marcel Decker Inc., 1979. ['Functors and Categories of Banach Spaces'] 10.1007/bfb0067828 Deligne, P., {Cat\'{e}gories Tannakiennes,} The Grothendieck Festschrift, %vol. 2, edited by P. Cartier et al., Progr. Math. 87, Birkh\"{a}user, 1990. ['Catégories tannakiennes'] 10.1007/978-0-8176-4575-5_3 Dugger, D., Hollander, S., Isaksen, D.C., {Hypercovers and Simplicial Presheaves,} Math. Proc. Cambridge Philos. Soc. 131 (1) , 2004. ['Hypercovers and simplicial presheaves'] 10.1017/s0305004103007175 A. Grothendieck. \'El\'ements de g\'eom\'etrie %alg\'ebrique. I. \'Etude locale des sch\'emas et des morphismes %de sch\'emas I. Inst. Hautes \'Etudes Sci. Publ. Math. None A. Grothendieck. \'El\'ements de g\'eom\'etrie %alg\'ebrique. IV. \'Etude locale des sch\'emas et des morphismes %de sch\'emas IV. Inst. Hautes \'Etudes Sci. Publ. Math., (32):361, 1967 None Eschmeier, J., Putinar, M., {Spectral %Decompositions and Analytic Sheaves,} Oxford University Press, 1996. None Forster, O., {Zur Theorie der Steinschen %Algebren und Moduln,} Mathematische Zeitschrift %97, 376-405, 1967. None Fischer, G., {Complex Analytic Geometry,} %Lecture Notes in Mathematics 538, Springer-Verlag, 1976. ['Complex Analytic Geometry'] 10.1007/bfb0080338 Goerss, P., Jardine, J., {Simplicial Homotopy Theory,} Brikhauser Verlag, Reprint of the 1999 Edition. ['Simplicial Homotopy Theory'] 10.1007/978-3-0348-8707-6 Geigle, W., Lenzing, H., {Perpendicular categories with applications to representations and sheaves,} J. Alg., 144, 1991. ['Perpendicular categories with applications to representations and sheaves'] 10.1016/0021-8693(91)90107-j Goerss, P., Schemmerhorn, K., {Model Categories and Simplicial Methods,} Contemporary Mathematics ['Model categories and simplicial methods'] 10.1090/conm/436/08403 Gruson, L., {Th\'{e}orie de Fredholm p-adique,} Bulletin de la S.M.F., tome 94, 1966. ['Théorie de Fredholm $p$-adique'] 10.24033/bsmf.1635 Grauert, H., Remmert, R., {Theory of Stein Spaces,} %(reprint of 1979 edition) Springer, 2004. ['Theory of Stein Spaces'] 10.1007/978-3-642-18921-0 Hakim, M., {Topos anneles et schemas realtifes,} Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer-Verlag, 1972. None Elmendorf, A.D., Kriz, I., Mandell, M.A., May J.P., {Rings, Modules, and Algebras in Stable Homotopy Theory} None Houzel, C., {Espaces analytiques relatifs et th\'{e}or\`{e}mes de finitude, } Math. Ann. 205, 13-54, 1973. ['Espaces analytiques relatifs et th�or�me de finitude'] 10.1007/bf01432513 Helemskii, A. Ya., {Lectures and Exercises on Functional Analysis,} Translations of Mathematical Monographs, Volume 233, AMS, 2006. None D.A. Higgs, K.A. Rowe, {Nuclearity in the category of complete semilattices,} Journal of Pure and Applied Algebra, 57, 67-78m 1989. ['Nuclearity in the category of complete semilattices'] 10.1016/0022-4049(89)90028-5 Hovey, {Model categories,} Mathematical surveys and monographs, Vol. 63, Amer. Math. %Soc., Providence, 1998. ['Model categories'] 10.1090/surv/063/01 Huber, R., {Continuous valuations,} Mathematische Zeitschrift 212, Springer-Verlag, 1993. ['Continuous valuations'] 10.1007/bf02571668 Kontsevich, M., Rosenberg, A., {Noncommutative smooth spaces,} {\tt arXiv:math/9812158}. ['Noncommutative Smooth Spaces'] 10.1007/978-1-4612-1340-6_5 Lenzing, H., {Endlich pr\"{a}sentierbare Moduln,} Arch. der Math., 20, 262-266, 1969. ['Endlich präsentierbare Moduln'] 10.1007/bf01899297 Meyer, R., {Local and Analytic Cyclic Homology,} European Mathematical Society, 2007. ['Local and Analytic Cyclic Homology'] 10.4171/039 Meyer, R., {Embeddings of Derived Categories of Bornological Modules,} {\tt arXiv:math/0410596}. None Neeman, A. and Ranicki, A., {Noncommutative Localization and Chain Complexes I. Algebraic K. and L-Theory} None Orlov, D., {Quasi-coherent sheaves in commutative and noncommutative geometry} ['Quasi-coherent sheaves in commutative and non-commutative geometry'] 10.1070/im2003v067n03abeh000437 Palamodov, V.P., Homological methods in the theory of locally convex spaces (Russian), Uspehi Mat. Nauk 26 (1971), no. 1 (157), English transl.: Russian Math. Surveys 26, 1971. ['Homological methods in the theory of locally convex spaces'] 10.1070/rm1971v026n01abeh003815 Paugam, F., {Global analytic geometry,} {\tt http://arxiv.org/pdf/0803.0148v3.pdf} ['Global analytic geometry'] 10.1016/j.jnt.2009.05.001 Paugam, F., {Overconvergent Global Analytic Geometry,} see author's webpage. ['Global analytic geometry'] 10.1016/j.jnt.2009.05.001 Poineau, J., {Les espaces de Berkovich sont ang\'{e}liques,} Bulletin de la SMF 141 (2), p. 267-297, 2013. ['Les espaces de Berkovich sont angéliques'] 10.24033/bsmf.2648 Pirkovskii, A. Yu., {On Certain Homological Properties of Stein Algebras,} Journal of Mathematical Sciences, Vol. 95, No. 6, 1999. ['On certain homological properties of Stein algebras'] 10.1007/bf02169288 Prosmans, F., {Alg\`{e}bre Homologique Quasi-Ab\'{e}lienne,} Laboratoire Analyse, G\'{e}om\'{e}trie et Applications, URA CNRS 742 None Prosmans, F., {Derived Projective Limits of Topological Abelian Groups,} Journal of Functional Analysis 162, 1999. ['Derived Projective Limits of Topological Abelian Groups'] 10.1006/jfan.1998.3370 Prosmans, F., Schneiders, J.-P., A Topological Reconstruction Theorem for $D^{\infty}$-Modules, Duke Math. J. 102, 39-86, 2000. None Ramis, J.-P., Ruget, G., {R\'{e}sidus et dualit\'{e},} Invent. Math. 26, 89, 1974. None Schneider, P., {Nonarchimedean Functional Analysis,} Expanded and Revised course notes from Winter 1997/1998 course at the University of M\"unster 2005, available online. ['Nonarchimedean Functional Analysis'] 10.1007/978-3-662-04728-6 Schneiders, J-P., {Quasi-Abelian Categories and Sheaves,} M\'{e}moires de la S.M.F. deuxieme s\'{e}rie, tome 76, 1999. ['Quasi-abelian categories and sheaves'] 10.24033/msmf.389 Schwede, S., {Spectra in model categories and applications to the algebraic cotangent complex,} Journal of Pure and Applied Algebra 120, 77-104, 1997. ['Spectra in model categories and applications to the algebraic cotangent complex'] 10.1016/s0022-4049(96)00058-8 Soibelman, Y., {On non-commutative analytic spaces over non-archimedean fields,} {\tt arXiv:math/0606001}. ['On Non-Commutative Analytic Spaces Over Non-Archimedean Fields'] 10.1007/978-3-540-68030-7_7 Taylor, J. L., {A general framework for a multi-operator functional calculus,} Adv. Math., 9, 1972. ['A general framework for a multi-operator functional calculus'] 10.1016/0001-8708(72)90017-5 Temkin, M., {Introduction to Berkovich Analytic Spaces,} available from author's homepage. ['Introduction to Berkovich Analytic Spaces'] 10.1007/978-3-319-11029-5_1 Temkin, M., {A new proof of the Gerritzen-Grauert theorem,} Math. Ann. 333, 261-269 (2005). ['A new proof of the Gerritzen-Grauert theorem'] 10.1007/s00208-005-0660-4 Thuillier, A.: {G\'eom\'etrie toro\"idale et g\'eom\'etrie analytique non archim\'edienne. Application au type d'homotopie de certains sch\'emas formels}, Manuscripta Math. 123, 381-451, 2007. ['Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels'] 10.1007/s00229-007-0094-2 To\"{e}n, B. {Simplicial presheaves and derived algebraic geometry,} lecture notes available from author's homepage, May 2009. None To\"{e}n, Vaqui\'{e}, {Under Spec Z}, J. K-Theory 3, no. 3, 437--500, 2009. None To\"{e}n, Vaqui\'{e}, Alg\'ebisation des vari\'et\'es analytiques complexes et cat\'egories d\'eriv\'ees ['Algébrisation des variétés analytiques complexes et catégories dérivées'] 10.1007/s00208-008-0257-9 To\"{e}n, B., Vezzosi, G., {From HAG to DAG: derived moduli stacks, in Axiomatic, %enriched and motivic homotopy theory,} NATO Sci. Ser. II Math. Phys. Chem., %131, Kluwer Acad. Publ., Dordrecht, 2004. ['From Hag To Dag: Derived Moduli Stacks'] 10.1007/978-94-007-0948-5_6 To\"{e}n, B., Vezzosi, G., {Homotopical algebraic geometry I: Topos theory,} Adv. Math. 193, no. 2, 2005. ['Homotopical algebraic geometry I: topos theory'] 10.1016/j.aim.2004.05.004 To\"{e}n, B., Vezzosi, G., {Homotopical algebraic geometry II: Geometric stacks and applications,} Mem. Amer. Math. Soc. 193, no. 902, (2008). ['Homotopical algebraic geometry. II. Geometric stacks and applications'] 10.1090/memo/0902 To\"{e}n, B., Vezzosi, G., {A sketchy note on enriched %homotopical topologies and enriched homotopical %stacks,} math.CT/0507447. None To\"{e}n, B., Vezzosi, G., {"Brave New" algebraic geometry and global derived moduli spaces of ring spectra, Proceedings of the Euroworkshop "Elliptic Cohomology and Higher Chromatic Phenomenon",} H. Miller, D. Ravenel, editors. ['Elliptic Cohomology'] 10.1017/cbo9780511721489 Wengenroth, J., {Derived functors in Functional Analysis,} Springer, 2003. \end{thebibliography} \end{document} ['Derived Functors in Functional Analysis'] 10.1007/b80165 -------------- ['Presentations for Globular Operads'] author [{'given': 'Rhiannon', 'family': 'Griffiths'}] publication date 2023-03-08 volume 39 issue 10 page range ('267', '321') url http://www.tac.mta.ca/tac/volumes/39/10/39-10abs.html abstract In this paper we develop the theory of presentations for globular operads and construct presentations for the globular operads corresponding to several key theories of n-category for n ⩽ 4. keywords n-categories, algebraic higher categories, operads, presentations ams class 08C05; 18C15; 18M60; 18N20 dois [] DOI citations: Michael Barr and Charles Wells. \textit{Toposes, triples and theories.} Reprints in Theory and Applications of Categories, No. 12, 2005, pp. 1-288. ['Toposes, Triples and Theories'] 10.1007/978-1-4899-0021-0 Michael Batanin. \textit{Computads and slices of operads.} Preprint, arXiv:math/0209035 [math.CT], 2002. None M. A. Batanin. \textit{Monoidal globular categories as a natural environment for the theory of weak n-categories.} Advances in Mathematics, 136(1):39–103, 1998. ['Monoidal Globular Categories As a Natural Environment for the Theory of Weakn-Categories'] 10.1006/aima.1998.1724 Rhiannon Griffiths. \textit{Higher categories and slices of globular operads.} In preparation. None Rhiannon Griffiths. \textit{Doctoral thesis.} 2021. None Nick Gurski. \textit{Coherence in Three-Dimensional Category Theory.} Cambridge Tracts in Mathematics, vol. 21, Cambridge University Press, Cambridge, 2013. None Vladimir Hinich. \textit{Homological algebra of homotopy algebras.} Communications in algebra, 25(10): 3291-3323, 1997; erratum math.AG/0309453, 2003. ['Homological algebra of homotopy algebras'] 10.1080/00927879708826055 Alexander E. Hoffnung. \textit{Spans in 2-Categories: A monoidal tricategory.} arXiv:1112.0560 [math.CT], 2011. None Masaki Kashiwara and Pierre Schapira. \textit{Categories and Sheaves.} volume 332 of Grundlehren der Mathematischen Wissenschaften, Springer Heidelberg, 2006. ['Categories and Sheaves'] 10.1007/3-540-27950-4 Tom Leinster. \textit{Higher Operads, Higher Categories,} volume 298 of London Mathematical Society Lecture Note Series. Cambridge University Press, 2004. ['Higher Operads, Higher Categories'] 10.1017/cbo9780511525896 Tom Leinster. \textit{Operads in higher-dimensional category theory.} arXiv:math/0011106v1 [math.CT], 2000. None Tom Leinster. \textit{Basic Bicategories.} arXiv:math/9810017v1 [math.CT], 1998. None Martin Markl, Steve Shnider and Jim Stasheff. \textit{Operads in Algebra, Topology and Physics.} volume 96 of Mathematical Surveys and Monographs, American Mathematical Society, 2002. ['Algebra'] 10.1090/surv/096/04 Peter May. \textit{The Geometry of Iterated Loop Spaces.} volume 271 of Lecture Notes in Mathematics, Springer-Verlag Berlin Heidelberg, 1972. ['The Geometry of Iterated Loop Spaces'] 10.1007/bfb0067491 Bruno Vallette. \textit{Algebra + Homotopy = Operad.} arXiv:math/1202.3245 [math,AT], 2012. \end{thebibliography} \end{document} None -------------- ['Partial evaluations and the compositional structure of the bar construction'] author [{'given': 'Carmen', 'family': 'Constantin'}, {'given': 'Tobias', 'family': 'Fritz'}, {'given': 'Paolo', 'family': 'Perrone'}, {'given': 'Brandon T.', 'family': 'Shapiro'}] publication date 2023-03-17 volume 39 issue 11 page range ('322', '364') url http://www.tac.mta.ca/tac/volumes/39/11/39-11abs.html abstract The algebraic expression 3 + 2 + 6 can be evaluated to 11, but it can also be partially evaluated to 5 + 6. In categorical algebra, such partial evaluations can be defined in terms of the 1-skeleton of the bar construction for algebras of a monad. We show that this partial evaluation relation can be seen as the relation internal to the category of algebras generated by relating a formal expression to its total evaluation. The relation is transitive for many monads which describe commonly encountered algebraic structures, and more generally for BC monads on (which are those monads for which the underlying functor and the multiplication are weakly cartesian). We find that this is not true for all monads: we describe a finitary monad on for which the partial evaluation relation on the terminal algebra is not transitive. With the perspective of higher algebraic rewriting in mind, we then investigate the compositional structure of the bar construction in all dimensions. We show that for algebras of BC monads, the bar construction has fillers for all directed acyclic configurations in Δ^n, but generally not all inner horns. keywords partial evaluation, bar construction, monads, simplicial sets ams class 18C15 dois [] DOI citations: John Baez. \newblock Simplicial sets from algebraic gadgets. \newblock Notes available at \href{http://math.ucr.edu/home/baez/qg-spring2007/s07week06b.pdf}{http://math.ucr.edu/home/baez/qg-spring2007/s07week06b.pdf}. None Michael~A. Batanin. \newblock The {E}ckmann-{H}ilton argument and higher operads. \newblock {\em Adv. Math.}, 217(1):334--385, 2008. \newblock \href{https://arxiv.org/abs/math/0207281}{arXiv:math/0207281}. None Francis Borceux. \newblock {\em Handbook of categorical algebra. 2}, volume~51 of {\em Encyclopedia of Mathematics and its Applications}. \newblock Cambridge University Press, Cambridge, 1994. \newblock Categories and structures. ['Handbook of Categorical Algebra'] 10.1017/cbo9780511525865 Francis Borceux and Dominique Bourn. \newblock {\em Mal’cev, protomodular, homological and semi-abelian categories}, volume 566 of {\em Mathematics and Its Applications}. \newblock Springer, 2004. ['Mal’cev, Protomodular, Homological and Semi-Abelian Categories'] 10.1007/978-1-4020-1962-3 Michael Ching. \newblock Bar constructions for topological operads and the {G}oodwillie derivatives of the identity. \newblock {\em Geom. Topol.}, 9:833--934, 2005. \newblock \href{https://arxiv.org/abs/math/0501429}{arXiv:math/0501429}. ['Bar constructions for topological operads and the Goodwillie derivatives of the identity'] 10.2140/gt.2005.9.833 Maria~Manuel Clementino, Dirk Hofmann, and George Janelidze. \newblock The monads of classical algebra are seldom weakly {C}artesian. \newblock {\em J. Homotopy Relat. Struct.}, 9(1):175--197, 2014. \newblock \href{http://www.mat.uc.pt/preprints/ps/p1246.pdf}{http://www.mat.uc.pt/preprints/ps/p1246.pdf}. ['The monads of classical algebra are seldom weakly cartesian'] 10.1007/s40062-013-0063-2 Carmen Constantin, Tobias Fritz, Paolo Perrone, and Brandon Shapiro. \newblock Weak cartesian properties of simplicial sets, 2021. \newblock \href{https://arxiv.org/abs/2105.04775}{arXiv:2105.04775}. ['Weak cartesian properties of simplicial sets'] 10.1007/s40062-023-00334-1 Tobias Fritz and Paolo Perrone. \newblock Monads, partial evaluations, and rewriting. \newblock In {\em The 36th {M}athematical {F}oundations of {P}rogramming {S}emantics {C}onference, 2020}, volume 352 of {\em Electron. Notes Theor. Comput. Sci.}, pages 129--148. Elsevier Sci. B. V., Amsterdam, 2020. \newblock \href{https://arxiv.org/abs/1810.06037}{arXiv:1810.06037}. ['Monads, Partial Evaluations, and Rewriting'] 10.1016/j.entcs.2020.09.007 Imma G{\'a}lvez-Carrillo, Joachim Kock, and Andrew Tonks. \newblock Decomposition spaces, incidence algebras and {M}{\"o}bius inversion {I}: Basic theory. \newblock {\em Adv. Math.}, 331:952--1015, 2018. \newblock \href{https://arxiv.org/abs/1512.07577}{arXiv:1512.07577}. None Imma G{\'a}lvez-Carrillo, Joachim Kock, and Andrew Tonks. \newblock Decomposition spaces, incidence algebras and {M}{\"o}bius inversion {II}: Completeness, length filtration, and finiteness. \newblock {\em Adv. Math.}, 333:1242--1292, 2018. \newblock None Jonathan~S. Golan. \newblock {\em Semirings and their applications}. \newblock Kluwer Academic Publishers, Dordrecht, 1999. ['Semirings and their Applications'] 10.1007/978-94-015-9333-5 Christopher~D. Hollings and Mark~V. Lawson. \newblock {\em Wagner's Theory of Generalised Heaps}. \newblock Springer, 2017. ['Wagner’s Theory of Generalised Heaps'] 10.1007/978-3-319-63621-4 Andr{\'e} Joyal. \newblock Foncteurs analytiques et esp{\`e}ces de structures. \newblock In Gilbert Labelle and Pierre Leroux, editors, {\em Combinatoire {\'e}num{\'e}rative}, volume 1234 of {\em Lecture Notes in Math.}, pages 126--159. Springer Berlin Heidelberg, 1986. ['Combinatoire énumérative'] 10.1007/bfb0072503 Andr\'e Joyal. \newblock Quasi-categories and {K}an complexes. \newblock {\em J. Pure Appl. Algebra}, 175(1-3):207--222, 2002. \newblock Special volume celebrating the 70th birthday of Professor Max Kelly. ['Quasi-categories and Kan complexes'] 10.1016/s0022-4049(02)00135-4 Joachim Kock. \newblock The incidence comodule bialgebra of the {B}aez-{D}olan construction. \newblock {\em Adv. Math.}, 383:Paper No. 107693, 79, 2021. \newblock \href{https://arxiv.org/abs/1912.11320}{arXiv:1912.11320}. ['The incidence comodule bialgebra of the Baez–Dolan construction'] 10.1016/j.aim.2021.107693 Tom Leinster. \newblock {\em Higher operads, higher categories}, volume 298 of {\em London Mathematical Society Lecture Note Series}. \newblock Cambridge University Press, Cambridge, 2004. \newblock \href{https://arxiv.org/abs/math/0305049}{arXiv:math/0305049}. ['Higher Operads, Higher Categories'] 10.1017/cbo9780511525896 Alex Simpson. \newblock Category-theoretic structure for independence and conditional independence. \newblock In {\em The {T}hirty-third {C}onference on the {M}athematical {F}oundations of {P}rogramming {S}emantics ({MFPS} {XXXIII})}, volume 336 of {\em Electron. Notes Theor. Comput. Sci.}, pages 281--297. Elsevier Sci. B. V., Amsterdam, 2018. \newblock \href{https://coalg.org/mfps-calco2017/mfps-papers/6-simpson.pdf}{https://coalg.org/mfps-calco2017/mfps-papers/6-simpson.pdf}. ['Category-theoretic Structure for Independence and Conditional Independence'] 10.1016/j.entcs.2018.03.028 Stanis{\l}aw Szawiel and Marek Zawadowski. \newblock Theories of analytic monads. \newblock {\em Math. Structures Comput. Sci.}, 24(6):e240604, 33, 2014. \newblock \href{https://arxiv.org/abs/1204.2703}{arXiv:1204.2703}. ['Theories of analytic monads'] 10.1017/s0960129513000868 Todd Trimble. \newblock On the bar construction, 2007. \newblock $n$-Category Caf\'e blog post. \href{https://golem.ph.utexas.edu/category/2007/05/on\\_the_bar_construction.html}{https://golem.ph.utexas.edu/category/2007/05/on\_the\_bar\_construction.html}. None Mark Weber. \newblock Generic morphisms, parametric representations and weakly cartesian monads. \newblock {\em Theory Appl. Categ.}, 13(14):191--234, 2004. None Mark Weber. \newblock Internal algebra classifiers as codescent objects of crossed internal categories. \newblock {\em Theory Appl. Categ.}, 30(50):1713--1792, 2015. \end{thebibliography} \end{document} None -------------- ['Colimits in enriched ∞-categories and Day convolution'] author [{'given': 'Vladimir', 'family': 'Hinich'}] publication date 2023-03-21 volume 39 issue 12 page range ('365', '422') url http://www.tac.mta.ca/tac/volumes/39/12/39-12abs.html abstract Let be a monoidal ∞-category with colimits. In this paper we study colimits of -functors → where is left-tensored over and is an -enriched category. We prove that the enriched Yoneda embedding Y:→ P_() yields a universal -functor. In case when has a certain monoidal structure, the category of enriched presheaves P_() inherits the same monoidal structure and the enriched Yoneda embedding acquires the structure of universal monoidal -functor. keywords enriched categories, Day convolution, left-tensored categories ams class 18D20, 18N60, 18N70 dois [] DOI citations: G.~Arone, I.~Barnea, T.~Schlank, Noncommutative CW spectra as enriched presheaves on matrix algebras, arXiv:2101.09775 ['Noncommutative CW-spectra as enriched presheaves on matrix algebras'] 10.4171/jncg/481 D.~Gepner, R.~Haugseng, Enriched $\infty$-categories via non-symmetric operads, Adv. Math., 279(2015), 575--716. ['Enriched ∞-categories via non-symmetric ∞-operads'] 10.1016/j.aim.2015.02.007 D.~Gepner, R.~Hauseng, T.~Nikolaus, Lax colimits and free fibrations in $\infty$-categories, Doc. Math. 22(2017), 1225--1266. None R.~Haugseng, F.~Hebestrijt, S.~Linskens, J.~Nuiten, Lax monoidal adjunctions, two-variable fibrations and the calculus of mates, arXiv:2011.08808v2. ['Lax monoidal adjunctions, two‐variable fibrations and the calculus of mates'] 10.1112/plms.12548 V.~Hinich, So, what is the derived functor?, arXiv:1811.12255, Homology, homotopy, applications, 22(2020), no. 2, 279--293. ['So, what is a derived functor?'] 10.4310/hha.2020.v22.n2.a18 V.~Hinich, Yoneda lemma for enriched infinity categories, arXiv:1805.0507635, Adv. Math., 367(2020), 107129. None V.~Hinich, Lectures on $\infty$-categories, preprint arXiv:1709.06271. None V.