['Monoidal centres and groupoid-graded categories'] author [{'given': 'Branko', 'family': 'Nikolić'}, {'given': 'Ross', 'family': 'Street'}] publication date 2024-03-21 volume 40 issue 01 page range ('3', '31') url http://www.tac.mta.ca/tac/volumes/40/1/40-01abs.html abstract We denote the monoidal bicategory of two-sided modules (also called profunctors, bimodules and distributors) between categories by Mod; the tensor product is cartesian product of categories. For a groupoid , we study the monoidal centre ZPs(,Mod^op) of the monoidal bicategory Ps(,Mod^op) of pseudofunctors and pseudonatural transformations; the tensor product is pointwise. Alexei Davydov defined the full centre of a monoid in a monoidal category. We define a higher dimensional version: the full monoidal centre of a monoidale (= pseudomonoid) in a monoidal bicategory , and it is a braided monoidale in the monoidal centre Z of . Each fibration π : → between groupoids provides an example of a full monoidal centre of a monoidale in Ps(,Mod^op). For a group G, we explain how the G-graded categorical structures, as considered by Turaev and Virelizier in order to construct topological invariants, fit into this monoidal bicategory context. We see that their structures are monoidales in the monoidal centre of the monoidal bicategory of k-linear categories on which G acts. keywords monoidal centre; graded center; graded category; Day convolution; bidual ams class 18M15; 18N10; 18D15; 18D20; 18D60; 57K16; 57K31 dois [] DOI citations: {\sc J.C. Baez and M. Neuchl,} \newblock Higher dimensional algebra, I. Braided monoidal 2-categories. \newblock {\em Advances in Math. 121} (1996) 196--244. ['Higher Dimensional Algebra'] 10.1006/aima.1996.0052 {\sc T. Barmeier and C. 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This property is also showed to be equivalent to a property of the category Grd𝔼 of internal groupoids in 𝔼 which is almost opposite, for the monomorphic internal functors, of the comprehensive factorization. keywords equivalence relation, equivalence class, normal subobject, normalizers, Mal'tsev and protomodular categories, internal categories and groupoids, comprehensive factorization, non-pointed additive categories ams class 18A05, 18B99, 18E13, 08C05, 08A30, 08A99 dois [] DOI citations: M.\ Barr, \emph{Exact categories}, Springer L.N. in Math., \textbf{236}, 1971, 1-120. ['Exact categories'] 10.1007/bfb0058580 F. Borceux and D. Bourn, \emph{Mal'cev, Protomodular, Homological and Semi-Abelian Categories}, Mathematics and Its Applications. vol. \textbf{566}, 2004, Kluwer Acad. Publ. 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['Obstruction theory in algebraic categories, I'] 10.1016/0022-4049(72)90008-4 R.\ Street and R.F.C.\ Walters, \emph{The comprehensive factorization of a functor}, Bull. of the A.M.S., \textbf{75}, 1975, 936-941. \endrefs \end{document} ['The comprehensive factorization of a functor'] 10.1090/s0002-9904-1973-13268-9 -------------- ['Cartesian Closed Double Categories'] author [{'given': 'Susan', 'family': 'Niefield'}] publication date 2024-03-26 volume 40 issue 03 page range ('63', '79') url http://www.tac.mta.ca/tac/volumes/40/3/40-03abs.html abstract We consider two approaches to cartesian closed double categories generalizing two definitions which are equivalent for 1-categories, and give examples to show that the two differ in the double category case. One approach, previously considered in , requires the lax functor (-)× Y on 𝔻 to have a right adjoint (-)^Y, for every object Y, while the other supposes that the exponentials are given by a lax bifunctor 𝔻^ op×𝔻→<125> 𝔻 also involving vertical (i.e., loose) morphisms of 𝔻. Examples include the double categories , , , and , whose objects are small categories, posets, topological spaces, locales, and commutative quantales, respectively; as well as, the double categories ( D) and Q-, whose vertical morphisms are spans in a category D with pullback and relations valued in a locale Q, respectively. keywords double categories, cartesian closed, spans/cospans, quantales, relations ams class 18N10, 18D15, 18B10, 18F75, 54C35 dois [] DOI citations: E.~Aleiferi, Cartesian Double Categories with an Emphasis on Characterizing Spans, Ph.D. Thesis, Dalhousie University, 2018 (https://arxiv.org/abs/1809.06940). % None M.~Bunge and M.