~Hinich, Dwyer-Kan localization, revisited, preprint arXiv:1311.4128, Homology, homotopy, applications, 18(2016), 27--48. ['Dwyer–Kan localization revisited'] 10.4310/hha.2016.v18.n1.a3 V.~Hinich, Rectification of algebras and modules, preprint arXiv:1311.4130, Doc. Math. 20 (2015), 879–-926. ['Rectification of algebras and modules'] 10.4171/dm/508 H.~Heine, An equivalence between enriched $\infty$-categories and $\infty$-categories with a weak action, arXiv:2009.02428. ['An equivalence between enriched ∞-categories and ∞-categories with weak action'] 10.1016/j.aim.2023.108941 Lurie, Higher algebra, preprint September 18, 2017, available at http://www.math.harvard.edu/~lurie/papers/HA.pdf. None Lurie, Higher topos theory, Annals of Mathematics Studies, 170. Princeton University Press, Princeton, NJ, 2009. xviii+925 pp, also available at http://www.math.harvard.edu/~lurie/papers/croppedtopoi.pdf. \end{thebibliography} \end{document} None -------------- ['Centrality and the commutativity of finite products with coequalisers'] author [{'given': 'Michael', 'family': 'Hoefnagel'}] publication date 2023-03-24 volume 39 issue 13 page range ('423', '443') url http://www.tac.mta.ca/tac/volumes/39/13/39-13abs.html abstract We study centrality of morphisms in a setting derived from that of a pointed category in which finite products commute with coequalisers. The main results of this paper show that much of the behaviour of central morphisms for unital categories is retained in our setting, including categories which are (weakly) unital, but also categories outside of the unital setting. keywords centrality, commutativity, finite products, coequalisers, unital category ams class 18E13, 18A30, 08B10, 18E05, 08B25 dois [] DOI citations: J.~Ad\'amek and J.~Rosick\'y. \newblock On sifted colimits and generalised varieties. \newblock {\em Theory and Applications of Categories}, 8:33--53, 2001. None J.~Ad\'amek, J.~Rosick\'y, and E.~M. Vitale. \newblock What are sifted colimits? \newblock {\em Theory and Applications of Categories}, 23:251--260, 2010. None M.~Barr, P.~A. Grillet, and {D. H. van} Osdol. \newblock Exact categories and categories of sheaves. \newblock {\em Springer, Lecture Notes in Mathematics}, 236, 1971. ['Exact Categories and Categories of Sheaves'] 10.1007/bfb0058579 F.~Borceux and D.~Bourn. \newblock {\em Mal'cev, Protomodular, Homological and Semi-Abelian Categories}. \newblock Kluwer, 2004. ['Mal’cev, Protomodular, Homological and Semi-Abelian Categories'] 10.1007/978-1-4020-1962-3 D.~Bourn. \newblock Mal'cev categories and fibration of pointed objects. \newblock {\em Applied Categorical Structures}, 4:307--327, 1996. ["Mal'cev categories and fibration of pointed objects"] 10.1007/bf00122259 D.~Bourn. \newblock Intrinsic centrality and associated classifying properties. \newblock {\em Journal of Algebra}, 256:126--145, 2002. ['Intrinsic centrality and associated classifying properties'] 10.1016/s0021-8693(02)00149-7 D.~Bourn. \newblock Fibration of points and congruence modularity. \newblock {\em Algebra Universalis}, 52:403--429, 2005. ['Fibration of points and congruence modularity'] 10.1007/s00012-004-1880-2 D.~Bourn. \newblock On congruence modular varieties and {G}umm categories. \newblock {\em Communications in Algebra}, 50:2377--2407, 2021. ['On congruence modular varieties and Gumm categories'] 10.1080/00927872.2021.2006679 D.~Bourn and M.~Gran. \newblock Categorical aspects of modularity. \newblock {\em Fields Institute Communications}, 43:77--100, 2004. ['Categorical aspects of modularity'] 10.1090/fic/043/04 B.~Eckmann and P.~J. Hilton. \newblock Group-like structures in general categories {I}. \newblock {\em Mathematische Annalen}, 145:227--255, 1962. None P.~Freyd and A.~Scendrov. \newblock {\em Categories, Allegories}. \newblock North-Holland Mathematical Library, 39, 1990. None M.~Gran. \newblock Applications of categorical {G}alois theory in universal algebra. \newblock {\em Fields Institute Communications}, 43:243 -- 280, 2004. ['Applications of categorical Galois theory in universal algebra'] 10.1090/fic/043/10 M.~Gran. \newblock An introduction to regular categories. \newblock {\em In New perspectives in Algebra, Topology and Categories Coimbra Mathematical Texts, Vol. 1, Eds. M.M. Clementino, A. Facchini and M. Gran}, Springer, 113 -- 145, 2021. ['New Perspectives in Algebra, Topology and Categories'] 10.1007/978-3-030-84319-9 J.~R.~A. Gray. \newblock Algebraic exponentiation and internal homology in general categories ({P}h{D} {T}hesis). \newblock {\em University of Cape Town}, 2010. None J.~R.~A. Gray. \newblock Algebraic exponentiation in general categories. \newblock {\em Applied Categorical Structures}, 20(6):543 -- 567, 2012. ['Algebraic Exponentiation in General Categories'] 10.1007/s10485-011-9251-6 H.~P. Gumm. \newblock Geometrical methods in congruence modular algebras. \newblock {\em Memoirs of the American Mathematical Society}, 45(286), 1983. ['Geometrical methods in congruence modular algebras'] 10.1090/memo/0286 M.~Hoefnagel. \newblock Majority categories. \newblock {\em Theory and Applications of Categories}, 34:249--268, 2019. None M.~Hoefnagel. \newblock Products and coequalisers in pointed categories. \newblock {\em Theory and Applications of Categories}, 34:1386–1400, 2019. None M.~Hoefnagel. \newblock Characterizations of majority categories. \newblock {\em Applied Categorical Structures}, 28(1):113--134, 2020. ['Characterizations of Majority Categories'] 10.1007/s10485-019-09571-z M.~Hoefnagel. \newblock Anticommutativity and the triangular lemma. \newblock {\em Algebra Universalis}, 82:403--429, 2021. ['Anticommutativity and the triangular lemma'] 10.1007/s00012-021-00710-z M.~Hoefnagel. \newblock On the commutativity of products with coequalisers in general varieties of algebras ({\em preprint available}), 2023. None M.~Hoefnagel and Z.~Janelidze. \newblock Isomorphism formulas for products and quotients in pointed categories ({\em in preparation}). None S.~A. Huq. \newblock Commutator, nilpotency, and solvability in categories. \newblock {\em The Quarterly Journal of Mathematics}, 19(1):363--389, 1968. ['COMMUTATOR, NILPOTENCY, AND SOLVABILITY IN CATEGORIES'] 10.1093/qmath/19.1.363 P.~A. Jacqmin. \newblock A class of exactness properties characterized via left {K}an extensions. \newblock {\em Journal of Pure and Applied Algebra}, (106784), 2021. ['A class of exactness properties characterized via left Kan extensions'] 10.1016/j.jpaa.2021.106784 P.~A. Jacqmin and Z.~Janelidze. \newblock On stability of exactness properties under the procompletion. \newblock {\em Advances in Mathematics}, (107484), 2021. ['On stability of exactness properties under the pro-completion'] 10.1016/j.aim.2020.107484 S.~{Mac Lane}. \newblock {\em Categories for the Working Mathematician}. \newblock Graduate Texts in Mathematics. Springer New York, 1998. ['Categories for the Working Mathematician'] 10.1007/978-1-4612-9839-7 N.~Martins-Ferreira. \newblock Low-dimensional internal categorical structures in weakly {M}al’cev sesquicategories (PhD Thesis). \newblock {\em University of Cape Town}, 2008. None N.~Martins-Ferreira. \newblock Weakly {M}al’cev categories. \newblock {\em Theory and Applications of Categories}, 21:97--117, 2008. None N.~Martins-Ferreira. \newblock Weakly {M}al’tsev categories and distributive lattices. \newblock {\em Journal of Pure and Applied Algebra}, 216:1961–1963, 2012. None N.~Martins-Ferreira. \newblock New wide classes of weakly {M}al’tsev categories. \newblock {\em Applied Categorical Structures}, 23:741--751, 2015. \end{thebibliography} \end{document} ['New Wide Classes of Weakly Mal’tsev Categories'] 10.1007/s10485-014-9377-4 -------------- ['Addendum to “Rank-based persistence”'] author [{'given': 'Mattia G.', 'family': 'Bergomi'}, {'given': 'Pietro', 'family': 'Vertechi'}] publication date 2023-03-31 volume 39 issue 14 page range ('444', '446') url http://www.tac.mta.ca/tac/volumes/39/14/39-14abs.html abstract The Rank-based persistence framework and the generalization of topological persistence introduced in  yield overlapping applications. We discuss the similarities and differences between the two approaches from theoretical and applied standpoints. keywords rank, persistence, categorification, regular category, abelian category, semisimple category, classification, group action, point cloud, poset, bottleneck, interleaving ams class 18E10, 18A35, 55N35, 68U05 dois [] DOI citations: Generalized persistence diagrams Patel, Amit ['Generalized persistence diagrams'] 10.1007/s41468-018-0012-6 Bottleneck stability for generalized persistence diagrams McCleary, Alex and Patel, Amit ['Bottleneck stability for generalized persistence diagrams'] 10.1090/proc/14929 Rank-based persistence Bergomi, Mattia G and Vertechi, Pietro None Steady and ranging sets in graph persistence Bergomi, Mattia G and Ferri, Massimo and Tavaglione, Antonella ['Steady and ranging sets in graph persistence'] 10.1007/s41468-022-00099-1 Comparing Neural Networks via Generalized Persistence Bergomi, Mattia G and Vertechi, Pietro None Persistent Cup Product Structures and Related Invariants M{\'e}moli, Facundo and Stefanou, Anastasios and Zhou, Ling ['Persistent cup product structures and related invariants'] 10.1007/s41468-023-00138-5 -------------- ['The over-topos at a model'] author [{'given': 'Olivia', 'family': 'Caramello'}, {'given': 'Axel', 'family': 'Osmond'}] publication date 2023-04-12 volume 39 issue 15 page range ('447', '492') url http://www.tac.mta.ca/tac/volumes/39/15/39-15abs.html abstract With a model of a geometric theory in an arbitrary topos, we associate a site obtained by endowing a category of generalized elements of the model with a Grothendieck topology, which we call the antecedent topology. Then we show that the associated sheaf topos, which we call the over-topos at the given model, admits a canonical totally connected morphism to the given base topos and satisfies a universal property generalizing that of the colocalization of a topos at a point. We first treat the case of the base topos of sets, where global elements are sufficient to describe our site of definition; in this context, we also introduce a geometric theory classified by the over-topos, whose models can be identified with the model homomorphisms towards the (internalizations of the) model. Then we formulate and prove the general statement over an arbitrary topos, which involves the stack of generalized elements of the model. keywords Over-topos, totally connected topos, Giraud topology, colocalization ams class 18F10, 18C10, 3G30 dois [] DOI citations: A. Abbes and M. Gros, \emph{Topos co-{\'e}vanescents et g{\'e}n{\'e}ralisations}, Ann. of Math. Stud. volume 1107, 2011. None M. Artin, A. Grothendieck and J. L. Verdier, \emph{Th\'eorie des topos et cohomologie \'etale des sch\'emas} - (SGA 4), S\'eminaire de G\'eom\'etrie Alg\'ebrique du Bois-Marie, ann\'ee 1963-64; second edition published as Lecture Notes in Math., vols 269, 270 and 305 (Springer-Verlag, 1972). ['Théorie des Topos et Cohomologie Etale des Schémas'] 10.1007/bfb0081551 M. Bunge and J. Funk, \emph{Singular coverings of toposes}, Springer, 2006. None O. Caramello, \emph{Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic `bridges'} (Oxford University Press, 2017). ['Theories, Sites, Toposes'] 10.1093/oso/9780198758914.001.0001 O. Caramello, Denseness conditions, morphisms and equivalences of toposes, \emph{arxiv:math.CT/1906.08737v3} (2020). None J. Cole, The bicategory of topoi and spectra, \emph{Reprints in Theory and Applications of Categories}, No. 25 (2016) pp. 1-16. None P. F. Faul, G. Manuell and J. Siqueira, Artin glueings of toposes as adjoint split extensions, \emph{https://arxiv.org/abs/2012.04963}.\, ['Artin glueings of toposes as adjoint split extensions'] 10.1016/j.jpaa.2022.107273 J. Giraud, Classifying topos, in \emph{Toposes, Algebraic Geometry and Logic}, pp. 43-56 (Springer, 1972). ['Classifying topos'] 10.1007/bfb0073964 P. T. Johnstone, \emph{Sketches of an Elephant: a topos theory compendium. Vols. 1 and 2}, vols. 43 and 44 of \emph{Oxford Logic Guides} (Oxford University Press, 2002). None S. Zoghaib, Théorie homotopique de Morel-Voevodsky et applications, 2020, available at \texttt{http://www.normalesup.org/~zoghaib/math/dea.pdf}.\, \end{thebibliography} \vspace{1cm} \textsc{Olivia Caramello} \vspace{0.2cm} None -------------- ['Exponentiability in categories of relational structures'] author [{'given': 'Jason', 'family': 'Parker'}] publication date 2023-05-01 volume 39 issue 16 page range ('493', '518') url http://www.tac.mta.ca/tac/volumes/39/16/39-16abs.html abstract For a relational Horn theory , we provide useful sufficient conditions for the exponentiability of objects and morphisms in the category of -models; well-known examples of such categories, which have found recent applications in the study of programming language semantics, include the categories of preordered sets and (extended) metric spaces. As a consequence, we obtain useful sufficient conditions for to be cartesian closed, locally cartesian closed, and even a quasitopos; in particular, we provide two different explanations for the cartesian closure of the categories of preordered and partially ordered sets. Our results recover (the sufficiency of) certain conditions that have been shown by Niefield and Clementino–Hofmann to characterize exponentiability in the category of partially ordered sets and the category 𝒱-𝖢𝖺𝗍 of small -categories for certain commutative unital quantales . keywords relational Horn theory; relational structure; exponentiability; cartesian closed; locally cartesian closed; partial product; quasitopos ams class 06A06, 06F07, 08A02, 18C35, 18D15 dois [] DOI citations: Ji\v{r}\'{\i} Ad\'{a}mek, Horst Herrlich, and George~E. Strecker, \emph{Abstract and concrete categories: the joy of cats}, Repr. Theory Appl. Categ. (2006), no.~17, 1--507, Reprint of the 1990 original [Wiley, New York]. None Ji\v{r}\'{\i} Ad\'{a}mek and Ji\v{r}\'{\i} Rosick\'{y}, \emph{Locally presentable and accessible categories}, London Mathematical Society Lecture Note Series, vol. 189, Cambridge University Press, Cambridge, 1994. None Maria~Manuel Clementino and Dirk Hofmann, \emph{Exponentiation in {$V$}-categories}, Topology Appl. \textbf{153} (2006), no.~16, 3113--3128. ['Exponentiation in V-categories'] 10.1016/j.topol.2005.01.038 Maria~Manuel Clementino and Dirk Hofmann, \emph{The rise and fall of {$V$}-functors}, Fuzzy Sets and Systems \textbf{321} (2017), 29--49. ['The rise and fall of V-functors'] 10.1016/j.fss.2016.09.005 Maria~Manuel Clementino, Dirk Hofmann, and Isar Stubbe, \emph{Exponentiable functors between quantaloid-enriched categories}, Appl. Categ. Structures \textbf{17} (2009), 91--101. ['Exponentiable Functors Between Quantaloid-Enriched Categories'] 10.1007/s10485-007-9104-5 Maria~Manuel Clementino, Dirk Hofmann, and Walter Tholen, \emph{Exponentiability in categories of lax algebras}, Theory Appl. Categ. \textbf{11} (2003), No. 15, 337--352. None B.~J. Day and G.~M. Kelly, \emph{On topological quotient maps preserved by pullbacks or products}, Proc. Cambridge Philos. Soc. \textbf{67} (1970), 553--558. ['On topological quotient maps preserved by pullbacks or products'] 10.1017/s0305004100045850 Roy Dyckhoff and Walter Tholen, \emph{Exponentiable morphisms, partial products and pullback complements}, J. Pure Appl. Algebra \textbf{49} (1987), no.~1-2, 103--116. ['Exponentiable morphisms, partial products and pullback complements'] 10.1016/0022-4049(87)90124-1 Chase Ford, Stefan Milius, and Lutz Schr\"{o}der, \emph{{Monads on Categories of Relational Structures}}, 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021), Leibniz International Proceedings in Informatics (LIPIcs), vol. 211, 2021, pp.~14:1--14:17. None Dirk Hofmann, Gavin J.~Seal, and Walter Tholen (eds.), \emph{Monoidal topology}, Encyclopedia of Mathematics and its Applications, vol. 153, Cambridge University Press, Cambridge, 2014. ['Monoidal Topology'] 10.1017/cbo9781107517288 G.~M. Kelly, \emph{Basic concepts of enriched category theory}, Repr. Theory Appl. Categ. (2005), no.~10, Reprint of the 1982 original [Cambridge Univ. Press, Cambridge]. None Radu Mardare, Prakash Panangaden, and Gordon Plotkin, \emph{Quantitative algebraic reasoning}, Proceedings of the 31st {A}nnual {ACM}-{IEEE} {S}ymposium on {L}ogic in {C}omputer {S}cience, 2016, pp.~700--709. ['Quantitative Algebraic Reasoning'] 10.1145/2933575.2934518 Susan Niefield, \emph{Cartesianness: topological spaces, uniform spaces, and affine schemes}, J. Pure Appl. Algebra \textbf{23} (1982), no.~2, 147--167. ['Cartesianness: topological spaces, uniform spaces, and affine schemes'] 10.1016/0022-4049(82)90004-4 Susan Niefield, \emph{Exponentiable morphisms: posets, spaces, locales, and {G}rothendieck toposes}, Theory Appl. Categ. \textbf{8} (2001), 16--32. None Jason Parker, \emph{Extensivity of categories of relational structures}, Theory Appl. Categ. \textbf{38} (2022), No. 23, 898--912. None Kimmo~I. Rosenthal, \emph{Quantales and their applications}, Pitman Research Notes in Mathematics Series, vol. 234, Longman Scientific \& Technical, Harlow, 1990. None Ji\v{r}\'{\i} Rosick\'{y}, \emph{Concrete categories and infinitary languages}, J. Pure Appl. Algebra \textbf{22} (1981), no.~3, 309--339. ['Concrete categories and infinitary languages'] 10.1016/0022-4049(81)90105-5 Walter Tholen and Jiyu Wang, \emph{Metagories}, Topology Appl. \textbf{273} (2020), 106965, 24. \end{references*} \end{document} ['Metagories'] 10.1016/j.topol.2019.106965 -------------- ['Quasi-uniform structures and functors'] author [{'given': 'Minani', 'family': 'Iragi'}, {'given': 'David', 'family': 'Holgate'}] publication date 2023-05-15 volume 39 issue 17 page range ('519', '534') url http://www.tac.mta.ca/tac/volumes/39/17/39-17abs.html abstract We study a number of categorical quasi-uniform structures induced by functors. We depart from a category 𝒞 with a proper (ℰ, ℳ)-factorization system, then define the continuity of a 𝒞-morphism with respect to two syntopogenous structures (in particular with respect to two quasi-uniformities) on 𝒞 and use it to describe the quasi-uniformities induced by pointed and copointed endofunctors of 𝒞. In particular, we demonstrate that every quasi-uniformity on a reflective subcategory of 𝒞 can be lifted to a coarsest quasi-uniformity on 𝒞 for which every reflection morphism is continuous. Thinking of categories supplied with quasi-uniformities as large “spaces”, we generalize the continuity of 𝒞-morphisms (with respect to a quasi-uniformity) to functors. We prove that for an ℳ-fibration or a functor that has a right adjoint, we can obtain a concrete construction of the coarsest quasi-uniformity for which the functor is continuous. The results proved are shown to yield those obtained for categorical closure operators. Various examples considered at the end of the paper illustrate our results. keywords Closure operator, Syntopogenous structure, Quasi-uniform structure, (co)pointed endofunctor and Adjoint functor ams class 18A05, 18F60, 54A15, 54B30 dois [] DOI citations: J. Ad\'{a}mek, H. Herrlich, and G. E. Strecker. Abstract and concrete categories: the joy of cats. Repr. Theory Appl. Categ.,(17), 1-507, 2006. Reprint of the 1990 original [Wiley, New York]. None N. Bourbaki. General Topology: Chapters 1-4, volume 18. Springer Science and Business Media, 1998. ['General Topology'] 10.1007/978-3-642-61701-0 G. Br{\"u}mmer. Categorical aspects of the theory of quasi-uniform spaces. Università degliStudi di Trieste. Dipartimento di Scienze Matematiche, 1987. None D. Dikrajan and W. Tholen. Categorical structure of closure operators with Applications to Topology, Algebra and Discrete Mathematics. Volume 346 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1995. None D. Dikranjan and E. Giuli. Closure operators I. Topology and its Applications, 27(2):129-143, 1987. ['Closure operators I'] 10.1016/0166-8641(87)90100-3 D. Dikranjan and H. P. K{\"u}nzi. Separation and epimorphisms in quasi-uniform spaces. Applied Categorical Structures, 8(1):175-207, 2000. ['Separation and Epimorphisms in Quasi-Uniform Spaces'] 10.1023/a:1008743408583 D. Dikranjan. Semiregular closure operators and epimorphisms in topological categories. In V International Meeting on Topology in Italy (Italian)(Lecce, 1990/Otranto, 1990). Rend. Circ. Mat. Palermo (2) Suppl, volume 29, pages 105-160, 1992. None C. Dowker. Mappings of proximity structures. General Topology and its Relations to Modern Analysis and Algebra, pages 139-141, 1962. None P. Fletcher and W. F. Lindgren. Quasi-uniform spaces. Lectures notes in Pure Apll.Math.77, Dekker, New York, 1982. None D. Holgate and M. Iragi. Quasi-uniform and Syntopogenous structures on categories. Topology and its Applications.(263):16-25, 2019. ['Quasi-uniform and syntopogenous structures on categories'] 10.1016/j.topol.2019.05.024 D. Holgate and M. Iragi. Quasi-uniform structures determined by closure operators. Topology and its Applications, (295):107669, 2021. ['Quasi-uniform structures determined by closure operators'] 10.1016/j.topol.2021.107669 D. Holgate. The pullback closure, perfect morphisms and completions. PhD Thesis, University of Cape Town, 1995. None D. Holgate. The pullback closure operator and generalisations of perfectness. Applied Categorical Structures, 4(1):107-120, 1996. ['The Pullback Closure Operator and Generalisations of Perfectness'] 10.1007/978-94-009-0263-3_10 D. Holgate, M. Iragi, and A. Razafindrakoto. Topogenous and nearness structures on categories. Appl. Categor. Struct., (24):447-455, 2016. ['Topogenous and Nearness Structures on Categories'] 10.1007/s10485-016-9455-x D. Holgate and J.{\v{S}}lapal.Categorical neighborhood operators. Topology Appl.,158(17):2356-2365, 2011. ['Categorical neighborhood operators'] 10.1016/j.topol.2011.06.031 M. Iragi. Topogenous structures on categories. MSc Thesis, University of the Western Cape, 2016 None M. Iragi. Quasi-uniform and syntopogenous structures on categories. PhD Thesis, University of the Western Cape, 2019. None L. Stramaccia. Classes of spaces defined by an epireflector. Rendiconti del Circolo, Matematico di Palermo. Serie II. Supplemento,(18), pages 423-432, 1988. None S. J. R. Vorster. Interior operators in general categories. Quaest. Math., 23(4):405-416,2000. \endrefs \end{document} ['Interior Operators in General Categories'] 10.2989/16073600009485987 -------------- ['Operads for Symmetric Monoidal Categories'] author [{'given': 'A. D.', 'family': 'Elmendorf'}] publication date 2023-05-25 volume 39 issue 18 page range ('535', '544') url http://www.tac.mta.ca/tac/volumes/39/18/39-18abs.html abstract This paper gives an explicit description of the categorical operad whose algebras are precisely symmetric monoidal categories. This allows us to place the operad in a sequence of four, thus generating a sequence of four successively stricter concepts of symmetric monoidal category. A companion paper will use this operadic presentation to describe a vast array of underlying multicategories for a symmetric monoidal category. keywords symmetric monoidal category, operad ams class 18M05 dois [] DOI citations: Gerald Dunn, \emph{$E_n$-monoidal categories and their group completions}, J.\ Pure Appl.\ Algebra {\bf95} (1994), 27-39. ['En-monoidal categories and their group completions'] 10.1016/0022-4049(94)90116-3 A.\ D.\ Elmendorf and M.\ A.\ Mandell, \emph{Permutative categories, multicategories, and algebraic K-theory}, Algebr.\ Geom.\ Topol. {\bf 9} (2009), 2391-2441. None Saunders Mac Lane, \emph{Categories for the Working Mathematician}, 2nd ed., {Graduate Texts in Mathematics} {\bf5}, Springer-Verlag New York, 1998. ['Categories for the Working Mathematician'] 10.1007/978-1-4612-9839-7 J.\ Peter May, \emph{The Geometry of Iterated Loop Spaces}, Lecture Notes in Mathematics {\bf 271}: Springer-Verlag, Berlin, Heidelberg, New York 1972. ['The Geometry of Iterated Loop Spaces'] 10.1007/bfb0067491 J.\ Peter May, \emph{$E_\infty$ spaces, group completions, and permutative categories}, in: \emph{New Developments in Topology}, London Mathematical Society Lecture Note Series {\bf11}, G.\ Segal, ed., Cambridge University Press, Cambridge, 1974. ['New Developments in Topology'] 10.1017/cbo9780511662607 J.\ D.\ Stasheff, \emph{Homotopy associativity of $H$-spaces. I}, Trans.\ Amer.\ Math.\ Soc.\ {\bf108} (1963), 275-292. \end{thebibliography} ['Homotopy associativity of 𝐻-spaces. I'] 10.1090/s0002-9947-1963-99936-3 -------------- ['On Rota-Baxter Lie 2-algebras'] author [{'given': 'Shilong', 'family': 'Zhang'}, {'given': 'Jiefeng', 'family': 'Liu'}] publication date 2023-05-30 volume 39 issue 19 page range ('545', '566') url http://www.tac.mta.ca/tac/volumes/39/19/39-19abs.html abstract In this paper, we introduce the notion of Rota-Baxter Lie 2-algebras, which is a categorification of Rota-Baxter Lie algebras. We prove that the category of Rota-Baxter Lie 2-algebras and the category of 2-term Rota-Baxter L_∞-algebras are equivalent. We introduce the notion of a crossed module of Rota-Baxter Lie algebras, and show that there is a one-to-one correspondence between strict 2-term Rota-Baxter L_∞-algebras and crossed modules of Rota-Baxter Lie algebras. At last, as applications of the crossed modules of Rota-Baxter Lie algebras, we give constructions of crossed modules of pre-Lie algebras and crossed modules of Lie algebras from them. keywords Rota-Baxter Lie 2-algebra, 2-term Rota-Baxter L_∞-algebra, crossed module ams class 17B38, 18N25 dois [] DOI citations: G. Baxter, An analytic problem whose solution follows from a simple algebraic identity. \emph{Pacific J. Math.} {\bf 10} (1960), 731-742. ['An analytic problem whose solution follows from a simple algebraic identity'] 10.2140/pjm.1960.10.731 J. Baez and A.S. Crans, Higher-Dimensional Algebra VI: Lie $2$-Algebras. \emph{Theory Appl. Categ.} {\bf 12} (2004), 492-528. None J. Baez, A. Hoffnung and C. Rogers, Categorified symplectic geometry and the classical string. \emph{Comm. Math. Phys.} {\bf 293} (2010), 701-725. ['Categorified Symplectic Geometry and the Classical String'] 10.1007/s00220-009-0951-9 J. Baez and C. Rogers, Categorified symplectic geometry and the string Lie $2$-algebra. \emph{Homology, Homotopy Appl.}, {\bf 12} (2010), 221-236. ['Categorified symplectic geometry and the string Lie 2-algebra'] 10.4310/hha.2010.v12.n1.a12 C. Bai, An introduction to pre-Lie algebras. In: Algebra and Applications 1: Non-associative Algebras and Categories. Wiley Online Library (2021), 245-273. ['An Introduction to Pre‐Lie Algebras'] 10.1002/9781119818175.ch7 C. Bai, O. Bellier, L. Guo and X. Ni, Spliting of operations, Manin products and Rota-Baxter operators. \emph{Int. Math. Res. Not.} {\bf 3} (2013), 485-524. ['Splitting of Operations, Manin Products, and Rota–Baxter Operators'] 10.1093/imrn/rnr266 C. Bai, Y. Sheng and C. Zhu, Lie $2$-bialgebras. \emph{Comm. Math. Phys.} {\bf 320} (2013), 149-172. ['Lie 2-Bialgebras'] 10.1007/s00220-013-1712-3 R. J. Baxter, One-dimensional anisotropic Heisenberg chain. \emph{Ann. Physics} {\bf 70} (1972), 323-337. ['One-dimensional anisotropic Heisenberg chain'] 10.1016/0003-4916(72)90270-9 D. Burde, Left-symmetric algebras and pre-Lie algebras in geometry and physics. {\em Cent. Eur. J. Math.} {\bf 4} (2006), 323-357. ['Left-symmetric algebras, or pre-Lie algebras in geometry and physics'] 10.2478/s11533-006-0014-9 P. Cartier, On the structure of free Baxter algebras. \emph{Adv. in Math.} {\bf 9}(1972), 253-265. ['On the structure of free baxter algebras'] 10.1016/0001-8708(72)90018-7 V. Chari and A. Pressley, A guide to quantum groups. \emph{Cambridge University Press, Cambridge,} 1994. xvi+651 pp. None A. Connes and D. Kreimer, { Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem.} {\em Comm. Math. Phys.} {\bf 210} (2000), 249-273. ['Renormalization in Quantum Field Theory and the Riemann-Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem'] 10.1007/s002200050779 K Ebrahimi-Fard, D. Manchon and F. Patras, A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's counterterm recursion. {\em J. Noncommut. Geom.} {\bf 3} (2009), 181-222. ['A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion'] 10.4171/jncg/35 L. Guo, An introduction to Rota-Baxter algebra. Surveys of Modern Mathematics, 4. \emph{International Press, Somerville, MA; Higher Education Press, Beijing,} 2012. xii+226 pp. None L. Guo, Properties of free Baxter algebras. \emph{Adv. Math.} {\bf 151} (2000), 346-374. ['Properties of Free Baxter Algebras'] 10.1006/aima.1999.1898 L. Guo, H. Lang and Y. Sheng, Integration and geometrization of Rota-Baxter Lie algebras. \emph{ Adv. Math.} {\bf 387} (2021), 107834. ['Integration and geometrization of Rota-Baxter Lie algebras'] 10.1016/j.aim.2021.107834 L. Guo, J.-Y. Thibon and H. Yu, The Hopf algebras of signed permutations, of weak quasi-symmetric functions and of Malvenuto-Reutenauer. \emph{ Adv. Math.} {\bf 374} (2020), 107341. ['The Hopf algebras of signed permutations, of weak quasi-symmetric functions and of Malvenuto-Reutenauer'] 10.1016/j.aim.2020.107341 J. Jiang and Y. Sheng, Representations and cohomologies of relative Rota-Baxter Lie algebras and applications. {\em J. Algebra} {\bf 602} (2022), 637-670. ['Representations and cohomologies of relative Rota-Baxter Lie algebras and applications'] 10.1016/j.jalgebra.2022.03.027 G.-C. Rota, Baxter algebras and combinatorial identities I, II. \emph{Bull. Amer. Math. Soc.} {\bf 75} (1969), 325-329, 330-334. ['Baxter algebras and combinatorial identities. II'] 10.1090/s0002-9904-1969-12158-0 T. Lada and J. Stasheff, Introduction to sh Lie algebras for physicists. \emph{Internat. J. Theoret. Phys.} {\bf 32} (1993), 1087-1103. ['Introduction to SH Lie algebras for physicists'] 10.1007/bf00671791 A. Lazarev, Y. Sheng and R. Tang, Deformations and homotopy theory of relative Rota-Baxter Lie algebras. \emph{Comm. Math. Phys.} {\bf 383} (2021), 595-631. ['Deformations and Homotopy Theory of Relative Rota–Baxter Lie Algebras'] 10.1007/s00220-020-03881-3 J. Pei, C. Bai and L. Guo, Splitting of operads and Rota-Baxter operators on operads. \emph{Appl. Categor. Struct.} {\bf 25} (2017), 505-538. ['Splitting of Operads and Rota-Baxter Operators on Operads'] 10.1007/s10485-016-9431-5 D. Roytenberg, Courant algebroids and strongly homotopy Lie algebras. \emph{Lett. Math. Phys.} {\bf46} (1998), 81-93. None M. Semonov-Tian-Shansky, What is a classical R-matrix? \emph{Funct. Anal. Appl.} {\bf 17}(1983), 259-272 . None Y. Sheng, Categorification of pre-Lie Algebras and solutions of 2-graded Classical Yang-Baxter equations. \emph{Theory Appl. Categ.} {\bf 34} (2019), 269-294. None R. Tang, C. Bai, L. Guo and Y. Sheng, Deformations and their controlling cohomologies of $\huaO$-operators. \emph{Commun. Math. Phys.} {\bf 368} (2019), 665-700. None B. Vallette, Homology of generalized partition posets. \emph{J. Pure Appl. Algebra} {\bf 208} (2007), 699-725. ['Homology of generalized partition posets'] 10.1016/j.jpaa.2006.03.012 C. N. Yang, Some exact results for the many-body problem in one dimension with repulsive deltafunction interaction. \emph{Phys. Rev. Lett.} {\bf 19} (1967), 1312-1315. ['Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction'] 10.1103/physrevlett.19.1312 H. Yu, L. Guo and J.-Y. Thibon, Weak quasi-symmetric functions, Rota-Baxter algebras and Hopf algebras. \emph{Adv. Math.} {\bf 344} (2019), 1-34. \end{thebibliography} \end{document} ['Weak quasi-symmetric functions, Rota–Baxter algebras and Hopf algebras'] 10.1016/j.aim.2018.12.001 -------------- ['Generalized principal bundles and quotient stacks'] author [{'given': 'Elena', 'family': 'Caviglia'}] publication date 2023-06-01 volume 39 issue 20 page range ('567', '597') url http://www.tac.mta.ca/tac/volumes/39/20/39-20abs.html abstract We consider the internalization of the usual notion of principal bundle in a site that has all pullbacks and a terminal object. We use this notion to consider the explicit construction of quotient prestacks via presheaves of categories of principal bundles equipped with equivariant morphisms in this abstract context. We then prove that, if the site is subcanonical and the underlying category satisfies some mild conditions, these quotient prestacks satisfy descent in the sense of stacks. keywords principal bundles, quotient stacks, classifying stacks, canonical topology ams class 18F20, 18F10, 18C40, 14A20, 18F15 dois [] DOI citations: B.~Eckmann and P.~J. Hilton. \newblock Group-like structures in general categories. {I}. {Multiplications} and comultiplications. \newblock {\em Math. Ann.}, 145:227--255, 1962. ['Group-like structures in general categories I multiplications and comultiplications'] 10.1007/bf01451367 B.~Eckmann and P.~J. Hilton. \newblock Group-like structures in general categories. {II}: {Equalizers}, limits, lengths. {III}: {Primitive} categories. \newblock {\em Math. Ann.}, 151:150--186, 1963. ['Group-like structures in general categories II equalizers, limits, lengths'] 10.1007/bf01344176 A.~Grothendieck. \newblock Technique de descente et th\'eor\`emes d'existence en g\'eom\'etrie alg\'ebrique. {I.} {G\'en\'eralit\'es.} {Descente} par morphismes fid\`element plats. \newblock In {\em S\'eminaire Bourbaki : ann\'ees 1958/59 - 1959/60, expos\'es 169-204}, number~5 in S\'eminaire Bourbaki. Soci\'et\'e math\'ematique de France, 1960. \newblock talk:190. None A.~Grothendieck. \newblock Techniques de construction et th\'eor\`emes d'existence en g\'eom\'etrie alg\'ebrique {III} : pr\'esch\'emas quotients. \newblock In {\em S\'eminaire Bourbaki : ann\'ees 1960/61, expos\'es 205-222}, number~6 in S\'eminaire Bourbaki. Soci\'et\'e math\'ematique de France, 1961. \newblock talk:212. None J.~Heinloth. \newblock Notes on differentiable stacks. \newblock In {\em Mathematisches Institut, Georg-August-Universit\"at G\"ottingen: Seminars Winter Term 2004/2005.}, pages 1--32. G{\"o}ttingen: Universit{\"a}tsverlag G{\"o}ttingen, 2005. None A.~Heyting. \newblock Die formalen {Regeln} der intuitionistischen {Logik}. {I}, {II}, {III}. \newblock {\em Sitzungsber. Preu{{\ss}}. Akad. Wiss., Phys.-Math. Kl.}, 1930:42--56, 57--71, 158--169, 1930. None D.~Husemoller. \newblock {\em Fibre bundles}. \newblock Springer-Verlag New York, [2d ed.] edition, 1975. ['Fibre Bundles'] 10.1007/978-1-4757-4008-0 P.~Ion and J.~Serre. \newblock {\em Galois Cohomology}. \newblock Springer Monographs in Mathematics. Springer Berlin Heidelberg, 2001. ['Galois Cohomology'] 10.1007/978-3-642-59141-9 P.~T. Johnstone. \newblock {\em Sketches of an elephant. {A} topos theory compendium. {II}}, volume~44 of {\em Oxf. Logic Guides}. \newblock Oxford: Clarendon Press, 2002. None A.~Kock. \newblock Fibre bundles in general categories. \newblock {\em J. Pure Appl. Algebra}, 56(3):233--245, 1989. ['Fibre bundles in general categories'] 10.1016/0022-4049(89)90059-5 G.~Laumon and L.~Moret-Bailly. \newblock {\em Champs alg{\'e}briques}, volume~39 of {\em Ergeb. Math. Grenzgeb., 3. Folge}. \newblock Berlin: Springer, 2000. ['Champs algébriques'] 10.1007/978-3-540-24899-6 C.~Lester. \newblock {\em The Canonical Grothendieck Topology and a Homotopical Analog}. \newblock https://arxiv.org/abs/1909.03188, 2019. None J.~W. Milnor and J.~D. Stasheff. \newblock {\em Characteristic classes}. \newblock Annals of Mathematics Studies, no. 76. Princeton University Press, Princeton, N.J, 1974. None S.~A. Mitchell. \newblock Notes on principal bundles and classifying spaces. \newblock 2006. None F.~{Neumann}. \newblock {\em {Algebraic Stacks and Moduli of Vector Bundles}}. \newblock IMPA Lecture Notes, 2011. None T.~Nikolaus, U.~Schreiber, and D.~Stevenson. \newblock Principal infinity-bundles: general theory. \newblock {\em Journal of Homotopy and Related Structures}, 10:749--801, 12 2015. None J.~Penon. \newblock Sur les quasi-topos. \newblock {\em Cah. Topologie G{\'e}om. Diff{\'e}r. Cat{\'e}goriques}, 18:181--218, 1977. None H.~Sati and U.~Schreiber. \newblock Equivariant principal infinity-bundles, 2022. None J.~M. Souriau. \newblock Groupes diff{\'e}rentiels. \newblock Differential geometrical methods in mathematical physics, {Proc}. {Conf}. {Aix}-en-{Provence} and {Salamanca} 1979, {Lect}. {Notes} {Math}. 836, 91-128 (1980)., 1980. None N.~Steenrod. \newblock {\em The Topology of Fibre Bundles. (PMS-14)}. \newblock Princeton University Press, Princeton, 1951. \end{references*} None -------------- ['The category of L-algebras'] author [{'given': 'Wolfgang', 'family': 'Rump'}] publication date 2023-06-07 volume 39 issue 21 page range ('598', '624') url http://www.tac.mta.ca/tac/volumes/39/21/39-21abs.html abstract The category of L-algebras is shown to be complete and cocomplete, regular with a zero object and a projective generator, normal and subtractive, ideal determined, but not Barr-exact. Originating from algebraic logic, L-algebras arise in the theory of Garside groups, measure theory, functional analysis, and operator theory. It is shown that the category is far from protomodular, but it has natural semidirect products which have not been described in category-theoretic terms. keywords L-algebra, regular category, Barr-exact, protomodular, semidirect product ams class 08C05, 18D30, 06F05, 08A55, 18B10, 18C10 dois [] DOI citations: J. Ad\'amek, J. Rosick\'y: Locally presentable and accessible categories, Cambridge University Press, 1994 None J. Ad\'amek, J. Rosick\'y, E. M. Vitale: Algebraic theories, A Categorical %Introduction to General Algebra, Cambridge Tracts in Mathematics~184, Cambridge %University Press, 2010 None E. Artin: Theory of braids, Ann. of Math.~48 (1947), 101-126 ['Theory of Braids'] 10.2307/1969218 M. Artin, A. Grothendieck and J.L. Verdier: Th\'eorie des topos et cohomologie %\'etale des schemas, Lecture Notes in Math.~269 and 270, Springer, New York, 1972 ['Topos'] 10.1007/bfb0081555 M. Barr: Exact categories, in: Exact Categories and Sheaves, Lecture Notes in Math.~236, Springer, New York (1971), l-120 ['Exact categories'] 10.1007/bfb0058580 J. B\'enabou: Introduction to bicategories, Lecture Notes in Mathematics~47 (Springer, Berlin, 1967), 1-77 ['Introduction to bicategories'] 10.1007/bfb0074299 J. B\'enabou, J. Roubaud: Monades et descente, C. R. Acad. Sci.~270 (1970), 96-98 None A. Bigard, K. Keimel, and S. Wolfenstein: Groupes et anneaux r\'eticul\'es, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin-New York, 1977 ['Groupes et Anneaux Réticulés'] 10.1007/bfb0067004 B. Bosbach: Rechtskomplementäre Halbgruppen, Math. Z.~124 % (1972), 273-288 None W. J. Blok, D. Pigozzi: Algebraizable logics, Mem. Amer. Math. Soc.~336 (1989), 85pp. ['Algebraizable logics'] 10.1090/memo/0396 D. Bourn: Normalization equivalence, kernel equivalence and affine categories, Springer Lecture Notes Math.~1488 (1991), 43–62 ['Normalization equivalence, kernel equivalence and affine categories'] 10.1007/bfb0084212 D. Bourn: Mal'cev categories and fibrations of pointed objects, Appl. Categ. Struct.~4 (1996), 307-327 ["Mal'cev categories and fibration of pointed objects"] 10.1007/bf00122259 D. Bourn: Protomodular aspect of the dual of a topos, Adv. Math.~187 (2004), 240-255 ['Protomodular aspect of the dual of a topos'] 10.1016/j.aim.2003.09.004 D. Bourn, G. Janelidze: Protomodularity, descent, and semidirect products, Theor. Appl. Cat~4 (1998), 37–46 None D. Bourn, G. Janelidze: Characterization of protomodular varieties of universal algebras, Theor. Appl. Categories~11 (2003), 143-147 None D. Bourn, N. Martins-Ferreira, A. Montoli, M. Sobral: Monoids and pointed S-protomodular categories, Homology, Homotopy, Appl.~18 (2016), 151-172 ['Monoids and pointed $S$-protomodular categories'] 10.4310/hha.2016.v18.n1.a9 E. Brieskorn, K. Saito: Artin-Gruppen und Coxeter-Gruppen, Invent. Math.~17 (1972), 245-271 ['Artin-Gruppen und Coxeter-Gruppen'] 10.1007/bf01406235 T. Bühler: Exact categories, Expo. Math.~28 (2010), no.~1, 1-69 ['Exact categories'] 10.1016/j.exmath.2009.04.004 F. Buekenhout, E. Shult: On the foundations of polar geometry, Geometriae Dedicata~3 (1974), 155-170 ['On the foundations of polar geometry'] 10.1007/bf00183207 A. Carboni: Categories of affine spaces, J. Pure Appl. Algebra~61 (1989), %243-250 ['Categories of affine spaces'] 10.1016/0022-4049(89)90074-1 A. Carboni, J. Lambek, M. C. Pedicchio: Diagram chasing in Mal'cev categories, J. Pure Appl. Algebra~69 (1991), 271-284 ["Diagram chasing in Mal'cev categories"] 10.1016/0022-4049(91)90022-t A. Carboni, E. M. Vitale: Regular and exact completions, J. Pure Appl. Algebra~125 (1998), 79-116 ['Regular and exact completions'] 10.1016/s0022-4049(96)00115-6 A. Carboni, R. F. C. Walters: Cartesian bicategories, I, J. Pure Appl. Algebra~49 (1987), 11-32 ['Cartesian bicategories I'] 10.1016/0022-4049(87)90121-6 C. C. Chang: Algebraic analysis of many valued logics, Trans. Amer. Math. Soc.~88 (1958), 467-490 ['Algebraic analysis of many valued logics'] 10.1090/s0002-9947-1958-0094302-9 C. C. Chang: A new proof of the completeness of the \L ukasiewicz axioms, Trans. Amer. Math. Soc.~93 (1959), 74-80 ['A new proof of the completeness of the Łukasiewicz axioms'] 10.1090/s0002-9947-1959-0122718-1 P. M. Cohn: Universal Algebra, Dordrecht, Netherlands: D.Reidel Publishing, 1981 ['Universal Algebra'] 10.1007/978-94-009-8399-1 M. R. Darnel: Theory of lattice-ordered groups, Monographs and Textbooks in Pure and Applied Mathematics, 187, Marcel Dekker, Inc., New York, 1995 None P. Dehornoy: Groupes de Garside, Ann. Sci. \'Ecole Norm. Sup. (4)~35 (2002), no. 2, 267-306 ['Groupes de Garside'] 10.1016/s0012-9593(02)01090-x P. Dehornoy, L. Paris: Gaussian groups and Garside groups, two generalisations of Artin groups, Proc. London Math. Soc. (3)~79 (1999), no.~3, 569-604 ['Gaussian Groups and Garside Groups, Two Generalisations of Artin Groups'] 10.1112/s0024611599012071 P. Dehornoy, F. Digne, E. Godelle, D. Krammer, J. Michel: Foundations of Garside Theory, EMS Tracts in Math. 22, European Mathematical Society, 2015 ['Foundations of Garside Theory'] 10.4171/139 P. Deligne: Les immeubles des groupes de tresses g\'en\'eralis\'es, Invent. Math.~17 (1972), 273-302 ['Les immeubles des groupes de tresses g�n�ralis�s'] 10.1007/bf01406236 A. Dvure\v{c}enskij: Pseudo MV-algebras are intervals in $l$-groups, J.~Austral. Math. Soc.~72 (2002), 427-445 ['Pseudo MV-algebras are intervals in ℓ-groups'] 10.1017/s1446788700036806 A. Dvure\v censkij, S. Pulmannov\'a: New trends in quantum structures, Mathematics and its Applications, 516, Kluwer Academic Publishers, Dordrecht; Ister Science, Bratislava, 2000 ['New Trends in Quantum Structures'] 10.1007/978-94-017-2422-7 P. Etingof, T. Schedler, A. Soloviev: Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J.~100 (1999), 169-209 ['Set-theoretical solutions to the quantum Yang-Baxter equation'] 10.1215/s0012-7094-99-10007-x Tomas Everaert: Effective descent morphisms of regular epimorphisms, J. Pure Appl. Algebra~216 (2012), 1896-1904 ['Effective descent morphisms of regular epimorphisms'] 10.1016/j.jpaa.2012.02.027 D. H. Fremlin: Measure theory, Vol.