~Fiori, Unique factorization lifting and categories of processes, Math.~Str.~Comp.~Sci.~10 (2000) 137--163. % None M.~Bunge and S.~Niefield, Exponentiability and single universes, {J. Pure Appl. 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More specifically, we consider properties of geometric morphisms featuring in factorization systems, namely: surjections, inclusions, localic morphisms, hyperconnected morphisms, terminal-connected morphisms, étale morphisms, pure morphisms and complete spreads. We end with an application of topos-theoretic Galois theory to the special case of toposes of the form (M). keywords topos, monoid, factorization, terminal-connected, étale, pure, complete spread ams class 18B25, 20M30 dois [] DOI citations: M.~Barr and R.~Diaconescu, \emph{On locally simply connected toposes and their fundamental groups}, Cahiers Topologie G\'{e}om. Diff\'{e}rentielle \textbf{22} (1981), no.~3, 301--314, Third Colloquium on Categories, Part IV (Amiens, 1980). None J.~B{\'{e}}nabou, \emph{Some geometric aspects of the calculus of fractions}, vol.~4, 1996, The European Colloquium of Category Theory (Tours, 1994), pp.~139--165. None M.~Bunge and J.~Funk, \emph{Spreads and the symmetric topos}, J. Pure Appl. 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['Topos points of quasi-coherent sheaves over monoid schemes'] 10.1017/s0305004119000069 M.~Rogers, \emph{{Toposes} of {Discrete} {Monoid} {Actions}}, preprint (2019), \href {http://arxiv.org/abs/1905.10277} {\path{arXiv:1905.10277}}. None \bysame, \emph{{Toposes} of {Topological} {Monoid} {Actions}}, preprint (2021), to appear in Compositionality, \href {http://arxiv.org/abs/2105.00772} {\path{arXiv:2105.00772}}. None A.~Sagnier, \emph{An arithmetic site at the complex place}, J. Number Theory \textbf{212} (2020), 173--202. ['An arithmetic site at the complex place'] 10.1016/j.jnt.2019.11.001 \emph{Th\'{e}orie des topos et cohomologie \'{e}tale des sch\'{e}mas. {T}ome 1: {T}h\'{e}orie des topos}, Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin-New York, 1972, S\'{e}minaire de G\'{e}om\'{e}trie Alg\'{e}brique du Bois-Marie 1963--1964 (SGA 4), Dirig\'{e} par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. ['Théorie des Topos et Cohomologie Etale des Schémas'] 10.1007/bfb0061319 The {Stacks project authors}, \emph{The stacks project}, \url{https://stacks.math.columbia.edu}, 2022. None R.~Street and R.~F.~C. Walters, \emph{The comprehensive factorization of a functor}, Bull. Amer. Math. Soc. \textbf{79} (1973), 936--941. ['The comprehensive factorization of a functor'] 10.1090/s0002-9904-1973-13268-9 V.~Zoonekynd, \emph{Th\'{e}or\`eme de van {K}ampen pour les champs alg\'{e}briques}, Ann. Math. Blaise Pascal \textbf{9} (2002), no.~1, 101--145. \end{references} \end{document} ['Théorème de Van Kampen pour les champs algébriques'] 10.5802/ambp.153 -------------- ['Retrocells'] author [{'given': 'Robert', 'family': 'Paré'}] publication date 2024-04-02 volume 40 issue 05 page range ('130', '179') url http://www.tac.mta.ca/tac/volumes/40/5/40-05abs.html abstract The notion of retrocell in a double category with companions is introduced and its basic properties established. Explicit descriptions in some of the usual double categories are given. Monads in a double category provide an important example where retrocells arise naturally. Cofunctors appear as a special case. The motivating example of vertically closed double categories is treated in some detail. keywords Double category, companion, retrocell, cofunctor, closed bicategory ams class 18D05, 18C15, 18D15 dois [] DOI citations: Marcelo Aguiar. \newblock {\em Internal categories and quantum groups}. \newblock PhD thesis, Cornell University, 1997. None Bryce Clarke. \newblock Internal split opfibrations and cofunctors. \newblock {\em Theory and Applications of Categories}, 35(44):1608--1633, 2020. None Bryce Clarke. \newblock {\em The double category of lenses}. \newblock PhD thesis, Macquarie University, 2022. None Bryce Clarke and Matthew~Di Meglio. \newblock An introduction to enriched cofunctors. \newblock Preprint, 2022. \newblock arXiv:2209.01144. None Robert Dawson and Robert Par\'e. \newblock General associativity and general composition for double categories. \newblock {\em Cahiers de Topologie et G{\'e}om{\'e}trie Diff{\'e}rentielle Cat{\'e}goriques}, 34(1):57--79, 1993. None Brian Day. \newblock Limit spaces and closed span categories. \newblock In G.