~3, Measure algebras, Corrected second printing of the 2002 original, Torres Fremlin, Colchester, 2004 None P. Freyd, A. Scedrov: Categories, Allegories, North-Holland, Amsterdam, 1990 None F. A. Garside: The braid group and other groups, Quart. J. Math. Oxford Ser. (2)~20 (1969) 235-254 ['THE BRAID GROUP AND OTHER GROUPS'] 10.1093/qmath/20.1.235 J. Gispert, D. Mundici: MV-algebras: a variety for magnitudes with Archimedean units, Algebra univers.~53 (2005), 7-43 ['MV-algebras: a variety for magnitudes with archimedean units'] 10.1007/s00012-005-1905-5 P. A. Grillet: Regular categories, in: Exact Categories and Categories of Sheaves, Lecture Notes in Math.~236, Springer Verlag, Berlin-Heidelberg-NewYork, 1971 ['Regular categories'] 10.1007/bfb0058581 M. Gran, Z. Janelidze, A. Ursini: A good theory of ideals in regular multi-pointed categories, J. Pure Appl. Algebra~216 (2012), 1905-1919 ['A good theory of ideals in regular multi-pointed categories'] 10.1016/j.jpaa.2012.02.028 H. P. Gumm, A. Ursini: Ideals in universal algebras, Algebra Univ.~19 (1984), 45-54 ['Ideals in universal algebras'] 10.1007/bf01191491 S. S. Holland, Jr.: Orthomodularity in infinite dimensions; a theorem of M. Solèr, Bull. Amer. Math. Soc.~32 (1995), no. 2, 205-234 ['Orthomodularity in infinite dimensions; a theorem of M. Solèr'] 10.1090/s0273-0979-1995-00593-8 Z. Janelidze: Subtractive categories, Appl. Categ. Struct.~13 (2005), 343-350 ['Subtractive Categories'] 10.1007/s10485-005-0934-8 Z. Janelidze: The pointed subobject functor, $3\times 3$ lemmas, and subtractivity of spans, Theor. Appl. Categories~23 (2010), 221-242 None G. Janelidze, W. Tholen: Facets of descent II, Applied Categ. Struct.~5 (1997), 229-248 ['Facets of Descent, II'] 10.1023/a:1008697013769 G. Janelidze, L. M\'arki, W. Tholen: Semi-abelian categories, J. Pure Appl. Algebra~168 (2002), 367-386 ['Semi-abelian categories'] 10.1016/s0022-4049(01)00103-7 G. Janelidze, L. M\'arki, A. Ursini: Ideals and clots in pointed regular categories, Appl. Categ. Struct.~17 (2009), 345-350 ['Ideals and Clots in Pointed Regular Categories'] 10.1007/s10485-008-9135-6 G. Janelidze, L. M\'arki, W. Tholen, A. Ursini: Ideal determined categories, Cahiers de Topol. G\'eom. Diff\'erentielle Catég.~51 (2010), 115-125 None G. Kalmbach: Orthomodular lattices, London Mathematical Society Monographs, 18, Academic Press, Inc., London, 1983 None B. Keller: Chain complexes and stable categories, Manuscr. math.~67 (1990), 379-417 ['Chain complexes and stable categories'] 10.1007/bf02568439 G. M. Kelly: Monomorphisms, Epimorphisms, and Pull-Backs, J. Austral. Math. Soc.~9 (1969), 124-142 ['Monomorphisms, Epimorphisms, and Pull-Backs'] 10.1017/s1446788700005693 P. Köhler: The semigroup of varieties of Brouwerian semilattices, Trans. Amer. Math. Soc.~241 (1978), 331-342 ['The semigroup of varieties of Brouwerian semilattices'] 10.1090/s0002-9947-1978-0480230-6 F. W. Lawvere: Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. USA~50 (1963), 869-872 ['FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES'] 10.1073/pnas.50.5.869 F. W. Lawvere: Theory of Categories over a Base Topos, Ist. Mat. Univ. Perugia (1972-73) None W. A. J. Luxemburg, A. C. Zaanen: Riesz spaces, Vol. I., North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., New York, 1971 None S.~Mac Lane: Duality for groups, Bull. Amer. Math. Soc.~56 (1950), 485-516 None S.~Mac Lane: Categories for the Working Mathematician, %Springer-Verlag, New York - Heidelberg - Berlin 1971 ['Categories for the Working Mathematician'] 10.1007/978-1-4612-9839-7 S. Mac Lane, I. Moerdijk: Sheaves in geometry and logic, A first introduction to topos theory, Corrected reprint of the 1992 edition, Universitext, Springer-Verlag, New York, 1994 ['Sheaves in Geometry and Logic'] 10.1007/978-1-4612-0927-0 D. Maharam: The representation of abstract measure functions, Trans. Amer. Math. Soc.~65 (1949), 279-330 ['The representation of abstract measure functions'] 10.1090/s0002-9947-1949-0028923-0 N. Martins-Ferreira, M. Sobral: On categories with semidirect products, J. Pure Appl. Algebra~216 (2012), 1968-1975 ['On categories with semidirect products'] 10.1016/j.jpaa.2012.02.035 N. Martins-Ferreira, A. Montoli, M. Sobral: Semidirect products and crossed modules in monoids with operations, J. Pure Appl. Algebra~217 (2013), 334-347 ['Semidirect products and crossed modules in monoids with operations'] 10.1016/j.jpaa.2012.06.022 N. Martins-Ferreira, A. Montoli, M. Sobral: Semidirect products and Split Short Five Lemma in normal categories, Applied Categ. Struct~22 (2014), 687–697 ['Semidirect Products and Split Short Five Lemma in Normal Categories'] 10.1007/s10485-013-9344-5 J. C. C. McKinsey, A. Tarski: On closed elements in closure algebras, Ann. of Math.~47 (1946), 122-162 ['On Closed Elements in Closure Algebras'] 10.2307/1969038 B. Mitchell: Theory of Categories, Academic Press, New York and London, 1965 None D. Mundici: Interpretation of AFC$^\ast$ algebras in \L ukasiewicz sentential calculus, J. Funct. Anal.~65 (1986), 15-63 ['Interpretation of AF C∗-algebras in Łukasiewicz sentential calculus'] 10.1016/0022-1236(86)90015-7 M. C. Pedicchio, E. M. Vitale: On the abstract characterization of quasi-varieties, Algebra univ.~43 (2000), 269–278 ['On the abstract characterization of quasi-varieties'] 10.1007/s000120050158 D. Quillen: Higher algebraic $K$-theory, I, in: Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85-147, Lecture Notes in Math.~341, Springer, Berlin, 1973 ['Higher algebraic K-theory: I'] 10.1007/bfb0067053 L. R\'edei: Die Verallgemeinerung der Schreierschen Erweiterungstheorie, Acta Sci. Math. (Szeged)~14 (1952), 252–273 None Z. Rie\v{c}anov\'a: Generalization of blocks for D-lattices and lattice-ordered effect algebras, Internat. J. Theoret. Phys.~39 (2000), no.~2, 231-237 ['Generalization of Blocks for D-Lattices and Lattice-Ordered Effect Algebras'] 10.1023/a:1003619806024 W. Rump: A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation, Adv. Math.~193 (2005), 40-55 ['A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation'] 10.1016/j.aim.2004.03.019 W. Rump: Braces, radical rings, and the quantum Yang-Baxter %equation, J. Algebra~307 (2007), 153-170 ['Braces, radical rings, and the quantum Yang–Baxter equation'] 10.1016/j.jalgebra.2006.03.040 W. Rump: $L$-Algebras, Self-similarity, and $l$-Groups, J. Algebra~320 (2008), no.~6, 2328-2348 ['L-algebras, self-similarity, and l-groups'] 10.1016/j.jalgebra.2008.05.033 W. Rump: Semidirect products in algebraic logic and solutions of %the quantum Yang-Baxter equation, J. Algebra Appl.~7 (2008), no. 4, 471-490 ['SEMIDIRECT PRODUCTS IN ALGEBRAIC LOGIC AND SOLUTIONS OF THE QUANTUM YANG–BAXTER EQUATION'] 10.1142/s0219498808002904 W. Rump: A general Glivenko theorem, Algebra Univ.~61 (2009), no.~3-4, 455-473 ['A general Glivenko theorem'] 10.1007/s00012-009-0018-y W. Rump: Right $l$-Groups, Geometric Garside Groups, and Solutions of the Quantum Yang-Baxter Equation, J.~Algebra~439 (2015), 470-510 ['Right l-groups, geometric Garside groups, and solutions of the quantum Yang–Baxter equation'] 10.1016/j.jalgebra.2015.04.045 W. Rump: Von Neumann algebras, $L$-algebras, Baer ${}^\ast$-monoids, and Garside groups, Forum Math.~30 (2018), no.~4, 973-995 None W. Rump: $L$-algebras with duality and the structure group of a %set-theoretic solution to the Yang-Baxter equation, J. Pure Appl. Algebra~224 %(2020), no.~8, 106314, 12 pp. ['L-algebras with duality and the structure group of a set-theoretic solution to the Yang-Baxter equation'] 10.1016/j.jpaa.2020.106314 W. Rump: Commutative $L$-algebras and measure theory, Forum Math.~33, no.~6 (2021), 1527-1548 None W. Rump: Symmetric quantum sets and L-algebras, IMRN, Volume~2022, Issue~3, February 2022, Pages 1770-1810 {\tt doi:10.1093/imrn/rnaa135} None W. Rump: $L$-algebras and three main non-classical logics, Ann. Pure Appl. Logic~173, Issue 7 (2022), 103121 ['L-algebras and three main non-classical logics'] 10.1016/j.apal.2022.103121 W. Rump: Structure groups of $L$-algebras and Hurwitz action, Geometriae Dedicata~216 (2022), DOI: 10.1007/s10711-022-00697-4 ['Structure groups of L-algebras and Hurwitz action'] 10.1007/s10711-022-00697-4 W. Rump, X. Zhang: L-effect algebras, Studia Logica~108 (2020), %no.~4, 725-750 ['L-effect Algebras'] 10.1007/s11225-019-09873-2 J.-P. Schneiders: Quasi-abelian categories and sheaves, M\'em. Soc. Math. France~1999, no.~76 ['Quasi-abelian categories and sheaves'] 10.24033/msmf.389 R. Succi Cruciani: La teoria delle relazioni nello studio di categorie regolari e di categorie esatte, Riv. Mat. Univ. Parma~1 (1975), 143-158 None A. Tarski: Grundzüge des Systemenkalküls, Erster Teil. In: Fundam. Math.~25 (1935), 503–526 ['Grundzüge der Systemenkalküls I'] 10.4064/fm-25-1-503-526 A. Ursini: Sulle variet\`a di algebre con buona teoria degli ideali, Boll. Unione Mat. Ital.~7 (1972), 90-95 None A. Ursini: On subtractive varieties, I, Algebra Univ.~31 (1994), 204-222 ['On subtractive varieties, I'] 10.1007/bf01236518 E. M. Vitale: On the characterization of monadic categories over set, Cah. topol. g\'eom. diff\'erentielle cat\'eg.~35, no.~4 (1994), 351-358 None Y. Wu, J. Wang, Y. Yang: Lattice-ordered effect algebras and L-algebras, Fuzzy Sets and Systems~369 (2019), 103-113 \end{thebibliography} \end{document} Now let $\mbox{\bf Cyc$^\ast$}$ be the category of unital cycloids. Thus $\mbox{\bf Cyc$^\ast$}$ is a variety. ['Lattice-ordered effect algebras and L-algebras'] 10.1016/j.fss.2018.08.013 -------------- ['Firm homomorphisms of rings and semigroups'] author [{'given': 'Leandro', 'family': 'Marín'}, {'given': 'Valdis', 'family': 'Laan'}] publication date 2023-07-18 volume 39 issue 22 page range ('625', '666') url http://www.tac.mta.ca/tac/volumes/39/22/39-22abs.html abstract In this paper we define firm homomorphisms between rings without identity in such a way that the category of rings with identity will become a full subcategory of the category of firm rings with firm homomorphisms as morphisms. We prove that firm homomorphisms are in one-to-one correspondence with pairs of compatible concrete functors between certain module categories. This correspondence is given by the restriction of scalars. We also prove the semigroup theoretic analogues of these results and give a list of examples of firm homomorphisms. keywords Concrete functor, firm homomorphism, firm ring, firm module, closed module, restriction of scalars. ams class 16B50,16D90,20M30 dois [] DOI citations: J. Ad\'amek, H. Herrlich, G.E. Strecker, Abstract and concrete categories: the joy of cats, Repr. Theory Appl. Categ. No. 17 (2006), 1--507. None P.N. \'Anh, L. M\'arki, Morita equivalence for rings without identity, Tsukuba J. Math. 11 (1987), 1--16. ['Morita equivalence for rings withot identity'] 10.21099/tkbjm/1496160500 F. Borceux, Handbook of categorical algebra 1, Encyclopedia of Mathematics and its Applications~50, Cambridge University Press, Cambridge, 1994, xvi+345 pp. ['Handbook of Categorical Algebra'] 10.1017/cbo9780511525865 J.L. Garc\'ia, L. Mar\'in, Rings having a Morita-like equivalence, Comm. Algebra 27 (1999), 665--680. ['Rings having a morita-like equivalence'] 10.1080/00927879908826455 J.L. Garc\'ia, L. Mar\'in, Some properties of tensor-idempotent rings, Contemp. Math. 259 (2000), 223--235. ['Some properties of tensor-idempotent rings'] 10.1090/conm/259/04097 J.L. García, L. Mar\'in, Morita theory for associative rings, Comm. Algebra 29 (2001), 5835--5856. ['MORITA THEORY FOR ASSOCIATIVE RINGS'] 10.1081/agb-100107961 G. M. Kelly, R. Street, Review of the elements of $2$-categories. In: Category Seminar (Proc. Sem., Sydney, 1972/1973), pp. 75--103. Lecture Notes in Math., Vol. 420, Springer, Berlin, 1974. ['Category Seminar'] 10.1007/bfb0063096 M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, acts and categories. De Gruyter Expositions in Mathematics, 29. Walter de Gruyter \& Co., Berlin, 2000. xviii+529 pp. ['Monoids, Acts and Categories'] 10.1515/9783110812909 V. Laan, L. M\'arki, \"U. Reimaa, Morita equivalence of semigroups revisited: firm semigroups, J. Algebra 505 (2018), 247--270. ['Morita equivalence of semigroups revisited: Firm semigroups'] 10.1016/j.jalgebra.2018.02.018 V. Laan, \"U. Reimaa, Morita equivalence of factorizable semigroups, Internat. J. Algebra Comput. 29 (2019), 723--741. ['Morita equivalence of factorizable semigroups'] 10.1142/s0218196719500243 V. Laan, \"U. Reimaa, Monomorphisms in categories of firm acts, Studia Sci. Math. Hungar. 56 (2019), 267--279. ['Monomorphisms in categories of firm acts'] 10.1556/012.2019.56.3.1432 M. V. Lawson, Morita equivalence of semigroups with local units, J. Pure Appl. Algebra 215 (2011), 455--470. ['Morita equivalence of semigroups with local units'] 10.1016/j.jpaa.2010.04.030 S. Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics~5, Springer--Verlag, New York, 1998, xii+314 pp. ['Categories for the Working Mathematician'] 10.1007/978-1-4612-9839-7 L. Mar\'in, Categories of modules for idempotent rings and Morita equivalences, MSc thesis, University of Glasgow, UK; Publicaciones del Departamento de Matem\'aticas, num 23, Universidad de Murcia, Spain, 1998. None L. Mar\'in, The construction of a generator for $R-\mathsf{DMod}$, Interactions between ring theory and representations of algebras (Murcia), 287--296, Lecture Notes in Pure and Appl. Math., 210, Dekker, New York, 2000. None B. Pareigis, Vergessende Funktoren und Ringhomomorphismen, Math. Z. 93 (1966), 265--275. ['Vergessende Funktoren und Ringhomomorphismen'] 10.1007/bf01111937 D.G. Quillen, Module theory over nonunital rings, (1996), unpublished notes. None \"U. Reimaa, V. Laan, L. Tart, Morita equivalence of finite semigroups, Semigroup Forum 102 (2021), 842--860. %\endrefs \end{references*} \end{document} ['Morita equivalence of finite semigroups'] 10.1007/s00233-020-10153-y -------------- ['Symmetries of data sets and functoriality of persistent homology'] author [{'given': 'Wojciech', 'family': 'Chachólski'}, {'given': 'Alessandro De', 'family': 'Gregorio'}, {'given': 'Nicola', 'family': 'Quercioli'}, {'given': 'Francesca', 'family': 'Tombari'}] publication date 2023-08-04 volume 39 issue 23 page range ('667', '686') url http://www.tac.mta.ca/tac/volumes/39/23/39-23abs.html abstract The aim of this article is to describe a new perspective on functoriality of persistent homology and explain its intrinsic symmetry that is often overlooked. A data set for us is a finite collection of functions, called measurements, with a finite domain. Such a data set might contain internal symmetries which are effectively captured by the action of a set of the domain endomorphisms. Different choices of the set of endomorphisms encode different symmetries of the data set. We describe various category structures on such enriched data sets and prove some of their properties such as decompositions and morphism formations. We also describe a data structure, based on coloured directed graphs, which is convenient to encode the mentioned enrichment. We show that persistent homology preserves only some aspects of these collection of enriched data sets however not all. In other words persistent homology is not a functor on the entire category of enriched data sets. Nevertheless we show that persistent homology is functorial locally. We use the concept of set equivariant operator (SEO) to capture some of the information missed by persistent homology. Moreover, we provide examples and give ways to construct such SEOs. keywords persistent homology, equivariant operators ams class 55N31, 62R40, 68T09, 18D25 dois [] DOI citations: Bergomi, M. G., Frosini, P., Giorgi, D. and Quercioli, N., ``Towards a topological–geometrical theory of group equivariant non-expansive operators for data analysis and machine learning'', \emph{Nature Machine Intelligence}, 1(9):423-433,2019. doi:10.1038/s42256-019-0087-3. ['Towards a topological–geometrical theory of group equivariant non-expansive operators for data analysis and machine learning'] 10.1038/s42256-019-0087-3 Bergomi, M. G., Frosini, P., Giorgi, D. and Quercioli, N., ``Towards a topological–geometrical theory of group equivariant non-expansive operators for data analysis and machine learning'', arXiv:1812.11832, 2018. ['Towards a topological–geometrical theory of group equivariant non-expansive operators for data analysis and machine learning'] 10.1038/s42256-019-0087-3 Carlsson, G., Zomorodian, A., ``The theory of multidimensional persistence'', \emph{Discrete Comput. Geom.}, 42(1):71-93, 2009, doi:10.1007/s00454-009-9176-0 ['The Theory of Multidimensional Persistence'] 10.1007/s00454-009-9176-0 Cohen-Steiner, D., Edelsbrunner, H., and Harer, J., ``Extending persistence using Poincar\'{e} and Lefschetz duality'', \emph{Found. Comput. Math.}, 9(1):79--103, 2009, doi:10.1007/s10208-008-9027-z. ['Extending Persistence Using Poincaré and Lefschetz Duality'] 10.1007/s10208-008-9027-z Dwyer, W. G. and Henn, H.W., ``Homotopy theoretic methods in group cohomology'', \emph{Advanced Courses in Mathematics. CRM Barcelona}, Birkh\"{a}user Verlag, Basel, 2001, doi:10.1007/978-3-0348-8356-6. None Edelsbrunner, H., and Harer, J., ``Persistent homology---a survey'', \emph{Surveys on discrete and computational geometry}, volume 453 of \emph{Contemp. Math.}, pages 257--282. Amer. Math. Soc., Providence, RI, 2008. doi:10.1090/conm/453/08802. ['Persistent homology—a survey'] 10.1090/conm/453/08802 Giusti, C., Pastalkova, E., Curto, C., and Itskov, V., ``Clique topology reveals intrinsic geometric structure in neural correlations'', \emph{Proceedings of the National Academy of Sciences}, 112(44):13455-13460, 2015, doi:10.1073/pnas.1506407112. ['Clique topology reveals intrinsic geometric structure in neural correlations'] 10.1073/pnas.1506407112 De Gregorio, A., Fugacci, U., Memoli, F., and Vaccarino, F.,``On the notion of weak isometry for finite metric spaces'', 2020, arXiv:2005.03109. None Lesnick, M., ``The theory of the interleaving distance on multidimensional persistence modules", \emph{Found. Comput. Math.}, 15(3): 613-650, (2015), doi:10.1007/s10208-015-9255-y. ['The Theory of the Interleaving Distance on Multidimensional Persistence Modules'] 10.1007/s10208-015-9255-y Oudot, S. Y., ``Persistence theory: from quiver representations to data analysis", \emph{Mathematical Surveys and Monographs}, 209, (2015), https://doi.org/10.1090/surv/209. ['Persistence Theory: From Quiver Representations to Data Analysis'] 10.1090/surv/209 Scolamiero, M., Chach\'{o}lski, W. and Lundman, A. and Ramanujam, R. and \"{O}berg, S., ``Multidimensional persistence and noise", \emph{Found. Comput. Math.}, 17(6): 1367-1406, (2017), doi:110.1007/s10208-016-9323-y. ['Multidimensional Persistence and Noise'] 10.1007/s10208-016-9323-y Thomason, R. W., ``Homotopy colimits in the category of small categories", \emph{Math. Proc. Cambridge Philos. Soc.}, 85(1): 91-109, (1979), doi:10.1017/S0305004100055535. \end{thebibliography} \end{document} ['Homotopy colimits in the category of small categories'] 10.1017/s0305004100055535 -------------- ['Ideals and continuity forquantaloid-enriched categories'] author [{'given': 'Min', 'family': 'Liu'}, {'given': 'Shengwei', 'family': 'Han'}, {'given': 'Isar', 'family': 'Stubbe'}] publication date 2023-08-17 volume 39 issue 24 page range ('687', '713') url http://www.tac.mta.ca/tac/volumes/39/24/39-24abs.html abstract We study ideals in, and continuity of, quantaloid-enriched categories (-categories for short) as a `many-valued and many-typed' generalization of domain theory. Abstractly, for any (saturated) class Φ of presheaves, we define and study the Φ-continuity of -categories. Concretely, we compute three examples of such saturated classes of presheaves – the class of flat ideals, the class of irreducible ideals and the class of conical ideals – which are proper generalizations of ideals in domain theory. keywords Quantaloid, enriched category, domain theory, fuzzy order ams class 18B35, 18D20, 06F07 dois [] DOI citations: S. Abramsky, A. Jung, Domain Theory, in: S. Abramsky, D. M. Gabbay, T. S. E. Maibaum (Eds.), Handbook for Logic in Computer Science, vol. 3, Clarendon Press, Oxford, 1994. ['Handbook of Logic in Computer Science'] 10.1093/oso/9780198537359.001.0001 J. Ad\'{a}mek, H. Herrlich, G. E. Strecker, Abstract and Concrete Categories, Wiley, NewYork, 1990. None M. H. Albert, G. M. Kelly, The closure of a class of colimits, Journal of Pure and Applied Algebra 51 (1988) 1--17. ['The closure of a class of colimits'] 10.1016/0022-4049(88)90073-4 P. America, J. J. M. M. Rutten, Solving reflexive domain equations in a category of complete metric spaces, Journal of Computer and System Sciences 39 (1989) 343--375. ['Solving reflexive domain equations in a category of complete metric spaces'] 10.1016/0022-0000(89)90027-5 H. J. Bandelt, M. Ern\'{e}, The category of $Z$-continuous posets, Journal of Pure and Applied Algebra 30 (1983) 219--226. ['The category of Z-continuous posets'] 10.1016/0022-4049(83)90057-9 A. Baranga, $Z$-continuous posets, Discrete Mathematics 152 (1996) 33--45. ['Z-continuous posets'] 10.1016/0012-365x(94)00307-5 R. B\v{e}lohl\'{a}vek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic Publishers, Plenum Publishers, New York, 2002. None J. B\'{e}nabou, Introduction to bicategories, Lecture Notes in Mathematics 47 (1967) 1--77. ['Introduction to bicategories'] 10.1007/bfb0074299 G. J. Bird, G. M. Kelly, A. J. Power and R. H. Street, Flexible limits for 2-categories, Journal of Pure and Applied Algebra 61 (1989) 1--27. ['Flexible limits for 2-categories'] 10.1016/0022-4049(89)90065-0 M. M. Clementino, D. Hofmann, I. Stubbe, Exponentiable functors between quantaloid-enriched categories, Applied Categorical Structures 17 (2009) 91--101. ['Exponentiable Functors Between Quantaloid-Enriched Categories'] 10.1007/s10485-007-9104-5 M. Ern\'{e}, $Z$-Continuous posets and their topological manifestation, Applied Categorical Structures 7 (1999) 31--70. ['Z-Continuous Posets and Their Topological Manifestation'] 10.1023/a:1008657800278 L. Fan, A new approach to quantitative domain theory, Electronic Notes in Theoretical Computer Science 45 (2001) 77--87. ['A New Approach to Quantitative Domain Theory'] 10.1016/s1571-0661(04)80956-3 R. C. Flagg, R. Kopperman, Continuity spaces: Reconciling domains and metric spaces, Theoretical Computer Science 177 (1997) 111--138. ['Continuity spaces: Reconciling domains and metric spaces'] 10.1016/s0304-3975(97)00236-3 G. Gierz, K. H. Hofmann, K. Keimel, et al., Continuous Lattices and Domains, Cambridge University Press, Cambridge, 2003. None J. He, H. Lai, L. Shen, Towards probabilistic partial metric spaces: Diagonals between distance distributions, Fuzzy Sets and Systems 370 (2019) 99--119. ['Towards probabilistic partial metric spaces: Diagonals between distance distributions'] 10.1016/j.fss.2018.07.011 H. Heymans, I. Stubbe, Elementary characterisation of quantaloids of closed cribles, Journal of Pure and Applied Algebra 216 (2012) 1952--1960. ['Elementary characterisation of small quantaloids of closed cribles'] 10.1016/j.jpaa.2012.02.032 D. Hofmann, L. Sousa, Aspects of algebraic algebras, Logical Methods in Computer Science 13 (2017) 1--25. ['Aspects of algebraic Algebras'] 10.23638/lmcs-13(3:4)2017 D. Hofmann, I. Stubbe, Topology from enrichment: the curious case of partial metrics, Cahiers Topologie G\'{e}om\'{e}trie Diff\'{e}rentielle Cat\'{e}goriques 59 (2018) 307--353. None D. Hofmann, P. L. Waszkiewicz, Approximation in quantale-enriched categories, Topology and its Applications 158 (2011) 963--977. ['Approximation in quantale-enriched categories'] 10.1016/j.topol.2011.02.003 U. H\"{o}hle, Commutative, residuated l-monoids, in: U. H\"{o}hle, E.P. Klement (Eds.), Non-Classical Logics and Their Applications to Fuzzy Subsets: A Handbook on the Mathematical Foundations of Fuzzy Set Theory, Kluwer Academic Publishers, Dordrecht, 1995, pp. 53--105. ['Non-Classical Logics and their Applications to Fuzzy Subsets'] 10.1007/978-94-011-0215-5 U. H\"{o}hle, Categorical foundations of topology with applications to quantaloid enriched topological spaces, Fuzzy Sets and Systems 256 (2014) 166--210. ['Categorical foundations of topology with applications to quantaloid enriched topological spaces'] 10.1016/j.fss.2013.03.010 U. H\"{o}hle, T. Kubiak, A non-commutative and non-idempotent theory of quantale sets, Fuzzy Sets and Systems 166 (2011) 1--43. ['A non-commutative and non-idempotent theory of quantale sets'] 10.1016/j.fss.2010.12.001 G. M. Kelly, Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Notes Series vol. 64, Cambridge University Press, Cambridge, 1982. None G. M. Kelly, V. Schmitt, Notes on enriched categories with colimits of some class, Theory and Applications of Categories 14 (2005) 399--423. None R. Kopperman, All topologies come from generalized metrics, The American Mathematical Monthly 95 (1988) 89--97. ['All Topologies Come From Generalized Metrics'] 10.1080/00029890.1988.11971974 H. Lai, L. Shen, Fixed points of adjoint functors enriched in a quantaloid, Fuzzy Sets and Systems 321 (2017) 1--28. ['Fixed points of adjoint functors enriched in a quantaloid'] 10.1016/j.fss.2016.12.001 H. Lai, D. Zhang, Complete and directed complete $\Omega$-categories, Theoretical Computer Science 388 (2007) 1--25. None H. Lai, D. Zhang, G. Zhang, A comparative study of ideals in fuzzy orders, Fuzzy Sets and Systems 382 (2020) 1--28. ['A comparative study of ideals in fuzzy orders'] 10.1016/j.fss.2018.11.019 M. Liu, B. Zhao, Closure operators and closure systems on quantaloid-enriched categories, Journal of Mathematical Research with Applications 33 (2013) 23--34. None M. Liu, B. Zhao, Cartesian closed categories of $F\mathcal{Z}$-domains, Acta Mathematica Sinica, English Series 29 (2013) 2373--2390. ['Cartesian closed categories of FƵ-domains'] 10.1007/s10114-013-1240-2 M. Liu, B. Zhao, Two cartesian closed subcategories of fuzzy domains, Fuzzy Sets and Systems 238 (2014) 102--112. ['Two cartesian closed subcategories of fuzzy domains'] 10.1016/j.fss.2013.07.015 S. G. Matthews, Partial metric topology, in Proceedings of the 8th Summer Conference on Topology and Its Applications, Vol. 728, 1992, pp. 176--185. None S. J. O'Neill, A fundamental study into the theory and application of the partial metric spaces, PhD thesis, University of Warwick, Department of Computer Science, March 1998. None Q. Pu, D. Zhang, Preordered sets valued in a GL-monoid, Fuzzy Sets and Systems 187 (2012) 1--32. ['Preordered sets valued in a GL-monoid'] 10.1016/j.fss.2011.06.012 A. Pitts, Applications of sup-lattice enriched category theory to sheaf theory, Proceedings of the London Mathematical Society 57 (1988) 433--480. ['Applications of Sup-Lattice Enriched Category Theory to Sheaf Theory'] 10.1112/plms/s3-57.3.433 S. P. Rao, Q. G. Li, Fuzzy $Z$-continuous posets, Abstract and Applied Analysis, vol. 2013, Article ID 607934, 14 pages, 2013. ['Fuzzy -Continuous Posets'] 10.1155/2013/607934 K. I. Rosenthal, Quantales and Their Applications, Pitman Research Notes in Mathematics Series, Vol. 234, Longman, Essex, 1990. None K. I. Rosenthal, Quantaloids, enriched categories and automata theory, Applied Categorical Structures 3 (1995) 279--301. ['Quantaloids, enriched categories and automata theory'] 10.1007/bf00878445 K. I. Rosenthal, The Theory of Quantaloids, Pitman Research Notes in Mathematics Series, Vol.348, Longman, Essex, 1996. None J. J. M. M. Rutten, Elements of generalized ultrametric domain theory, Theoretical Computer Science 170 (1996) 349--381. ['Elements of generalized ultrametric domain theory'] 10.1016/s0304-3975(96)80711-0 D. S. Scott, Outline of a mathematical theory of computation, in: The Fourth Annual Princeton Conf. on Information Sciences and Systems, Princeton University Press, Princeton, NJ, 1970, pp. 169--176. None D. S. Scott, Continuous lattices, topos, algebraic geometry and logic, in: Lecture Notes in Mathematics, Springer, Berlin, Vol. 274, 1972, pp. 97--136. ['Continuous lattices'] 10.1007/bfb0073967 L. Shen, ${\Q}$-closure spaces, Fuzzy Sets and Systems 300 (2016) 102--133. None L. Shen, Y. Tao, D. Zhang, Chu connections and back diagonals between ${\Q}$-distributors, Journal of Pure and Applied Algebra 220 (2016) 1858--1901. None L. Shen, W. Tholen, Topological categories, quantaloids and Isbell adjunctions, Topology and its Applications 200 (2016) 212--236. ['Topological categories, quantaloids and Isbell adjunctions'] 10.1016/j.topol.2015.12.020 L. Shen, D. Zhang, Categories enriched over a quantaloid: Isbell adjunctions and Kan adjunctions, Theory and Applications of Categories 28 (2013) 577--615. None M. B. Smyth, Quasi-uniformities: Reconciling domains and metric spaces, Lecture Notes in Comput. Sci. 298 (1988) 236--253. ['Quasi-uniformities: Reconciling domains with metric spaces'] 10.1007/3-540-19020-1_12 R. H. Street, Enriched categories and cohomology, Quaestiones Mathematicae 6 (1983) 265--283. ['ENRICHED CATEGORIES AND COHOMOLOGY'] 10.1080/16073606.1983.9632304 I. Stubbe, Categorical structures enriched in a quantaloid: Categories, distributors, functors, Theory and Applications of Categories 14 (2005) 1--45. None I. Stubbe, Categorical structures enriched in a quantaloid: Tensored and cotensored categories, Theory and Applications of Categories 16 (2005) 283--306. None I. Stubbe, Categorical structures enriched in a quantaloid: Orders and ideals over a base quantaloid, Applied Categorical Structures 13 (2005) 235--255. ['Categorical Structures Enriched in a Quantaloid: Orders and Ideals over a Base Quantaloid'] 10.1007/s10485-004-7421-5 I. Stubbe, Towards ``dynamic domains": Totally continuous cocomplete ${\Q}$-categories, Theoretical Computer Science 373 (2007) 142--160. None I. Stubbe, ``Hausdorff distance" via conical cocompletion, Cahiers Topologie G\'{e}o\-m\'{e}\-trie Diff\'{e}rentielle Cat\'{e}goriques 51 (2010) 51--76. None I. Stubbe, An introduction to quantaloid-enriched categories, Fuzzy Sets and Systems 256 (2014) 95-116. ['An introduction to quantaloid-enriched categories'] 10.1016/j.fss.2013.08.009 Y. Tao, H. Lai, D. Zhang, Quantale-valued preorders: globalization and cocompleteness, Fuzzy Sets and Systems 256 (2014) 236--251. ['Quantale-valued preorders: Globalization and cocompleteness'] 10.1016/j.fss.2012.09.013 K. R. Wagner, Solving recursive domain equations with enriched categories, PhD thesis, Carnegie Mellon University, 1994. None K. R. Wagner, Liminf convergence in $\Omega$-categories, Theoretical Computer Science 184 (1997) 61--104. ['Liminf convergence in Ω-categories'] 10.1016/s0304-3975(96)00223-x R. F. C. Walters, Sheaves and Cauchy-complete categories, Cahiers Topologie G\'{e}o\-m\'{e}trie Diff\'{e}rentielle Cat\'{e}goriques 22 (1981) 283--286. None R. F. C. Walters, Sheaves on sites as Cauchy complete categories, Journal of Pure and Applied Algebra 24 (1982) 95--102. ['Sheaves on sites as Cauchy-complete categories'] 10.1016/0022-4049(82)90061-5 P. Waszkiewicz, Quantitative continuous domains, Applied Categorical Structures 11 (2003) 41--67. ['Quantitative Continuous Domains'] 10.1023/a:1023012924892 J. B. Wright, E. G. Wagner, J. W. Thatcher, A uniform approach to inductive posets and inductive closure, Theoretical Computer Science 7 (1978) 57--77. ['A uniform approach to inductive posets and inductive closure'] 10.1016/0304-3975(78)90040-3 W. Yao, Quantitative domains via fuzzy sets: Part I: continuity of fuzzy directed-complete poset, Fuzzy Sets and Systems 161 (2010) 983--987. ['Quantitative domains via fuzzy sets: Part I: Continuity of fuzzy directed complete posets'] 10.1016/j.fss.2009.06.018 W. Yao, F. G. Shi, Quantitative domains via fuzzy sets: Part II: Fuzzy Scott topology on fuzzy directed-complete posets, Fuzzy Sets and Systems 173 (2011) 60--80. ['Quantitative domains via fuzzy sets: Part II: Fuzzy Scott topology on fuzzy directed-complete posets'] 10.1016/j.fss.2011.02.003 W. Yao, B. Zhao, A duality between $\Omega$-categories and algebraic $\Omega$-categories, Electronic Notes in Theoretical Computer Science 301 (2014) 153--168. ['A Duality Between Ω-categories and Algebraic Ω-categories'] 10.1016/j.entcs.2014.01.013 Q. Y. Zhang, L. Fan, Continuity in quantitative domains, Fuzzy Sets and Systems 154 (2005) 118--131. \end{thebibliography} \end{document} ['Continuity in quantitative domains'] 10.1016/j.fss.2005.01.007 -------------- ['On Enriched Categories and Induced Representations'] author [{'given': 'Joshua A.', 'family': 'Leslie'}, {'given': 'Ralph A.', 'family': 'Twum'}] publication date 2023-08-23 volume 39 issue 25 page range ('714', '734') url http://www.tac.mta.ca/tac/volumes/39/25/39-25abs.html abstract We show that induced representations for a pair of diffeological Lie groups exist, in the form of an indexed colimit in the category of diffeological spaces. keywords diffeology induced representation, diffeological spaces, diffeological categories, diffeological lie groups, Kan extensions, enriched category, indexed limits, indexed colimits ams class Primary 18D20; Secondary 22D30 dois [] DOI citations: R.~Bott. \newblock Homogeneous vector bundles. \newblock {\em Annals of Mathematics}, 66(2):203--248, 1957. ['Homogeneous Vector Bundles'] 10.2307/1969996 M~Demazure. \newblock A very simple proof of {Bott's} theorem. \newblock {\em Inventiones Mathematicae}, 33:271 -- 272, 1976. ["A very simple proof of Bott's theorem"] 10.1007/bf01404206 Ivan Dimitrov and Ivan Penkov. \newblock Ind-varieties of generalized flags as homogeneous spaces for classical ind-groups. \newblock {\em International Mathematics Research Notices}, 55, 2004. None J.~Frey. \newblock Notes on 2-categorical limits. \newblock \url{https://drive.google.com/file/d/0B6cQeyZSnWMlcm1jNkd2eHlSWTA/view}, 2010. \newblock Accessed: 12th March, 2021. None P.~Iglesias-Zemmour. \newblock {\em Diffeology}. \newblock 185. AMS, Rhode Island, 2013. \newblock Mathematical Surveys and Monographs. None G.M. Kelly. \newblock {\em Basic Concepts of Enriched Category Theory}. \newblock Number~10 in Reprints in Theory and Applications of Categories. TAC, 2005. None B.~Kostant. \newblock Lie algebra cohomology and the generalized {Borel-Weil} theorem. \newblock {\em Annals of Mathematics}, 74(2):329--387, 1961. ['Lie Algebra Cohomology and the Generalized Borel-Weil Theorem'] 10.2307/1970237 S~Kumar. \newblock {\em Kac-Moody Groups, Their Flag Varieties and Representation Theory}. \newblock Number 204 in Progress in Mathematics. Birkhauser, 2002. ['Kac-Moody Groups, their Flag Varieties and Representation Theory'] 10.1007/978-1-4612-0105-2 M.~Laubinger. \newblock {\em Differential Geometry in Cartesian Closed Categories of Smooth Spaces}. \newblock PhD thesis, Louisiana State University, 2008. ['Differential geometry in cartesian closed categories of smooth spaces'] 10.31390/gradschool_dissertations.3981 J.~Leslie. \newblock On a diffeological group realization of certain generalized symmetrizable {Kac-Moody} {Lie} algebras. \newblock {\em Journal of Lie Theory}, 13:427--442, 2003. None J.~Leslie. \newblock Automorphisms of graded super symplectic manifolds. \newblock {\em Surveys in Differential Geometry}, 15:237--253, 2011. ['Automorphisms of graded super symplectic manifolds'] 10.4310/sdg.2010.v15.n1.a7 J.~Lurie. \newblock A proof of the {Borel-Weil-Bott} theorem. \newblock \url{https://www.math.ias.edu/~lurie/papers/bwb.pdf}. \newblock Accessed: 16th June, 2021. None Jean-Pierre Magnot. \newblock The group of diffeomorphisms of a non compact manifold is not regular. \newblock Hal-01653814, 2017. \newblock \url{https://hal.archives-ouvertes.fr/hal-01653814/document}. ['The group of diffeomorphisms of a non-compact manifold is not regular'] 10.1515/dema-2018-0001 J.~Milnor. \newblock Remarks on some infinite dimensional {Lie} groups. \newblock In B.S. {DeWitt} and {R.} {Stora}, editors, {\em Relativity, Groups and Topology II}, pages 1007--1057. Elsevier Science Publishers B.V., 1983. None S.~Mac~Lane. \newblock {\em Categories for the Working Mathematician}. \newblock Number~5 in Graduate Texts in Mathematics. Springer-Verlag, New York, 2 edition, 1998. ['Categories for the Working Mathematician'] 10.1007/978-1-4612-9839-7 A~Rogers. \newblock {\em Supermanifolds: Theory and Applications}. \newblock World Scientific, 2007. ['Supermanifolds - Theory and Applications'] 10.1142/9789812708854 R.~Twum. \newblock {\em An Application of Enriched Category Theory to the Borel-Weil Theorem}. \newblock PhD thesis, Howard University, 2012. \end{thebibliography} None Basic Concepts of Enriched Category Theory Kelly, G.M. None Homogeneous Vector Bundles Bott, R. ['Homogeneous Vector Bundles'] 10.2307/1969996 Lie Algebra Cohomology and the Generalized {Borel-Weil} Theorem Kostant, B. ['Lie Algebra Cohomology and the Generalized Borel-Weil Theorem'] 10.1007/b94535_13 Notes on 2-Categorical Limits Frey, J. None On a Diffeological Group Realization of certain Generalized Symmetrizable {Kac-Moody} {Lie} Algebras Leslie, J. None Automorphisms of Graded Super Symplectic Manifolds Leslie, J. ['Automorphisms of graded super symplectic manifolds'] 10.4310/sdg.2010.v15.n1.a7 Diffeology Iglesias-Zemmour, P. None Categories for the Working Mathematician Mac Lane, S. ['Categories for the Working Mathematician'] 10.1007/978-1-4612-9839-7 Supermanifolds: Theory and Applications Rogers, A ['Supermanifolds - Theory and Applications'] 10.1142/9789812708854 Kac-Moody Groups, Their Flag Varieties and Representation Theory Kumar, S ['Kac-Moody Groups, their Flag Varieties and Representation Theory'] 10.1007/978-1-4612-0105-2 An Application of Enriched Category Theory to the Borel-Weil Theorem Twum, R. None Differential Geometry in Cartesian Closed Categories of Smooth Spaces Laubinger, M. ['Differential geometry in cartesian closed categories of smooth spaces'] 10.31390/gradschool_dissertations.3981 Ind-Varieties of Generalized Flags as Homogeneous Spaces for Classical Ind-Groups Ivan Dimitrov and Ivan Penkov None A Very Simple Proof of {Bott's} Theorem Demazure, M ["A very simple proof of Bott's theorem"] 10.1007/bf01404206 A Proof of the {Borel-Weil-Bott} Theorem Lurie, J. None The Group of Diffeomorphisms of a Non Compact Manifold is not regular Magnot, Jean-Pierre ['The group of diffeomorphisms of a non-compact manifold is not regular'] 10.1515/dema-2018-0001 Remarks on Some Infinite Dimensional {Lie} Groups Milnor, J. None -------------- No match found for amsclass 39/39-26/39-26.tex ['On the homotopy hypothesis for 3-groupoids'] author [{'given': 'Simon', 'family': 'Henry'}, {'given': 'Edoardo', 'family': 'Lanari'}] publication date 2023-08-25 volume 39 issue 26 page range ('735', '768') url http://www.tac.mta.ca/tac/volumes/39/26/39-26abs.html abstract We show that if the canonical left semi-model structure on the category of Grothendieck n-groupoids exists, then it satisfies the homotopy hypothesis, i.e. the associated (∞,1)-category is equivalent to that of homotopy n-types, thus generalizing a result of the first-named author. As a corollary of the second named author's proof of the existence of the canonical left semi-model structure for Grothendieck 3-groupoids, we obtain a proof of the homotopy hypothesis for Grothendieck 3-groupoids. keywords Homotopy hypothesis, Grothendieck's ∞-groupoids, model categories ams class None dois [] DOI citations: D. Ara - ``\emph{Sur les $\infty$-groupoïdes de Grothendieck et une variante $\infty$-catégorique}'', PhD Thesis. None D. Ara - ``\emph{On the homotopy theory of Grothendieck $\infty$-groupoids}'', Journal of Pure and Applied Algebra, 217(7) (2013), 1237-1278. ['On the homotopy theory of Grothendieck -groupoids'] 10.1016/j.jpaa.2012.10.010 M. Batanin - ``\emph{Monoidal globular categories as natural environement for the theory of weak n-categories}'', Advances in Mathematics 136 (1998), pp.39-103. ['Monoidal Globular Categories As a Natural Environment for the Theory of Weakn-Categories'] 10.1006/aima.1998.1724 C.Barwick - ``\emph{On left and right model categories and left and right Bousfield localizations}, Homology, Homotopy and Applications, vol. 1(1), 2010, pp.1–76. ['On left and right model categories and left and right Bousfield localizations'] 10.4310/hha.2010.v12.n2.a9 C. Barwick, D.M. Kan - ``\emph{Relative categories: Another model for the homotopy theory of homotopy theories}'', Indagationes Mathematicae, Volume 23, Issues 1–2, March 2012, Pages 42-68. ['Relative categories: Another model for the homotopy theory of homotopy theories'] 10.1016/j.indag.2011.10.002 C.Berger - ``\emph{A cellular nerve for higher categories}'', Advances in Mathematics, Volume 169, Issue 1, 15 July 2002, Pages 118-175. ['A Cellular Nerve for Higher Categories'] 10.1006/aima.2001.2056 C.Berger - ``\emph{Double loop spaces, braided monoidal categories and algebraic 3-type of space}, Higher homotopy structures in topology and mathematical physics, Contemp. Math., vol. 227, Amer. Math. Soc., 1999,p. 49-66. ['Double loop spaces, braided monoidal categories and algebraic 3-type of space'] 10.1090/conm/227/03252 C. Berger, P.A. Mellies, M. Weber - ``\emph{Monads with arities and their associated theories.}'', Journal of Pure and Applied Algebra, 2012, vol 216, no 8-9, p2029-2048. None J. Bourke - ``\emph{Note on the construction of globular weak omega-groupoids from types, topological spaces etc % }'', https://arxiv.org/abs/1602.07962. % Bourke J. arXiv preprint arXiv:1811.09532 None J. Bourke - ``\emph{Iterated algebraic injectivity and the faithfulness conjecture.}'' ``\emph{Grothendieck $\omega$-groupoids as iterated injectives}'', talk given at CT2016, slides available at http://mysite.science.uottawa.ca/phofstra/CT2016/slides/Bourke.pdf ['Iterated algebraic injectivity and the faithfulness conjecture'] 10.21136/hs.2020.13 J. Bourke, S. Henry - ``\emph{Algebraically cofibrant and fibrant objects revisited}'', Homology, Homotopy \& Applications. 