~M. Kelly, editor, {\em Category Seminar}, volume 420 of {\em Lecture Notes in Mathematics}, pages 65--74, 1974. ['Category Seminar'] 10.1007/bfb0063096 Thomas Fiore, Nicola Gambino, and Joachim Kock. \newblock Double adjunctions and free monads. \newblock {\em Cahiers de Topologie et G{\'e}om{\'e}trie Diff{\'e}rentielle Cat{\'e}goriques}, 53(4):242--306, 2012. None Thomas~M. Fiore, Nicola Gambino, and Joachim Kock. \newblock Monads in double categories. \newblock {\em Journal of Pure and Applied Algebra}, 215(6):1174--1197, 2011. ['Monads in double categories'] 10.1016/j.jpaa.2010.08.003 Marco Grandis. \newblock {\em Higher Dimensional Categories: From Double to Multiple Categories}. \newblock World Scientific, 2019. ['Higher Dimensional Categories'] 10.1142/11406 Marco Grandis and Robert Par\'e. \newblock Adjoint for double categories. \newblock {\em Cahiers de Topologie et G{\'e}om{\'e}trie Diff{\'e}rentielle Cat{\'e}goriques}, 45(3):193--240, 2004. None Marco Grandis and Robert Par\'e. \newblock Kan extensions in double categories {(On weak double categories, Part III)}. \newblock {\em Theory and Applications of Categories}, 20(8):152--185, 2008. None Marco Grandis and Robert Par\'e. \newblock Intercategories. \newblock {\em Theory and Applications of Categories}, 30(38):1215--1255, 2015. None Seerp~Roald Koudenburg. \newblock On pointwise {Kan} extensions in double categories. \newblock {\em Theory and Applications of Categories}, 29(27):781--818, 2014. None Joachim Lambek. \newblock {\em Lectures on Rings and Modules}. \newblock Blaisdell Publishing, 1966. None Tom Leinster. \newblock {\em Higher Operads, Higher Categories}, volume 298 of {\em London Mathematical Society Lecture Note Series}. \newblock Cambridge University Press, 2004. ['Higher Operads, Higher Categories'] 10.1017/cbo9780511525896 Matthew~Di Meglio. \newblock The category of asymmetric lenses and its proxy pullbacks. \newblock Master's thesis, Macquarie University, 2022. None Robert Par\'e. \newblock Morphisms of rings. \newblock In Claudia Casadio and Philip~J. Scott, editors, {\em Joachim Lambek: The Interplay of Mathematics, Logic, and Linguistics}, volume~20 of {\em Outstanding Contributions to Logic}, pages 271--298. Springer, 2021. ['Joachim Lambek: The Interplay of Mathematics, Logic, and Linguistics'] 10.1007/978-3-030-66545-6 Michael Shulman. \newblock Framed bicategories and monoidal fibrations. \newblock {\em Theory and Applications of Categories}, 20(18):650--738, 2008. None Ross Street. \newblock The formal theory of monads. \newblock {\em Journal of Pure and Applied Algebra}, 2(2):149--168, 1972. \end{thebibliography} \end{document} ['The formal theory of monads'] 10.1016/0022-4049(72)90019-9 -------------- ['Lax comma 2-categories and admissible 2-functors'] author [{'given': 'Maria Manuel', 'family': 'Clementino'}, {'given': 'Fernando Lucatelli', 'family': 'Nunes'}] publication date 2024-04-05 volume 40 issue 06 page range ('180', '226') url http://www.tac.mta.ca/tac/volumes/40/6/40-06abs.html abstract This paper is a contribution towards a two dimensional extension of the basic ideas and results of Janelidze's Galois theory. In the present paper, we give a suitable counterpart notion to that of absolute admissible Galois structure for the lax idempotent context, compatible with the context of lax orthogonal factorization systems. As part of this work, we study lax comma 2-categories, giving analogue results to the basic properties of the usual comma categories. We show that each morphism of a 2-category induces a 2-adjunction between lax comma 2-categories and comma 2-categories, playing the role of the usual change-of-base functors. With these induced 2-adjunctions, we are able to show that each 2-adjunction induces 2-adjunctions between lax comma 2-categories and comma 2-categories, which are our analogues of the usual lifting to the comma categories used in Janelidze's Galois theory. We give sufficient conditions under which these liftings are 2-premonadic and induce a lax idempotent 2-monad, which corresponds to our notion of 2-admissible 2-functor. In order to carry out this work, we analyse when a composition of 2-adjunctions is a lax idempotent 2-monad, and when it is 2-premonadic. We give then examples of our 2-admissible 2-functors (and, in particular, simple 2-functors), especially using a result that says that all admissible (2-)functors in the classical sense are also 2-admissible (and hence simple as well). keywords change-of-base functor, comma object, Galois theory, Kock-Zöberlein monads, semi-left exact functor, lax comma 2-categories, simple 2-adjunctions, 2-admissible 2-functor ams class 18N10, 18N15, 18A05, 18A22, 18A40 dois [] DOI citations: R.~Blackwell, G.~M. Kelly, and A.~Power. \newblock Two-dimensional monad theory. \newblock {\em J. Pure Appl. Algebra}, 59(1):1--41, 1989. 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Structures}, 1(1):103--110, 1993. ['Galois theory in variable categories'] 10.1007/bf00872989 G.~Janelidze and W.~Tholen. \newblock How algebraic is the change-of-base functor? \newblock In {\em Category theory ({C}omo, 1990)}, volume 1488 of {\em Lecture Notes in Math.}, pages 174--186. Springer, Berlin, 1991. ['How algebraic is the change-of-base functor?'] 10.1007/bfb0084219 G.~M. Kelly. \newblock Monomorphisms, epimorphisms, and pull-backs. \newblock {\em J. Aust. Math. Soc.}, 9:124--142, 1969. ['Monomorphisms, Epimorphisms, and Pull-Backs'] 10.1017/s1446788700005693 G.~M. Kelly. \newblock Doctrinal adjunction. \newblock In {\em Category {S}eminar ({P}roc. {S}em., {S}ydney, 1972/1973)}, pages 257--280. Lecture Notes in Math., Vol. 420. Springer, Berlin, 1974. ['Category Seminar'] 10.1007/bfb0063096 G.~M. Kelly and S.~Lack. \newblock On property-like structures. \newblock {\em Theory Appl. Categ.}, 3:No. 9, 213--250, 1997. None G.~M. 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Soc., Providence, RI, 2004. \end{thebibliography} \end{document} ['Separable morphisms of categories via preordered sets'] 10.1090/fic/043/26 -------------- ['On the metrical and quantalic versions of the *-autonomous category of sup-lattices'] author [{'given': 'Walter', 'family': 'Tholen'}] publication date 2024-04-09 volume 40 issue 07 page range ('227', '248') url http://www.tac.mta.ca/tac/volumes/40/7/40-07abs.html abstract In 1984, Joyal and Tierney presented the category 𝖲𝗎𝗉 of complete lattices and their suprema-preserving maps as a *-autonomous category in the sense of Barr. Work on this paper was motivated by the question whether the Joyal-Tierney proof may be extended to a metrical context, so that the order of the lattice gets replaced by a generalized metric in the sense of Lawvere. The affirmative answer we give relies crucially on working with not necessarily symmetric metrics. It applies not only to small separated and cocomplete categories enriched in the Boolean quantale 2 (reproducing ), or in the Lawvere quantale [0,∞] (producing the category we were looking for), but in any commutative and unital quantale 𝒱. Benefitting from previous work by Stubbe, Hofmann, and others, with rather explicit constructions of its tensor product and the internal hom we give an alternative proof that the resulting category 𝒱-𝖲𝗎𝗉 is *-autonomous, a result first established by Eklund, Gutiérrez García, Höhle, and Kortelainen in 2018 from a predominantly order-theoretic perspective. keywords sup-lattice, quantale, enriched category, cocomplete, *-autonomous category, dualizing object. ams class 18D20, 18M10, 18B35, 18B10 dois [] DOI citations: %J. Ad\'amek, H. Herrlich and G.E. Strecker: %{\em Abstract and Concrete Categories: The Joy of Cats}. John Wiley \& Sons, New %York, 1990. Republished in {\em Reprints in Theory and Applications of %Categories} 17, 2006. None A. 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["Variations on Beck's tripleability criterion"] 10.1007/bfb0059143 P. Eklund, J. Guti\'{e}rrez Garc\'{i}a, U. H\"{o}hle, J. Kortelainen: {\it Semigroups in Complete Lattices. Quantales, Modules and Related Topics}. Springer, 2018. None %R.C. Flagg: Completeness in continuity spaces. In: R.A.G. Seely (editor): {\em Category Theory 1992}, American Mathematical Society, Providence RI, pp 183--200,1992. None %R.C. Flagg: Quantales and continuity spaces. {\em Algebra Universalis} 37:257--276, 1997. None M. Fr\'{e}chet: Sur quelqes points du calcul fonctionel. {\em Rendiconti del Circolo Matematico di Palermo} 22(1):1--72, 1906. ['Sur quelques points du calcul fonctionnel'] 10.1007/bf03018603 D. Hofmann: Topological theories and closed objects. {\em Advances in Mathematics} 215:789--824, 2018. ['Topological theories and closed objects'] 