2022 Jan 1;24(1). ['Algebraically cofibrant and fibrant objects revisited'] 10.4310/hha.2022.v24.n1.a14 J. Cartmell - ``\emph{Generalised algebraic theories and contextual categories}'', Annals of pure and applied logic 32 (1986): 209-243. ['Generalised algebraic theories and contextual categories'] 10.1016/0168-0072(86)90053-9 D.C.Cisinski - ``\emph{Higher Categories and Homotopical Algebra}'', published by Cambridge University Press. ['Higher Categories and Homotopical Algebra'] 10.1017/9781108588737 D.C. Cisinski - ``\emph{Les préfaisceux comme modèles des types d'homotopie}'', Astérisque 308 (2006). ["Les préfaisceaux comme modèles des types d'homotopie"] 10.24033/ast.715 W.G. Dwyer, D.M Kan - ``\emph{Simplicial localizations of categories}'', Journal of Pure and Applied Algebra 17 (1980) 267-284. ['Simplicial localizations of categories'] 10.1016/0022-4049(80)90049-3 B. Fresse - ``\emph{Modules over operads and functors}'', Lecture Notes in Mathematics, Springer (2009). ['Modules over Operads and Functors'] 10.1007/978-3-540-89056-0 A. Grothendieck ``\emph{Letter to Quillen}'', Manuscript, 1983. None S.Henry - ``\emph{Algebraic models of homotopy types and the homotopy hypothesis}'', available at https://arxiv.org/abs/1609.04622. None S.Henry - ``\emph{Weak model categories in classical and constructive mathematics}'', Theory \& Applications of Categories 35 (2020). None S. Henry - ``\emph{Combinatorial and accessible weak model categories }'', Journal of Pure and Applied Algebra, 227(2), p.107191. ['Combinatorial and accessible weak model categories'] 10.1016/j.jpaa.2022.107191 M. Hovey, ``\emph{Model Categories}'', Issue 63 of Mathematical Surveys and Monographs, American Mathematical Society, (1999). ['Model categories'] 10.1090/surv/063/01 S. Lack, ``\emph{A Quillen model structure for bicategories}'', K-Theory, 33 (2004), 185–197. ['A Quillen Model Structure for Bicategories'] 10.1007/s10977-004-6757-9 S. Lack, ``\emph{A Quillen model structure for Gray-categories}'', Journal of K-theory, 8(2):183-221, 2011. ['A Quillen model structure for Gray-categories'] 10.1017/is010008014jkt127 E.Lanari - ``\emph{Towards a globular path object for weak $\infty$-groupoids}'', Journal of Pure and Applied Algebra, Volume 224, Issue 2, February 2020, Pages 630-702. ['Towards a globular path object for weak ∞-groupoids'] 10.1016/j.jpaa.2019.06.004 E.Lanari - ``\emph{A semi-model structure for Grothendieck weak 3-groupoids }'', available at https://arxiv.org/abs/1809.07923. None I. Moerdiejk, B. Van den Berg - ``\emph{Exact completion of path categories and algebraic set theory: Part I: Exact completion of path categories}'', Journal of Pure and Applied Algebra, 222(10):3137-3181, 2018. None Y. Lafont, F. Métayer, K. Worytkiewicz - ``\emph{A folk model structure on omega-cat}, Advances in Mathematics, Volume 224, Issue 3, 20 June 2010, Pages 1183-1231. ['A folk model structure on omega-cat'] 10.1016/j.aim.2010.01.007 J.Lurie - ``\emph{Higher Topos Theory}'', Annals of Mathematics Studies, Princeton University Press, 2009. None G. Maltsiniotis - ``\emph{Grothendieck $\infty$-groupoids and still another definition of $\infty$-categories}'', https://arxiv.org/pdf/1009.2331.pdf. None T. Nikolaus - ``\emph{Algebraic models for higher categories}'', available at https://arxiv.org/abs/1003.1342 ['Algebraic models for higher categories'] 10.1016/j.indag.2010.12.004 A. Radulescu-Banu - ``\emph{Cofibrations in homotopy theory}'', available at https://arxiv.org/abs/math/0610009 None M. Spitzweck - ``\emph{Operads, algebras and modules in model categories and motives}'', PhD thesis, Universit{\"a}ts-und Landesbibliothek Bonn, 2004. None K. Szumiło - ``\emph{Homotopy theory of cofibration categories}'', Homology, Homotopy and Applications, vol. 18(2), 2016, pp.345–357. ['Homotopy theory of cofibration categories'] 10.4310/hha.2016.v18.n2.a19 B. Van Den Berg - ``\emph{Path categories and propositional identity types.}''. ACM Transactions on Computational Logic (TOCL) 19.2 (2018): 15. \end{thebibliography} \end{document} ['Path Categories and Propositional Identity Types'] 10.1145/3204492 -------------- ['On Multi-Determinant Functors for Triangulated Categories'] author [{'given': 'Ettore', 'family': 'Aldrovandi'}, {'given': 'Cynthia', 'family': 'Lester'}] publication date 2023-09-06 volume 39 issue 27 page range ('769', '803') url http://www.tac.mta.ca/tac/volumes/39/27/39-27abs.html abstract We extend Deligne's notion of determinant functor to tensor triangulated categories. Specifically, to account for the multiexact structure of the tensor, we define a determinant functor on the 2-multicategory of triangulated categories and we provide a multicategorical version of the universal determinant functor for triangulated categories whose multiexactness properties are conveniently captured by a certain complex modeled by cubical shapes, which we introduce along the way. We then show that for a tensor triangulated category whose tensor admits a Verdier structure the resulting determinant functor takes values in a categorical ring. keywords Tensor triangulated category, determinant functor, multicategory, Picard groupoid, categorical ring, K-Theory, cubical complex ams class 18G80, 18F25, 19D23, 18M65 dois [] DOI citations: Ettore Aldrovandi, \emph{Biextensions, bimonoidal functors, multilinear functor calculus, and categorical rings}, Theory and Applications of Categories \textbf{32} (2017), no.~27, 889--969, URL: \url{http://www.tac.mta.ca/tac/volumes/32/27/32-27abs.html}. None Paul Balmer, \emph{The spectrum of prime ideals in tensor triangulated categories}, J. Reine Angew. Math. \textbf{588} (2005), 149--168, \href {https://doi.org/10.1515/crll.2005.2005.588.149} {\path{doi:10.1515/crll.2005.2005.588.149}}. ['The spectrum of prime ideals in tensor triangulated categories'] 10.1515/crll.2005.2005.588.149 \bysame, \emph{Spectra, spectra, spectra---tensor triangular spectra versus {Z}ariski spectra of endomorphism rings}, Algebr. Geom. Topol. \textbf{10} (2010), no.~3, 1521--1563, \href {https://doi.org/10.2140/agt.2010.10.1521} {\path{doi:10.2140/agt.2010.10.1521}}. None \bysame, \emph{Tensor triangular geometry}, Proceedings of the {I}nternational {C}ongress of {M}athematicians. {V}olume {II}, Hindustan Book Agency, New Delhi, 2010, pp.~85--112. None Alexander~A. Beĭlinson, Joseph Bernstein, and Pierre Deligne, \emph{Faisceaux pervers}, Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100, Soc. Math. France, 1982, pp.~5--171. None Cornel Balteanu, Zbigniew Fiedorowicz, Roland Schw{\"a}nzl, and Rainer Vogt, \emph{Iterated monoidal categories}, Advances in Mathematics \textbf{176} (2003), no.~2, 277--349. ['Iterated monoidal categories'] 10.1016/s0001-8708(03)00065-3 José~I. Burgos~Gil, \emph{Hermitian vector bundles and characteristic classes}, The arithmetic and geometry of algebraic cycles ({B}anff, {AB}, 1998), CRM Proc. Lecture Notes, vol.~24, Amer. Math. Soc., Providence, RI, 2000, pp.~155--182. None Manuel Breuning, \emph{Determinant functors on triangulated categories}, J. K-Theory \textbf{8} (2011), no.~2, 251--291, \href {https://doi.org/10.1017/is010006009jkt120} {\path{doi:10.1017/is010006009jkt120}}. ['Determinant functors on triangulated categories'] 10.1017/is010006009jkt120 Pierre Deligne, \emph{Le déterminant de la cohomologie}, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemp. Math., vol.~67, Amer. Math. Soc., Providence, RI, 1987, pp.~93--177. ['Le déterminant de la cohomologie'] 10.1090/conm/067/902592 François Ducrot, \emph{Cube structures and intersection bundles}, J. Pure Appl. Algebra \textbf{195} (2005), no.~1, 33--73, \href {https://doi.org/10.1016/j.jpaa.2004.06.002} {\path{doi:10.1016/j.jpaa.2004.06.002}}. ['Cube structures and intersection bundles'] 10.1016/j.jpaa.2004.06.002 A.~D. Elmendorf and M.~A. Mandell, \emph{Permutative categories, multicategories and algebraic {$K$}-theory}, Algebr. Geom. Topol. \textbf{9} (2009), no.~4, 2391--2441, \href {https://doi.org/10.2140/agt.2009.9.2391} {\path{doi:10.2140/agt.2009.9.2391}}. None Nick Gurski, Niles Johnson, and Angélica~M. Osorno, \emph{The symmetric monoidal 2-category of permutative categories}, 2022, \href {http://arxiv.org/abs/2211.04464} {\path{arXiv:2211.04464}}. ['The symmetric monoidal 2-category of permutative categories'] 10.21136/hs.2024.06 Mamuka Jibladze and Teimuraz Pirashvili, \emph{Third {M}ac {L}ane cohomology via categorical rings}, J. Homotopy Relat. Struct. \textbf{2} (2007), no.~2, 187--216, \href {http://arxiv.org/abs/math/0608519} {\path{arXiv:math/0608519}}. None André Joyal and Ross Street, \emph{Braided tensor categories}, Adv. Math. \textbf{102} (1993), no.~1, 20--78, \href {https://doi.org/10.1006/aima.1993.1055} {\path{doi:10.1006/aima.1993.1055}}. ['Braided Tensor Categories'] 10.1006/aima.1993.1055 Niles Johnson and Donald Yau, \emph{Homotopy theory of enriched mackey functors}, 2022, \href {http://arxiv.org/abs/2212.04276} {\path{arXiv:2212.04276}}. ['Homotopy Theory of Enriched Mackey Functors'] 10.1017/9781009519564 Bernhard Keller and Amnon Neeman, \emph{The connection between {M}ay’s axioms for a triangulated tensor product and {H}appel’s description of the derived category of the quiver d4}, Doc. Math \textbf{7} (2002), 535--560. ["The connection between May's axioms for a triangulated tensor product and Happel's description of the derived category of the quiver $D_4$"] 10.4171/dm/131 Masaki Kashiwara and Pierre Schapira, \emph{Categories and sheaves}, Grundlehren der Mathematischen Wissenschaften, vol. 332, Springer-Verlag, Berlin, 2006. ['Categories and Sheaves'] 10.1007/3-540-27950-4 Tom Leinster, \emph{Higher operads, higher categories}, London Mathematical Society Lecture Note Series, vol. 298, Cambridge University Press, Cambridge, 2004, \href {https://doi.org/10.1017/CBO9780511525896} {\path{doi:10.1017/CBO9780511525896}}. ['Higher Operads, Higher Categories'] 10.1017/cbo9780511525896 Jean-Louis Loday, \emph{Spaces with finitely many nontrivial homotopy groups}, J. Pure Appl. Algebra \textbf{24} (1982), no.~2, 179--202, \href {https://doi.org/10.1016/0022-4049(82)90014-7} {\path{doi:10.1016/0022-4049(82)90014-7}}. ['Spaces with finitely many non-trivial homotopy groups'] 10.1016/0022-4049(82)90014-7 Jacob Lurie, \emph{Higher algebra}, 2017, URL: \url{https://www.math.ias.edu/~lurie/papers/HA.pdf}. None J.~P. May, \emph{The additivity of traces in triangulated categories}, Adv. Math. \textbf{163} (2001), 34--73, \href {https://doi.org/10.1006/aima.2001.1995} {\path{doi:10.1006/aima.2001.1995}}. ['The Additivity of Traces in Triangulated Categories'] 10.1006/aima.2001.1995 Randy McCarthy, \emph{A chain complex for the spectrum homology of the algebraic {$K$}-theory of an exact category}, Algebraic {$K$}-theory ({T}oronto, {ON}, 1996), Fields Inst. Commun., vol.~16, Amer. Math. Soc., Providence, RI, 1997, pp.~199--220, \href {https://doi.org/10.1016/j.ffa.2012.08.007} {\path{doi:10.1016/j.ffa.2012.08.007}}. ['A chain complex for the spectrum homology of the algebraic K-theory of an exact category'] 10.1090/fic/016/07 Saunders Mac~Lane, \emph{Homologie des anneaux et des modules}, Colloque de topologie algébrique, {L}ouvain, 1956, Georges Thone, Liège; Masson \& Cie, Paris, 1957, pp.~55--80. ['Homologie des anneaux et des modules'] 10.1007/978-1-4615-7831-4_18 Fernando Muro and Andrew Tonks, \emph{The 1-type of a {W}aldhausen {$K$}-theory spectrum}, Adv. Math. \textbf{216} (2007), no.~1, 178--211, \href {https://doi.org/10.1016/j.aim.2007.05.008} {\path{doi:10.1016/j.aim.2007.05.008}}. None Fernando Muro, Andrew Tonks, and Malte Witte, \emph{On determinant functors and {$K$}-theory}, Publ. Mat. \textbf{59} (2015), no.~1, 137--233, URL: \url{http://projecteuclid.org/euclid.pm/1421861996}. ['On determinant functors and $K$-theory'] 10.5565/publmat_59115_07 Amnon Neeman, \emph{The {$K$}-theory of triangulated categories}, Handbook of {$K$}-theory. {V}ol. 1, 2, Springer, Berlin, 2005, pp.~1011--1078, \href {https://doi.org/10.1007/978-3-540-27855-9\_20} {\path{doi:10.1007/978-3-540-27855-9\_20}}. None Olaf~M. Schnürer, \emph{Six operations on dg enhancements of derived categories of sheaves}, Selecta Mathematica \textbf{24} (2018), no.~3, 1805--1911, \href {https://doi.org/10.1007/s00029-018-0392-4} {\path{doi:10.1007/s00029-018-0392-4}}. ['Six operations on dg enhancements of derived categories of sheaves'] 10.1007/s00029-018-0392-4 Graeme Segal, \emph{Categories and cohomology theories}, Topology \textbf{13} (1974), 293--312, \href {https://doi.org/10.1016/0040-9383(74)90022-6} {\path{doi:10.1016/0040-9383(74)90022-6}}. \end{thebibliography} \end{document} %%% Local Variables: ['Categories and cohomology theories'] 10.1016/0040-9383(74)90022-6 -------------- ['Hopf Monads on Biproducts'] author [{'given': 'Masahito', 'family': 'Hasegawa'}, {'given': 'Jean-Simon Pacaud', 'family': 'Lemay'}] publication date 2023-09-20 volume 39 issue 28 page range ('804', '823') url http://www.tac.mta.ca/tac/volumes/39/28/39-28abs.html abstract A Hopf monad, in the sense of Bruguières, Lack, and Virelizier, is a special kind of monad that can be defined for any monoidal category. In this note, we study Hopf monads in the case of a category with finite biproducts, seen as a symmetric monoidal category. We show that for biproducts, a Hopf monad is precisely characterized as a monad equipped with an extra natural transformation satisfying three axioms, which we call a fusion invertor. We will also consider three special cases: representable Hopf monads, idempotent Hopf monads, and when the category also has negatives. In these cases, the fusion invertor will always be of a specific form that can be defined for any monad. Thus in these cases, checking that a monad is a Hopf monad is reduced to checking one identity. keywords Hopf Monads, Biproducts, Fusion Operators, Fusion Invertor ams class 18C15, 18M80, 18D99 dois [] DOI citations: [B, 1994] Borceux, F., Handbook of Categorical Algebra: Volume 2, Categories and Structures. \textit{Cambridge University Press}, 1994. ['Handbook of Categorical Algebra'] 10.1017/cbo9780511525865 [BLV, 2011] Brugui{\`e}res, A. and Lack, S. and Virelizier, A., Hopf Monads on Monoidal Categories. \textit{Advances in Mathematics}, Vol.227.(2), p.745--800, 2011. ['Hopf monads on monoidal categories'] 10.1016/j.aim.2011.02.008 [BV, 2007] Brugui{\`e}res, A. and Virelizier, A., Hopf Monads. \textit{Advances in Mathematics}, Vol.215.(2), p.679--733, 2007. ['Hopf monads'] 10.1016/j.aim.2007.04.011 [G, 2015] Guti{\'e}rrez, J. J., On Solid and Rigid Monoids in Monoidal Categories. \textit{Applied Categorical Structures}, Vol.23.(4), p.575--589, 2015. ['On Solid and Rigid Monoids in Monoidal Categories'] 10.1007/s10485-014-9370-y [HL, 2018] Hasegawa, M. and Lemay, J.-S. P., Linear Distributivity With Negation, Star-Autonomy, and {H}opf Monads. \textit{Theory and Applications of Categories}, Vol.33.(37), p.1145--1157, 2018. None [HL, 2022] Hasegawa, M. and Lemay, J.-S. P., Traced Monads and Hopf Monads. \textit{arXiv preprint arXiv:2208.06529}, 2022. ['Traced Monads and Hopf Monads'] 10.32408/compositionality-5-10 [Mc, 2002] McCrudden, P., Opmonoidal Monads. \textit{Theory and Applications of Categories}, Vol.10.(19), p.469--485, 2002. None [M, 2002] Moerdijk, I., Monads on Tensor Categories. \textit{Journal of Pure and Applied Algebra}, Vol.168.(2), p.189--208, 2002. ['Monads on tensor categories'] 10.1016/s0022-4049(01)00096-2 [S, 2003] Selinger, P., Order-incompleteness and finite lambda reduction models. \textit{Theoretical Computer Science}, Vol.309.(1-3), p.43--63, 2003. \endrefs \end{document} ['Order-incompleteness and finite lambda reduction models'] 10.1016/s0304-3975(02)00038-5 -------------- ['Bifunctor theorem and strictification tensor product for double categories with lax double functors'] author [{'given': 'Bojana', 'family': 'Femić'}] publication date 2023-10-03 volume 39 issue 29 page range ('824', '873') url http://www.tac.mta.ca/tac/volumes/39/29/39-29abs.html abstract We introduce a candidate for the inner hom for the category of double categories and lax double functors, and characterize a lax double functor into it obtaining a lax double quasi-functor. The latter consists of a pair of lax double functors with four 2-cells resembling distributive laws. We extend this characterization to a double category isomorphism. We show that instead of a Gray monoidal product we obtain a product that in a sense strictifies lax double quasi-functors. We explain why laxity of double functors hinders our candidate for the inner hom from making the category of double categories and lax double functors a closed and enriched category over 2-categories (or double categories). We prove a bifunctor theorem by which certain type of lax double quasi-functors give rise to lax double functors on the Cartesian product. We extend this theorem to a double functor between double categories and show how it restricts to a double equivalence. The (un)currying double functors are studied. We prove that a lax double functor from the trivial double category is a monad in the codomain double category, and show that our above double functor recovers the specification in that double category of the composition natural transformation on the monad functor. keywords bicategories, double categories, Gray monoidal product ams class 18N10 dois [] DOI citations: None J. B\'enabou, \emph{Introduction to bicategories}, Reports of the Midwest Category Seminar, Lecture Notes in Mathematics \textbf{47}, 1--77, Springer, Berlin 1967. ['Introduction to bicategories'] 10.1007/bfb0074299 G. B\"ohm, {\em The Gray monoidal product of double categories}, Appl. Categ. Struct. {\bf 28} (2020), 477--515. % https://doi.org/10.1007/s10485-019-09587-5 ['The Gray Monoidal Product of Double Categories'] 10.1007/s10485-019-09587-5 T. Cottrell, S. Fujii, J. Power, {\em Enriched and internal categories: an extensive relationship}, Tbilisi Math. J. {\bf 10} (2017), 239--254. ['Enriched and internal categories: an extensive relationship'] 10.1515/tmj-2017-0111 C. Douglas, {\em 2-dimensional algebra and quantum Chern-Simons field theory}, Talk at Conference on Topological Field Theories and Related Geometry and Topology, Northwestern University, May 2009. None P.F. Faul, G. Manuell, J. Siqueira, {\em 2-Dimensional bifunctor theorems and distributive laws}, Theory Appl. Categ. {\bf 37} (2021), 1149--1175. None B. Femi\'c, {\em Enrichment and internalization in tricategories, the case of tensor categories and alternative notion to intercategories}, arXiv:2101.01460v2. None B. Femi\'c, E. Ghiorzi, {\em Matrices and Spans and internalization and enrichment in tricategories}, %arxiv.org/abs/2203.16179. None T. M. Fiore, N. Gambino, J.Kock, {\em Monads in double categories}, J. Pure Appl. Algebra {\bf 215} (2011), 1174--1197. ['Monads in double categories'] 10.1016/j.jpaa.2010.08.003 R. Garner, N. Gurski, {\em The low-dimensional structures formed by tricategories}, Math. Proc. Cambridge Philos. Soc. {\bf 146} (2009), 551--589. ['The low-dimensional structures formed by tricategories'] 10.1017/s0305004108002132 R. Gordon, A. J. Power, R. Street, {\em Coherence for tricategories}, Memoirs Amer. Math. Soc. {\bf 117} (1995). ['Coherence for tricategories'] 10.1090/memo/0558 M. Grandis, {\em Higher Dimensional Categories: From Double to Multiple Categories}, World Scientific (2019). ['Higher Dimensional Categories'] 10.1142/11406 M. Grandis, R. Par\'e, {\em Limits in double categories}, Cahiers Topol. G\'eom. Diff. Cat\'eg. {\bf 40} (1999), 162--220. None M. Grandis, R. Par\'e, {\em Adjoint for double categories}, Cahiers Topol. G\'eom. Diff. Cat\'eg. {\bf 45} (2004), 193--240. None J. W. Gray, {\em Formal category theory: adjointness for 2-categories}, Lecture Notes in Mathematics {\bf 391}, Springer-Verlag, Berlin-New York (1974). ['Formal Category Theory: Adjointness for 2-Categories'] 10.1007/bfb0061280 S. Lack, R. Street, {\em The formal theory of monads II}, J. Pure Appl. Algebra {\bf 175}/(1-3) (2002), 243--365. ['The formal theory of monads II'] 10.1016/s0022-4049(02)00137-8 S. Mac Lane, {\em Categories for the Working Mathematicians}, Graduate Texts in Mathematics, Springer-Verlag (1971). ['Categories for the Working Mathematician'] 10.1007/978-1-4612-9839-7 B. Nikoli\'c, {\em Strictification tensor product of 2-categories}, Theory Appl. Categ. {\bf 34} (2019), 635--661. None M. Shulman {\em Constructing symmetric monoidal bicategories}, arXiv: 1004.0993 None M. Shulman, {\em Framed bicategories and monoidal fibrations}, Theory Appl. Categ. {\bf 20} (2008), 650--738. None R. Street, {\em The formal theory of monads}, J. Pure Appl. Algebra {\bf 2} (1972), 149--168. \endrefs \end{document} ['The formal theory of monads'] 10.1016/0022-4049(72)90019-9 -------------- ['Category-colored operads, internal operads, and Markl -operads'] author [{'given': 'Dominik', 'family': 'Trnka'}] publication date 2023-11-13 volume 39 issue 30 page range ('874', '915') url http://www.tac.mta.ca/tac/volumes/39/30/39-30abs.html abstract We present a Markl-style definition of operads colored by a small category. In the presence of a unit these are equivalent to substitudes of Day and Street. We show that operads colored by a category are internal algebras of a certain categorical operad of functors. We describe a groupoid-colored quadratic binary operad, whose algebras are non-unital Markl operads in the context of operadic categories. As a by-product we describe the free internal operad construction. keywords Colored operad, Internal operad, Operadic category, Markl operad, Hyperoperad ams class 18M60 dois [] DOI citations: M.A.~Batanin. \newblock The Eckmann--Hilton argument and higher operads. \newblock {\em Advances in Mathematics}, 217(1):334--385, 2008. ['The Eckmann–Hilton argument and higher operads'] 10.1016/j.aim.2007.06.014 M.A.~Batanin and C.~Berger. \newblock Homotopy theory for algebras over polynomial monads. \newblock {\em Theory and Applications of Categories}, 32(6):148--253, 2017. None M.A.~Batanin and M.~Markl. \newblock Operadic categories and duoidal Deligne's conjecture. \newblock {\em Advances in Mathematics}, 285:1630--1687, 2015. ["Operadic categories and duoidal Deligne's conjecture"] 10.1016/j.aim.2015.07.008 M.A.~Batanin and M.~Markl. \newblock Operadic categories as a natural environment for {K}oszul duality, \newblock {\em Compositionality}, 5(3), 2023. ['Operadic categories as a natural environment for Koszul duality'] 10.32408/compositionality-5-3 M.A.~Batanin and M.~Markl. \newblock Koszul duality for operadic categories, \newblock {\em Compositionality}, 5(4), 2023. ['Koszul duality for operadic categories'] 10.32408/compositionality-5-4 M.A.~Batanin, M.~Markl, and J.~Obradovi{\'c}. \newblock Minimal models for graph-related (hyper) operads. \newblock {\em Journal of Pure and Applied Algebra}, page 107329, 2023. ['Minimal models for graph-related (hyper)operads'] 10.1016/j.jpaa.2023.107329 M.A.~Batanin and D.~White. \newblock Homotopy theory of algebras of substitudes and their localisation. \newblock {\em Transactions of the American Mathematical Society}, 375(05):3569--3640, 2022. ['Homotopy theory of algebras of substitudes and their localisation'] 10.1090/tran/8600 D.~Calaque, R.~Campos, and J. Nuiten. \newblock Moduli problems for operadic algebras, \newblock {\em Journal of the London Mathematical Society}, 2022. ['Moduli problems for operadic algebras'] 10.1112/jlms.12666 B.~Day. \newblock On closed categories of functors. \newblock In {\em Reports of the Midwest Category Seminar IV}, pages 1--38. Springer, 1970. ['On closed categories of functors'] 10.1007/bfb0060438 B.~Day and R.~Street. \newblock Abstract substitution in enriched categories. \newblock {\em Journal of Pure and Applied Algebra}, 179(1-2):49--63, 2003. ['Abstract substitution in enriched categories'] 10.1016/s0022-4049(02)00291-8 M.~Dehling and B.~Vallette. \newblock Symmetric homotopy theory for operads. \newblock {\em Algebraic \& Geometric Topology}, 21(4):1595--1660, 2021. ['Symmetric homotopy theory for operads'] 10.2140/agt.2021.21.1595 V.~Dotsenko, S.~Shadrin, A.~Vaintrob, and B.~Vallette. \newblock Deformation theory of cohomological field theories, Preprint {\tt arXiv:2006.01649}, 2020. ['Deformation theory of cohomological field theories'] 10.1515/crelle-2023-0098 M.~Fiore, N.~Gambino, M.~Hyland, and G.~Winskel. \newblock The cartesian closed bicategory of generalised species of structures, \newblock {\em Journal of the London Mathematical Society}, 77(1):203--220, 2008. ['The cartesian closed bicategory of generalised species of structures'] 10.1112/jlms/jdm096 R.M. Kaufmann and B.C. Ward. \newblock Feynman categories, \newblock{\em Ast{\'e}risque}, 387, 2017. None M.~Markl. \newblock Models for operads. \newblock {\em Communications in Algebra}, 24(4):1471--1500, 1996. ['Models for operads'] 10.1080/00927879608825647 M.~Markl. \newblock Operads and props. \newblock {\em Handbook of algebra}, 5:87--140, 2008. ['Operads and PROPs'] 10.1016/s1570-7954(07)05002-4 M.~Markl, S.~Shnider, and J.~Stasheff. \newblock Operads in algebra, topology and physics. \newblock {\em Mathematical surveys and monographs}, 96, 2002. ['Algebra'] 10.1090/surv/096/04 D.~Petersen. \newblock The operad structure of admissible G-covers. \newblock {\em Algebra \& Number Theory}, 7(8):1953--1975, 2013. None K.~Stoeckl. \newblock Koszul Operads Governing Props and Wheeled Props, Preprint {\tt arXiv:2308.08718}, 2023. ['Koszul operads governing props and wheeled props'] 10.1016/j.aim.2024.109869 P.~van~der Laan. \newblock Coloured Koszul duality and strongly homotopy operads, Preprint {\tt arXiv:math/0312147}, 2003. None B.~C.~Ward. \newblock Massey Products for Graph Homology, \newblock {\em International Mathematics Research Notices}, 2022(11):8086-8161, 2022. \end{references*} \end{document} ['Massey Products for Graph Homology'] 10.1093/imrn/rnaa346 -------------- ['Categorical-Algebraic Properties of Lattice-ordered Groups'] author [{'given': 'Andrea', 'family': 'Cappelletti'}] publication date 2023-12-01 volume 39 issue 31 page range ('916', '948') url http://www.tac.mta.ca/tac/volumes/39/31/39-31abs.html abstract We study the categorical-algebraic properties of the semi-abelian variety of lattice-ordered groups. In particular, we show that this category is fiber-wise algebraically cartesian closed, arithmetical, and strongly protomodular. Moreover, we observe that is not action accessible, despite the good behaviour of centralizers of internal equivalence relations. Finally, we restrict our attention to the subvariety of lattice-ordered abelian groups, showing that it is algebraically coherent; this provides an example of an algebraically coherent category which is not action accessible. keywords lattice-ordered group, semi-abelian category, algebraically coherent category ams class 06F15, 18E13 dois [] DOI citations: Anderson, M. and Feil, T. (2012). \newblock {\em Lattice-ordered groups: an introduction}, volume~4. \newblock Springer Science \& Business Media. ['Lattice-Ordered Groups'] 10.1007/978-94-009-2871-8 Barr, M. (1971). \newblock Exact categories. \newblock {\em Exact categories and categories of sheaves}, 236:1--120. ['Exact categories'] 10.1007/bfb0058580 Birkhoff, G. (1942). \newblock Lattice-ordered groups. \newblock {\em Annals of Mathematics}, pages 298--331. ['Lattice-Ordered Groups'] 10.2307/1968871 Borceux, F. and Bourn, D. (2004). \newblock {\em Mal'cev, protomodular, homological and semi-abelian categories}, volume 566. \newblock Springer Science \& Business Media. ['Mal’cev, Protomodular, Homological and Semi-Abelian Categories'] 10.1007/978-1-4020-1962-3 Borceux, F., Janelidze, G., and Kelly, G.~M. (2005). \newblock On the representability of actions in a semi-abelian category. \newblock {\em Theory Appl. Categ}, 14(11):244--286. None Bourn, D. (1991). \newblock Normalization equivalence, kernel equivalence and affine categories. \newblock In {\em Category theory}, volume 1488, pages 43--62. Springer Lecture Notes in Mathematics. ['Normalization equivalence, kernel equivalence and affine categories'] 10.1007/bfb0084212 Bourn, D. (1996). \newblock Mal'cev categories and fibration of pointed objects. \newblock {\em Applied categorical structures}, 4(2):307--327. ["Mal'cev categories and fibration of pointed objects"] 10.1007/bf00122259 Bourn, D. (2000). \newblock Normal functors and strong protomodularity. \newblock {\em Theory and Applications of Categories}, 7(9):206--218. None Bourn, D. (2002). \newblock Intrinsic centrality and associated classifying properties. \newblock {\em Journal of Algebra}, 256(1):126--145. ['Intrinsic centrality and associated classifying properties'] 10.1016/s0021-8693(02)00149-7 Bourn, D. and Gran, M. (2002a). \newblock Centrality and connectors in maltsev categories. \newblock {\em Algebra Universalis}, 48(3):309--331. ['Centrality and connectors in Maltsev categories'] 10.1007/s000120200003 Bourn, D. and Gran, M. (2002b). \newblock Centrality and normality in protomodular categories. \newblock {\em Theory Appl. Categ}, 9(8):151--165. None Bourn, D. and Gray, J. R.~A. (2012). \newblock Aspects of algebraic exponentiation. \newblock {\em Bulletin of the Belgian Mathematical Society-Simon Stevin}, 19(5):821--844. ['Aspects of algebraic exponentiation'] 10.36045/bbms/1354031552 Bourn, D. and Janelidze, G. (1998). \newblock Protomodularity, descent, and semidirect products. \newblock {\em Theory Appl. Categ}, 4(2):37--46. None Bourn, D. and Janelidze, G. (2003). \newblock Characterization of protomodular varieties of universal algebras. \newblock {\em Theory and Applications of categories}, 11(6):143--447. None Bourn, D. and Janelidze, G. (2009). \newblock Centralizers in action accessible categories. \newblock {\em Cahiers de topologie et g{\'e}om{\'e}trie diff{\'e}rentielle cat{\'e}goriques}, 50(3):211--232. None Carboni, A., Lambek, J., and Pedicchio, M.~C. (1991). \newblock Diagram chasing in {M}al'cev categories. \newblock {\em Journal of Pure and Applied Algebra}, 69(3):271--284. ["Diagram chasing in Mal'cev categories"] 10.1016/0022-4049(91)90022-t Cignoli, R.~L., d'Ottaviano, I.~M., and Mundici, D. (2013). \newblock {\em Algebraic foundations of many-valued reasoning}, volume~7. \newblock Springer Science \& Business Media. ['Algebraic Foundations of Many-Valued Reasoning'] 10.1007/978-94-015-9480-6 Cigoli, A.~S., Gray, J. R.~A., and Van~der Linden, T. (2015a). \newblock Algebraically coherent categories. \newblock {\em Theory and Applications of Categories}, 30(54):1864--1905. None Cigoli, A.~S., Gray, J. R.~A., and Van~der Linden, T. (2015b). \newblock On the normality of {H}iggins commutators. \newblock {\em Journal of Pure and Applied Algebra}, 219(4):897--912. ['On the normality of Higgins commutators'] 10.1016/j.jpaa.2014.05.025 Clementino, M.~M., Montoli, A., and Sousa, L. (2015). \newblock Semidirect products of (topological) semi-abelian algebras. \newblock {\em Journal of Pure and Applied Algebra}, 219(1):183--197. ['Semidirect products of (topological) semi-abelian algebras'] 10.1016/j.jpaa.2014.04.019 Everaert, T. and Van~der Linden, T. (2012). \newblock Relative commutator theory in semi-abelian categories. \newblock {\em Journal of Pure and Applied Algebra}, 216(8-9):1791--1806. ['Relative commutator theory in semi-abelian categories'] 10.1016/j.jpaa.2012.02.018 Huq, S.~A. (1968). \newblock Commutator, nilpotency and solvability in categories. \newblock {\em Q. J. Math.}, 19(2):363--389. ['COMMUTATOR, NILPOTENCY, AND SOLVABILITY IN CATEGORIES'] 10.1093/qmath/19.1.363 Janelidze, G., M{\'a}rki, L., and Tholen, W. (2002). \newblock Semi-abelian categories. \newblock {\em Journal of Pure and Applied Algebra}, 168(2-3):367--386. ['Semi-abelian categories'] 10.1016/s0022-4049(01)00103-7 Kopytov, V. and Medvedev, N. (2013). \newblock {\em The theory of lattice-ordered groups}, volume 307. \newblock Springer Science \& Business Media. ['The Theory of Lattice-Ordered Groups'] 10.1007/978-94-015-8304-6 Mantovani, S. and Metere, G. (2010). \newblock Normalities and commutators. \newblock {\em Journal of Algebra}, 324(9):2568--2588. ['Normalities and commutators'] 10.1016/j.jalgebra.2010.07.043 Montoli, A. (2010). \newblock Action accessibility for categories of interest. \newblock {\em Theory Appl. Categ}, 23(1):7--21. None Mundici, D. (1986). \newblock Interpretation of {AF} {C}\text{*}-algebras in {\l}ukasiewicz sentential calculus. \newblock {\em Journal of Functional Analysis}, 65(1):15--63. ['Interpretation of AF C∗-algebras in Łukasiewicz sentential calculus'] 10.1016/0022-1236(86)90015-7 Orzech, G. (1972). \newblock Obstruction theory in algebraic categories, {I}. \newblock {\em Journal of Pure and Applied Algebra}, 2(4):287--314. ['Obstruction theory in algebraic categories, I'] 10.1016/0022-4049(72)90008-4 Pedicchio, M.~C. (1995). \newblock A categorical approach to commutator theory. \newblock {\em Journal of Algebra}, 177(3):647--657. ['A Categorical Approach to Commutator Theory'] 10.1006/jabr.1995.1321 Pedicchio, M.~C. (1996). \newblock Arithmetical categories and commutator theory. \newblock {\em Applied Categorical Structures}, 4:297--305. ['Arithmetical categories and commutator theory'] 10.1007/bf00122258 Smith, J. D.~H. (2006). \newblock {\em Mal'cev varieties}, volume 554. \newblock Springer. \endrefs \end{document} None -------------- ['Surjection-like classes of morphisms'] author [{'given': 'Pierre-Alain', 'family': 'Jacqmin'}] publication date 2023-12-04 volume 39 issue 32 page range ('949', '1013') url http://www.tac.mta.ca/tac/volumes/39/32/39-32abs.html abstract We characterize `good' classes of epimorphisms in a finitely complete category, i.e., those which `interact with finite limits as surjections do in the category of sets and functions'. More precisely, we prove that given a class E of morphisms in a small finitely complete category , there exists a faithful conservative (respectively fully faithful) embedding ↪^ into a presheaf category which preserves and reflects finite limits and which sends morphisms in E, and only those, to componentwise surjections if and only if E contains the identities, is closed under composition, has the strong right cancellation property, is stable under pullbacks and does not contain any proper monomorphisms (respectively any morphism in it is a regular epimorphism). The classes of split epimorphisms and descent morphisms are such examples and the corresponding full embedding theorems are given by Yoneda and Barr's embeddings. As new examples, we get a conservative embedding theorem for the class of pullback-stable strong epimorphisms and a full embedding theorem for the class of effective descent morphisms. The proof presented here is not based on transfinite inductions and is therefore rather explicit, in contrast with similar embedding theorems. keywords class of epimorphisms, embedding theorem, finite limit, pullback-stable strong epimorphism, descent morphism, effective descent morphism, split epimorphism, finite limit preserving functor ams class 18A20, 18B15, 18A30 (primary); 18A35, 18A25, 18A22, 18G05 (secondary) dois [] DOI citations: \textsc{M. Artin, A. Grothendieck and J.L. Verdier}, S\'eminaire de g\'eom\'etrie alg\'ebrique du Bois Marie 1963--1964, Th\'eorie des topos et cohomologie \'etale des sch\'emas (SGA4), \textit{Springer Lect. Notes Math.} \textbf{269} (1972). ['Théorie des Topos et Cohomologie Etale des Schémas'] 10.1007/bfb0081551 \textsc{M. Barr}, Embedding of categories, \textit{Proc. Amer. Math. Soc.} \textbf{37} (1973), 42--46. ['Embedding of categories'] 10.1090/s0002-9939-1973-0347932-4 \textsc{M. Barr}, Representation of categories, \textit{J. Pure Appl. Algebra} \textbf{41} (1986), 113--137. ['Representation of categories'] 10.1016/0022-4049(86)90105-2 \textsc{M. Barr, P.A. Grillet and D.H. van Osdol}, Exact categories and categories of sheaves, \textit{Springer Lect. Notes Math.} \textbf{236} (1971). ['Exact Categories and Categories of Sheaves'] 10.1007/bfb0058579 \textsc{M.A. Bednarczyk, A.M. Borzyszkowski and W. Pawlowski}, Generalized congruences --- epimorphisms in $\mathcal{C}at$, \textit{Theory Appl. Categ.} \textbf{5} (1999), 266--280. None \textsc{F. Borceux}, Handbook of Categorical Algebra 2, \textit{Encycl. Math. Appl. (Cambridge Uni. Press)} \textbf{51} (1994). ['Handbook of Categorical Algebra'] 10.1017/cbo9780511525865 \textsc{F. Borceux and D. Bourn}, Mal'cev, protomodular, homological and semi-abelian categories, \textit{Math. Appl. (Kluwer Acad. Publ.)} \textbf{566} (2004). ['Mal’cev, Protomodular, Homological and Semi-Abelian Categories'] 10.1007/978-1-4020-1962-3 \textsc{D. Hofmann}, An algebraic description of regular epimorphisms in topology, \textit{J. Pure Appl. Algebra} \textbf{199} (2005), 71--86. ['An algebraic description of regular epimorphisms in topology'] 10.1016/j.jpaa.2004.12.039 \textsc{G. Janelidze, M. Sobral and W. Tholen}, Beyond Barr exactness: effective descent morphisms, in: M.C. Pedicchio, W. Tholen (Eds), Categorical Foundations: Special Topics in Order, Topology, Algebra and Sheaf Theory, \textit{Encyclopedia Math. Appl. (Cambridge Uni. Press)} \textbf{97} (2004), 359--405. ['Beyond Barr Exactness: Effective Descent Morphisms'] 10.1017/cbo9781107340985.011 \textsc{G. Janelidze and W. Tholen}, Facets of descent, I, \textit{Appl. Categ. Structures} \textbf{2} (1994), 245--281. ['Facets of descent, I'] 10.1007/bf00878100 \textsc{J. Reiterman, M. Sobral and W. Tholen}, Composites of effective descent maps, \textit{Cah. Topol. G\'eom. Diff\'er. Cat\'eg.} \textbf{34} (1993), 193--207. None \textsc{M. Sobral and W. Tholen}, Effective descent morphisms and effective equivalence relations, \textit{CMS Conference Proceedings (Amer. Math. Soc.)} \textbf{13} (1992), 421--433. \end{thebibliography} \end{document} None -------------- ['Erratum to “The monotone-light factorization for 2-categories via 2-preorders"'] author [{'given': 'João J.', 'family': 'Xarez'}] publication date 2023-12-08 volume 39 issue 33 page range ('1014', '1017') url http://www.tac.mta.ca/tac/volumes/39/33/39-33abs.html abstract In this note we correct a proposition from the paper “The monotone-light factorization for 2-categories via 2-preorders". Moreover, we clarify what is meant by a 2-preorder generated by a 2-relation. keywords Monotone-light factorization, 2-categories ams class 18A32,18E50,18N10 dois [] DOI citations: Janelidze, G., Sobral, M., Tholen, W. \textit{Beyond Barr Exactness: Effective Descent Morphisms} in Categorical Foundations. Special Topics in Order, Topology, Algebra and Sheaf Theory, Cambridge University Press, 2004. ['Beyond Barr Exactness: Effective Descent Morphisms'] 10.1017/cbo9781107340985.011 Mac Lane, S. \textit{Categories for the Working Mathematician}, 2nd ed., Springer, 1998. ['Categories for the Working Mathematician'] 10.1007/978-1-4612-9839-7 Xarez, J. J. \textit{The monotone-light factorization for 2-categories via 2-preorders}, Theory Appl. Categories 38 (2022) 1209--1226. \end{thebibliography} \end{document} ['The monotone-light factorization for categories via preorders'] 10.1090/fic/043/25 -------------- ['Pseudocommutativity and lax idempotency for relative pseudomonads'] author [{'given': 'Andrew', 'family': 'Slattery'}] publication date 2023-12-11 volume 39 issue 34 page range ('1018', '1049') url http://www.tac.mta.ca/tac/volumes/39/34/39-34abs.html abstract We extend the classical work of Kock on strong and commutative monads, as well as the work of Hyland and Power for 2-monads, in order to define strong and pseudocommutative relative pseudomonads. To achieve this, we work in the more general setting of 2-multicategories rather than monoidal 2-categories. We prove analogous implications to the classical work: that a strong relative pseudomonad is a pseudo-multifunctor, and that a pseudocommutative relative pseudomonad is a multicategorical pseudomonad. Furthermore, we extend the work of López Franco with a proof that a lax-idempotent strong relative pseudomonad is pseudocommutative. We apply the results of this paper to the example of the presheaf relative pseudomonad. keywords category theory, monad theory, presheaf ams class Primary 18N15; Secondary 18D65, 18A05, 18M65 dois [] DOI citations: Altenkirch, T., Chapman, J. \& Uustalu, T. Monads need not be endofunctors. {\em Logical Methods In Computer Science}. \textbf{11} (2015) ['Monads need not be endofunctors'] 10.2168/lmcs-11(1:3)2015 Arkor, N. \& McDermott, D. The formal theory of relative monads. (2023),\\ https://arxiv.org/abs/2302.14014 ['The formal theory of relative monads'] 10.1016/j.jpaa.2024.107676 Blackwell, R., Kelly, G. \& Power, A. Two-dimensional monad theory. {\em Journal Of Pure And Applied Algebra}. \textbf{59}, 1-41 (1989) ['Two-dimensional monad theory'] 10.1016/0022-4049(89)90160-6 Bourke, J. Skew structures in 2-category theory and homotopy theory. {\em Journal Of Homotopy And Related Structures}. \textbf{12} (2015) ['Skew structures in 2-category theory and homotopy theory'] 10.1007/s40062-015-0121-z Bunge, M. Coherent extensions and relational algebras. {\em Trans. Am. Math. Soc.}. \textbf{197} (1974) ['Coherent extensions and relational algebras'] 10.1090/s0002-9947-1974-0344305-0 Day, B. \& Lack, S. Limits of small functors. {\em Journal Of Pure And Applied Algebra}. \textbf{210}, 651-663 (2007) ['Limits of small functors'] 10.1016/j.jpaa.2006.10.019 Day, B. \& Street, R. Monoidal Bicategories and Hopf Algebroids. {\em Advances In Mathematics}. \textbf{129}, 99-157 (1997) ['Monoidal Bicategories and Hopf Algebroids'] 10.1006/aima.1997.1649 Di Liberti, I., Lobbia, G. \& Sousa, L. KZ-pseudomonads and Kan Injectivity. (2023), https://arxiv.org/abs/2211.00380 None Fiore, M., Gambino, N., Hyland, M. \& Winskel, G. Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures. {\em Selecta Mathematica}. \textbf{24}, 2791-2830 (2018) ['Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures'] 10.1007/s00029-017-0361-3 Gambino, N. \& Lobbia, G. On the formal theory of pseudomonads and pseudodistributive laws. {\em Theory And Applications Of Categories}. \textbf{37}, 14-56 (2021,1) None Hermida, C. Representable Multicategories. {\em Advances In Mathematics}. \textbf{151}, 164-225 (2000) ['Representable Multicategories'] 10.1006/aima.1999.1877 Hyland, M. \& Power, J. Pseudo-commutative monads and pseudo-closed 2-categories. {\em Journal Of Pure And Applied Algebra}. \textbf{175}, 141-185 (2002) ['Pseudo-commutative monads and pseudo-closed 2-categories'] 10.1016/s0022-4049(02)00133-0 Kelly, G. On MacLane's conditions for coherence of natural associativities, commutativities, etc.. {\em Journal Of Algebra}. \textbf{1} pp. 397-402 (1964) ["On MacLane's conditions for coherence of natural associativities, commutativities, etc."] 10.1016/0021-8693(64)90018-3 Kelly, G. Coherence theorems for lax algebras and for distributive laws. {\em Category Seminar}. \textbf{420} pp. 281-375 (1974) ['Category Seminar'] 10.1007/bfb0063096 Kock, A. Monads on symmetric monoidal closed categories. {\em Archiv Der Mathematik}. \textbf{21} pp. 1-10 (1970) ['Monads on symmetric monoidal closed categories'] 10.1007/bf01220868 Kock, A. Monads for which structures are adjoint to units. {\em Journal Of Pure And Applied Algebra}. \textbf{104}, 41-59 (1995) ['Monads for which structures are adjoint to units'] 10.1016/0022-4049(94)00111-u Lack, S. A Coherent Approach to Pseudomonads. {\em Advances In Mathematics}. \textbf{152}, 179-202 (2000) ['A Coherent Approach to Pseudomonads'] 10.1006/aima.1999.1881 Lobbia, G. Distributive laws for relative monads. {\em Applied Categorical Structures}. \textbf{31}, 19 (2023,4) ['Distributive Laws for Relative Monads'] 10.1007/s10485-023-09716-1 López Franco, I. Pseudo-commutativity of KZ 2-monads. {\em Advances In Mathematics}. \textbf{228}, 2557-2605 (2011) ['Pseudo-commutativity of KZ 2-monads'] 10.1016/j.aim.2011.06.039 Marmolejo, F. Doctrines whose structure forms a fully faithful adjoint string. {\em Theory And Applications Of Categories}. \textbf{3} pp. 24-44 (1997) None Marmolejo, F. Distributive laws for pseudomonads.. {\em Theory And Applications Of Categories}. \textbf{5} pp. 91-147 (1999) ['Distributive laws for pseudomonads II'] 10.1016/j.jpaa.2004.04.008 Paquet, H. \& Saville, P. Strong pseudomonads and premonoidal bicategories. (2023), https://arxiv.org/abs/2304.11014 None Power, A., Cattani, G. \& Winskel, G. A Representation Result for Free Cocompletions. {\em Journal Of Pure And Applied Algebra}. \textbf{151}, 273-286 (2000) ['A representation result for free cocompletions'] 10.1016/s0022-4049(99)00063-8 Uustalu, T. Strong Relative Monads (Extended Abstract). (2010) \end{references*} \end{document} None -------------- ['Injective symmetric quantaloid-enriched categories'] author [{'given': 'Lili', 'family': 'Shen'}, {'given': 'Hang', 'family': 'Yang'}] publication date 2023-12-12 volume 39 issue 35 page range ('1050', '1076') url http://www.tac.mta.ca/tac/volumes/39/35/39-35abs.html abstract We characterize injective objects, injective hulls and essential embeddings in the category of symmetric categories enriched in a small, integral and involutive quantaloid. In particular, injective partial metric spaces are precisely formulated. keywords quantaloid, enriched category, symmetry, injective object, injective hull, essential embedding, Ω-set, partial metric space ams class 18D20, 18A20, 18F75 dois [] DOI citations: J.~Ad{\'a}mek, H.~Herrlich, and G.~E. Strecker. \newblock {\em Abstract and Concrete Categories: The Joy of Cats}. \newblock Wiley, New York, 1990. None N.~Aronszajn and P.~Panitchpakdi. \newblock Extension of uniformly continuous transformations and hyperconvex metric spaces. \newblock {\em Pacific Journal of Mathematics}, 6(3):405--439, 1956. ['Extension of uniformly continuous transformations and hyperconvex metric spaces'] 10.2140/pjm.1956.6.405 R.~Betti and R.~F.~C. Walters. \newblock The symmetry of the {Cauchy}-completion of a category. \newblock In K.~H. Kamps, D.~Pumpl{\"u}n, and W.~Tholen, editors, {\em Category Theory: Applications to Algebra, Logic and Topology, Proceedings of the International Conference Held at Gummersbach, July 6--10, 1981}, volume 962 of {\em Lecture Notes in Mathematics}, pages 8--12. Springer, Berlin--Heidelberg, 1982. ['Category Theory'] 10.1007/bfb0066878 F.~Borceux. \newblock {\em Handbook of Categorical Algebra: Volume 3, Sheaf Theory}, volume~52 of {\em Encyclopedia of Mathematics and its Application}. \newblock Cambridge University Press, Cambridge, 1994. ['Handbook of Categorical Algebra'] 10.1017/cbo9780511525865 M.~Bukatin, R.~Kopperman, S.~G. Matthews, and H.~Pajoohesh. \newblock Partial metric spaces. \newblock {\em American Mathematical Monthly}, 116(8):708--718, 2009. ['Partial Metric Spaces'] 10.4169/193009709x460831 C.~C. Chang. \newblock Algebraic analysis of many valued logics. \newblock {\em Transactions of the American Mathematical Society}, 88(2):467--490, 1958. ['Algebraic Analysis of Many Valued Logics'] 10.2307/1993227 A.~W.~M. Dress. \newblock Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: {A} note on combinatorial properties of metric spaces. \newblock {\em Advances in Mathematics}, 53(3):321--402, 1984. ['Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: A note on combinatorial properties of metric spaces'] 10.1016/0001-8708(84)90029-x P.~Eklund, J.~Guti{\'e}rrez~Garc{\'i}a, U.~H{\"o}hle, and J.~Kortelainen. \newblock {\em Semigroups in Complete Lattices: Quantales, Modules and Related Topics}, volume~54 of {\em Developments in Mathematics}. \newblock Springer, Cham, 2018. None R.~Esp{\'i}nola and M.~A. Khamsi. \newblock Introduction to hyperconvex spaces. \newblock In W.~A. Kirk and B.~Sims, editors, {\em Handbook of Metric Fixed Point Theory}, pages 391--435. Springer, Dordrecht, 2001. ['Handbook of Metric Fixed Point Theory'] 10.1007/978-94-017-1748-9 M.~P. Fourman and D.~S. Scott. \newblock Sheaves and logic. \newblock In M.~P. Fourman, C.~J. Mulvey, and D.~S. Scott, editors, {\em Applications of Sheaves: Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra, and Analysis, Durham, July 9--21, 1977}, volume 753 of {\em Lecture Notes in Mathematics}, pages 302--401. Springer, Berlin--Heidelberg, 1979. None S.~Fujii. \newblock Completeness and injectivity. \newblock {\em Topology and its Applications}, 301:107503, 2021. ['Completeness and injectivity'] 10.1016/j.topol.2020.107503 P.~H{\'a}jek. \newblock {\em Metamathematics of Fuzzy Logic}, volume~4 of {\em Trends in Logic}. \newblock Springer, Dordrecht, 1998. ['Metamathematics of Fuzzy Logic'] 10.1007/978-94-011-5300-3 H.~Herrlich. \newblock Hyperconvex hulls of metric spaces. \newblock {\em Topology and its Applications}, 44(1):181--187, 1992. ['Hyperconvex hulls of metric spaces'] 10.1016/0166-8641(92)90092-e H.~Heymans. \newblock {\em Sheaves on Quantales as Generalized Metric Spaces}. \newblock PhD thesis, Universiteit Antwerpen, Belgium, 2010. None H.~Heymans and I.~Stubbe. \newblock Symmetry and {Cauchy} completion of quantaloid-enriched categories. \newblock {\em Theory and Applications of Categories}, 25(11):276--294, 2011. None D.~Hofmann. \newblock Injective spaces via adjunction. \newblock {\em Journal of Pure and Applied Algebra}, 215(3):283--302, 2011. ['Injective spaces via adjunction'] 10.1016/j.jpaa.2010.04.021 D.~Hofmann and I.~Stubbe. \newblock Topology from enrichment: the curious case of partial metrics. \newblock {\em Cahiers de Topologie et G{\'e}om{\'e}trie Diff{\'e}rentielle Cat{\'e}goriques}, 59(4):307--353, 2018. None U.~H{\"o}hle and T.~Kubiak. \newblock A non-commutative and non-idempotent theory of quantale sets. \newblock {\em Fuzzy Sets and Systems}, 166:1--43, 2011. ['A non-commutative and non-idempotent theory of quantale sets'] 10.1016/j.fss.2010.12.001 J.~R. Isbell. \newblock Six theorems about injective metric spaces. \newblock {\em Commentarii Mathematici Helvetici}, 39(1):65--76, 1964. ['Six theorems about injective metric spaces'] 10.1007/bf02566944 E.~M. Jawhari, M.~Pouzet, and D.~Misane. \newblock Retracts: graphs and ordered sets from the metric point of view. \newblock In I.~Rival, editor, {\em Combinatorics and Ordered Sets}, volume~57 of {\em Contemporary Mathematics}, pages 175--226. American Mathematical Society, Providence, 1986. ['Retracts: graphs and ordered sets from the metric point of view'] 10.1090/conm/057/856237 A.~Joyal and M.~Tierney. \newblock An extension of the {Galois} theory of {Grothendieck}. \newblock {\em Memoirs of the American Mathematical Society}, 51(309), 1984. ['An extension of the Galois theory of Grothendieck'] 10.1090/memo/0309 M.~Kabil and M.~Pouzet. \newblock Geometric aspects of generalized metric spaces: {Relations} withgraphs, ordered sets and automata. \newblock In A.~H. Alkhaldi, M.~K. Alaoui, and M.~A. Khamsi, editors, {\em New Trends in Analysis and Geometry}, pages 319--377. Cambridge Scholars Publishing, Newcastle upon Tyne, 2020. None E.~P. Klement, R.~Mesiar, and E.~Pap. \newblock {\em Triangular Norms}, volume~8 of {\em Trends in Logic}. \newblock Springer, Dordrecht, 2000. ['Triangular Norms'] 10.1007/978-94-015-9540-7 H.~Lai and L.~Shen. \newblock Fixed points of adjoint functors enriched in a quantaloid. \newblock {\em Fuzzy Sets and Systems}, 321:1--28, 2017. ['Fixed points of adjoint functors enriched in a quantaloid'] 10.1016/j.fss.2016.12.001 H.~Lai, L.~Shen, Y.~Tao, and D.~Zhang. \newblock Quantale-valued dissimilarity. \newblock {\em Fuzzy Sets and Systems}, 390:48--73, 2020. ['Quantale-valued dissimilarity'] 10.1016/j.fss.2020.01.013 F.~W. Lawvere. \newblock Metric spaces, generalized logic and closed categories. \newblock {\em Rendiconti del Seminario Mat\'{e}matico e Fisico di Milano}, 43:135--166, 1973. ['Metric spaces, generalized logic, and closed categories'] 10.1007/bf02924844 J.~M. Maranda. \newblock Injective structures. \newblock {\em Transactions of the American Mathematical Society}, 110(1):98--135, 1964. ['Injective Structures'] 10.2307/1993639 S.~G. Matthews. \newblock Partial metric topology. \newblock {\em Annals of the New York Academy of Sciences}, 728(1):183--197, 1994. ['Partial Metric Topology'] 10.1111/j.1749-6632.1994.tb44144.x C.~J. Mulvey. \newblock {\&}. \newblock {\em Supplemento ai Rendiconti del Circolo Matematico di Palermo Series II}, 12:99--104, 1986. None Q.~Pu and D.~Zhang. \newblock Preordered sets valued in a {GL}-monoid. \newblock {\em Fuzzy Sets and Systems}, 187(1):1--32, 2012. ['Preordered sets valued in a GL-monoid'] 10.1016/j.fss.2011.06.012 K.~I. Rosenthal. \newblock {\em Quantales and their Applications}, volume 234 of {\em Pitman research notes in mathematics series}. \newblock Longman, Harlow, 1990. None K.~I. Rosenthal. \newblock {\em The Theory of Quantaloids}, volume 348 of {\em Pitman Research Notes in Mathematics Series}. \newblock Longman, Harlow, 1996. None L.~Shen and W.~Tholen. \newblock Topological categories, quantaloids and {Isbell} adjunctions. \newblock {\em Topology and its Applications}, 200:212--236, 2016. ['Topological categories, quantaloids and Isbell adjunctions'] 10.1016/j.topol.2015.12.020 L.~Shen and D.~Zhang. \newblock Categories enriched over a quantaloid: {Isbell} adjunctions and {Kan} adjunctions. \newblock {\em Theory and Applications of Categories}, 28(20):577--615, 2013. None I.~Stubbe. \newblock Categorical structures enriched in a quantaloid: categories, distributors and functors. \newblock {\em Theory and Applications of Categories}, 14(1):1--45, 2005. None I.~Stubbe. \newblock Categorical structures enriched in a quantaloid: tensored and cotensored categories. \newblock {\em Theory and Applications of Categories}, 16(14):283--306, 2006. None I.~Stubbe. \newblock Cocomplete {$\mathcal{Q}$}-categories are preciselythe injectives wrt. fully faithful functors. \newblock Preprint, 2006. None I.~Stubbe. \newblock An introduction to quantaloid-enriched categories. \newblock {\em Fuzzy Sets and Systems}, 256:95--116, 2014. ['An introduction to quantaloid-enriched categories'] 10.1016/j.fss.2013.08.009 I.~Stubbe. \newblock The double power monad is the composite power monad. \newblock {\em Fuzzy Sets and Systems}, 313:25--42, 2017. ['The double power monad is the composite power monad'] 10.1016/j.fss.2016.04.013 Y.~Tao, H.~Lai, and D.~Zhang. \newblock Quantale-valued preorders: Globalization and cocompleteness. \newblock {\em Fuzzy Sets and Systems}, 256:236--251, 2014. ['Quantale-valued preorders: Globalization and cocompleteness'] 10.1016/j.fss.2012.09.013 R.~F.~C. Walters. \newblock Sheaves and {Cauchy}-complete categories. \newblock {\em Cahiers de Topologie et G{\'e}om{\'e}trie Diff{\'e}rentielle Cat{\'e}goriques}, 22(3):283--286, 1981. None S.~Willerton. \newblock Tight spans, {Isbell} completions and semi-tropical modules. \newblock {\em Theory and Applications of Categories}, 28(22):696--732, 2013. \end{thebibliography} \end{document} None -------------- ['Differential Bundles in Commutative Algebra and Algebraic Geometry'] author [{'given': 'G.S.H.', 'family': 'Cruttwell'}, {'given': 'Jean-Simon Pacaud', 'family': 'Lemay'}] publication date 2023-12-17 volume 39 issue 36 page range ('1077', '1120') url http://www.tac.mta.ca/tac/volumes/39/36/39-36abs.html abstract In this paper, we explain how the abstract notion of a differential bundle in a tangent category provides a new way of thinking about the category of modules over a commutative ring and its opposite category. MacAdam previously showed that differential bundles in the tangent category of smooth manifolds are precisely smooth vector bundles. Here we provide characterizations of differential bundles in the tangent categories of commutative rings and (affine) schemes. For commutative rings, the category of differential bundles over a commutative ring is equivalent to the category of modules over that ring. For affine schemes, the category of differential bundles over the Spec of a commutative ring is equivalent to the opposite category of modules over said ring. Finally, for schemes, the category of differential bundles over a scheme is equivalent to the opposite category of quasi-coherent sheaves of modules over that scheme. keywords tangent categories, differential bundles, modules ams class 18F40, 13A99, 14A99 dois [] DOI citations: \textit{Stacks Project} The {Stacks Project Authors} None An introduction to {C}-infinity schemes and {C}-infinity algebraic geometry Joyce, D. ['An introduction to C-infinity schemes and C-infinity algebraic geometry'] 10.4310/sdg.2012.v17.n1.a7 Connections In Classical And Quantum Field Theory Mangiarotti, L. and Sardanashvily, G. ['Connections in Classical and Quantum Field Theory'] 10.1142/9789812813749 An introduction to manifolds Tu, L. W. ['An Introduction to Manifolds'] 10.1007/978-1-4419-7400-6 Handbook of Categorical Algebra, Volume I F. Bourceux None The rising sea: Foundations of algebraic geometry Vakil, R. None Synthetic differential geometry Kock, A. ['Synthetic Differential Geometry'] 10.1017/cbo9780511550812 Categorical Dynamics Lawvere, W. None Tangent infinity-categories and {G}oodwillie calculus Bauer, K. and Burke, M. and Ching, M. None The convenient setting of global analysis Kriegl, A. and Michor, P. ['The Convenient Setting of Global Analysis'] 10.1090/surv/053 The differential lambda-calculus T. Ehrhard and L. Regnier \doi{https://doi.org/10.1016/S0304-3975(03)00392-X} ['The differential lambda-calculus'] 10.1016/s0304-3975(03)00392-x The {J}acobi identity for tangent categories Cockett, J. R. B. and Cruttwell, G. S. H. None Connections in tangent categories Cockett, J. R. B. and Cruttwell, G. S. H. None A simplicial foundation for differential and sector forms in tangent categories Cruttwell, G. S. H. and Lucyshyn-Wright, R.B.B. ['A simplicial foundation for differential and sector forms in tangent categories'] 10.1007/s40062-018-0204-8 Differential Structure, Tangent Structure, and {SDG} Cockett, J. R. B. and Cruttwell, G. S. H. ['Differential Structure, Tangent Structure, and SDG'] 10.1007/s10485-013-9312-0 Differential Bundles and Fibrations for Tangent Categories Cockett, J. R. B. and Cruttwell, G. S. H. None Abstract tangent functors Rosick{\`y}, J. None {\'E}l{\'e}ments de g{\'e}om{\'e}trie alg{\'e}brique {IV} Grothendieck, A. ['Éléments de géométrie algébrique'] 10.1007/bf02684747 Cartesian differential categories Blute, R. F. and Cockett, J. R. B. and Seely, R. A. G. ['Differential categories'] 10.1017/s0960129506005676 An embedding theorem for tangent categories R. Garner \doi{https://doi.org/10.1016/j.aim.2017.10.039} ['An embedding theorem for tangent categories'] 10.1016/j.aim.2017.10.039 Vector bundles and differential bundles in the category of smooth manifolds MacAdam, B. ['Vector Bundles and Differential Bundles in the Category of Smooth Manifolds'] 10.1007/s10485-020-09617-7 Differential equations in a tangent category {I}: Complete vector fields, flows, and exponentials Cockett, J. R. B. and Cruttwell, G.S.H. and Lemay, J.-S.P. ['Differential Equations in a Tangent Category I: Complete Vector Fields, Flows, and Exponentials'] 10.1007/s10485-021-09629-x {Tangent Categories from the Coalgebras of Differential Categories} J. R. B. Cockett and J.-S. P. Lemay and R. B. B. Lucyshyn-Wright \doi{10.4230/LIPIcs.CSL.2020.17} None The tangent functor monad and foliations Jubin, B. None Derivations in codifferential categories Blute, R. F. and Lucyshyn-Wright, R. B. B. and O'Neill, K. None -------------- ['Q-system completeness of unitary connections'] author [{'given': 'Mainak', 'family': 'Ghosh'}] publication date 2023-12-22 volume 39 issue 37 page range ('1121', '1151') url http://www.tac.mta.ca/tac/volumes/39/37/39-37abs.html abstract A Q-system is a unitary version of a separable Frobenius algebra object in a C*-tensor category. In a recent joint work with P. Das, S. Ghosh and C. Jones, the author has categorified Bratteli diagrams and unitary connections by building a 2-category UC. We prove that every Q-system in UC splits. keywords Q-systems, Unitary Connections, Subfactors, C*-2-categories ams class 46M15, 46L37 dois [] DOI citations: L. Lamport, Latex User's Guide \& %Reference Manual. Addison-Wesley (fifth edition), 1986. None \ N. Afzaly, S. Morrison and D. Penneys 2015. The classification of subfactors with index at most $5 \frac{1}{4}$. https://doi.org/10.48550/arXiv.1509.00038 to appear Mem. Amer. Math. Soc. None Dietmar Bisch 1997. Bimodules, higher relative commutants and the fusion algebra associated to a subfactor, Operator algebras and their applications (Waterloo, ON, 1994/1995), 13-63, Fields Inst. Commun., 13, Amer. Math. Soc., Providence, RI, 1997. ['Bimodules, higher relative commutants, and the fusion algebra associated to a subfactor'] 10.1090/fic/013/02 \ Q. Chen, R.H. Palomares and C. Jones 2022. K-theoretic classification of inductive limit actions of fusion categories on AF-algebras. https://doi.org/10.48550/arXiv.2207.11854 None \ Q. Chen, R.H. Palomares, C. Jones and D. Penneys 2022. Q-System completion of C*-2-categories. \textit{Journal of Functional Analysis}, Volume 283, Issue 3, 2022, 109524, ISSN 0022-1236, https://doi.org/10.1016/j.jfa.2022.109524. (https://www.sciencedirect.com/science/article/pii/S0022123622001446) None \ P. Das, M. Ghosh, S. Ghosh and C. Jones 2022. Unitary connections on Bratteli diagrams. To appear in \textit{Journal of Topology and Analysis}. arXiv:2211.03822. ['Unitary connections on Bratteli diagrams'] 10.1142/s1793525323500589 \ C.L. Douglas and D.Reutter 2018. Fusion 2-categories and a state-sum invariant for 4-manifolds. arXiv:1812.11933. None \ D. Evans and Y. Kawahigashi 1998. Quantum symmetries and operator algebras. Oxford Mathematical Monographs, Oxford University Press. ['Quantum Symmetries on Operator Algebras'] 10.1093/oso/9780198511755.001.0001 \ D.Gaiotto and T.Johnson-Freyd 2019. Condensations in higher categories. arXiv:1905.09566. None \ P. Ghez, R. Lima and J. E. Roberts. W*-categories 1985. \textit{Pacific Journal of Mathematics}, 120(1): 79-109 1985. https://doi.org/pjm/1102703884. None \ C. Heunen and J. Vicary. Categories for quantum theory, volume 28 of \textit{Oxford Graduate Texts in Mathematics}. Oxford University Press, Oxford, 2019. An introduction, MR3971584 DOI:10.1093/oso/9780198739623.001.0001. ['Categories for Quantum Theory'] 10.1093/oso/9780198739623.001.0001 \ V.F.R. Jones. Index for Subfactors. \textit{Invent. Math}. 73, pp. 1-25. ['Index for subfactors'] 10.1007/bf01389127 \ V.F.R. Jones. Planar Algebras I. arxiv:math.QA/9909027. ['Planar algebras'] 10.53733/172 \ V.Jones, S.Morrison and N.Snyder. The classification of subfactors of index at most 5. \textit{Bull. Amer. Math. Soc.} (N.S.), 51(2):277–327, 2014. arXiv:1304.6141, doi :10.1090/S0273-0979-2013-01442-3. ['The classification of subfactors of index at most 5'] 10.1090/s0273-0979-2013-01442-3 \ C.Jones and D.Penneys. Realizations of algebra objects and discrete subfactors. \textit{Adv. Math.}, 350:588–661, 2019. MR3948170 DOI:10.1016/j.aim.2019.04.039 . arXiv:1704.02035. ['Realizations of algebra objects and discrete subfactors'] 10.1016/j.aim.2019.04.039 \ C.Jones and D.Penneys. Q-systems and compact W*-algebra objects. Topological phases of matter and quantum computation, volume 747 of Contemp. Math., pages 63–88. Amer. Math. Soc., Providence, RI, 2020. MR4079745 DOI:10.1090/conm/747/15039. arXiv:1707.02155. None \ N.Johnson and D.Yau. $2$-dimensional categories. Oxford University Press. https://doi.org/10.1093/oso/9780198871378.001.0001. arXiv:2002.06055. ['2-Dimensional Categories'] 10.1093/oso/9780198871378.001.0001 \ R.Longo 1994. A duality for Hopf algebras and for subfactors. I. \textit{Communications in Mathematical Physics}, 159(1) 133-150 1994. https://doi.org/cmp/1104254494. ['A duality for Hopf algebras and for subfactors. I'] 10.1007/bf02100488 \ E. Lance 1995. Hilbert C*-Modules: A Toolkit for Operator Algebraists (London Mathematical Society Lecture Note Series). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511526206. None R. Longo and J.E. Roberts 1997. A theory of dimension. \textit{K-theory} 11(2): pp. 103-159, 1997. ['A Theory of Dimension'] 10.1023/a:1007714415067 \ M. M{\"u}ger 2003. From subfactors to categories and topology I: Frobenius algebras and Morita equivalence of tensor categories. \textit{Journal of Pure and Applied Algebra}. 180.1, pp. 81-157. ['From subfactors to categories and topology I: Frobenius algebras in and Morita equivalence of tensor categories'] 10.1016/s0022-4049(02)00247-5 \ A. Ocneanu 1988. Quantized groups, string algebras and Galois theory for algebras. Operator algebras and applications, London Math. Soc. Lecture Note Ser., 136: pp. 119-172. ['Quantized groups, string algebras, and Galois theory for algebras'] 10.1017/cbo9780511662287.008 S. Popa 1989. Relative dimension, towers of projections and commuting squares of subfactors. \textit{Pacific Journal of Mathematics}. Vol. 137 (1989), No. 1, 181–207. ['Relative dimension, towers of projections and commuting squares of subfactors'] 10.2140/pjm.1989.137.181 \ S. Popa 1994. Classification of amenable Subfactors of type II. \textit{Acta. Math}. 172, pp. 163-225. ['Classification of amenable subfactors of type II'] 10.1007/bf02392646 \ S. Popa 1995. An axiomatization of the lattice of higher relative commutants. \textit{Invent. Math}. 120, pp. 237-252. None \ Pasquale A. Zito 2007. 2-C*-categories with non-simple units. \textit{Adv. Math.}, 210(1):122–164, 2007. doi :10.1016/j.aim.2006.05.017. arXiv:math/0509266 \endrefs \end{document} %sed -f sample.sed sample.tex> Qsystem.tex None -------------- --------------------- --------------------- No Tex File Found For 39/Erratum/Erratum.tex --------------------- ---------------------