10.1016/j.aim.2007.04.013 D. Hofmann and P. Nora: Enriched Stone-type dualities. {\em Advances in Mathematics} 330:307--360, 2018. ['Enriched Stone-type dualities'] 10.1016/j.aim.2018.03.010 D. Hofmann, G.J. Seal and W. Tholen (editors): {\em Monoidal Topology. A Categorical Approach to Order, Metric, and Topology}. Cambridge University Press, Cambridge, 2014. ['Monoidal Topology'] 10.1017/cbo9781107517288 A. Joyal and M. Tierney: {\em An extension of the Galois theory of Grothendieck}. Memoirs of the American Mathematical Society 309, Providence RI, 1984. ['An extension of the Galois theory of Grothendieck'] 10.1090/memo/0309 G.M. Kelly: {\em Basic Concepts of Enriched Category Theory}. Cambridge Universiy Press, Cambridge, 1982. Republished in {\em Reprints in Theory and Applications of Categories} 10, 2005. None F.W. Lawvere: Metric spaces, generalized logic, and closed categories. {\em Rendiconti del Seminario Matematico e Fisico di Milano} 43:135--166, 1973. Republished in {\em Reprints in Theory and Applications of Categories} 1, 2002. ['Metric spaces, generalized logic, and closed categories'] 10.1007/bf02924844 S. Mac Lane: {\em Categories for the Working Mathematician}. Springer-Verlag, Berlin-Heidelberg-New York, 1971; 2nd ed. 1994. ['Categories for the Working Mathematician'] 10.1007/978-1-4612-9839-7 E. Martinelli: {\it Actions, Injectives and Injective Hulls in Quantale-Enriched Multicategories}. Thesis, University of Aveiro, Aveiro, 2021. None %K.I. Rosenthal: {\em Quantales and Their Applications}. Addison Wesley Longman, Harlow, 1990. None %K.I. Rosenthal: {\em Theory of Quantaloids}. Addison Wesley Longman, Harlow, 1996. None I. Stubbe: Categorical structures enriched in a quantaloid: categories distributors, functors. {\em Theory and Applications of Categories} 14(1):1--45, 2005. None I. Stubbe: Categorical structures enriched in a quantaloid: tensored and cotensored categories. {\em Theory and Applications of Categories} 16(14):283--306, 2006. None I. Stubbe: $\mathcal Q$-modules and $\mathcal Q$-suplattices. {\em Theory and Applications of Categories} 19(4):50--60, 2007. None W. Tholen: Another categorical look at monoids, quantales, metrics, {\it etc.} Talk given at the {\em 5th Workshop on Categorical Algebra: Recent Developments and Future Perspectives}, Gargnano del Garda (Italy), 26-30 April 2022. Available at https://tholen.mathstats.yorku.ca . \end{document} None -------------- ['On coherent systems of subobjectswith application to torsion theories'] author [{'given': 'Francis', 'family': 'Borceux'}, {'given': 'Maria Manuel', 'family': 'Clementino'}] publication date 2024-04-11 volume 40 issue 08 page range ('249', '277') url http://www.tac.mta.ca/tac/volumes/40/8/40-08abs.html abstract In a coherent category, the posets of subobjects have very strong properties. We emphasize the validity of these properties, in general categories, for well-behaved classes of subobjects. As an example of application, we investigate the problem of the various torsion theories which can be universally associated with a pretorsion one. keywords coherent system of subobjects, (pre)torsion theory, torsion objects, torsion free objects, stable category, category of fractions ams class 18B25, 18C10, 18E13, 54D30 dois [] DOI citations: \mbox{}\putleft{\framebox{#1}}} \let\Bibitem=\bibitem \received{2022-10-24} \revised{2023-09-18} \num{8} \startpage{249} \published{2024-04-11} \tacyear{2024} \copyrightyear{2024} None -------------- ['Distributive idempotents in an order-enriched category'] author [{'given': 'Christopher', 'family': 'Townsend'}] publication date 2024-04-15 volume 40 issue 09 page range ('278', '300') url http://www.tac.mta.ca/tac/volumes/40/9/40-09abs.html abstract We introduce distributive maps between lattices and consider the categorical assumption that distributive idempotents split. We explore this assumption in the context of a categorical axiomatization of the category of locales. The assumption is shown to be stable under groupoids (this includes slice stability) and we further show that it implies that triquotient surjections are effective descent morphisms. This result follows even without assuming that the underlying (axiomatized) category of locales has coequalizers. keywords Locale, topos, categorical logic, powerlocales, order enriched, distributive lattice, axioms ams class 06D22, 03G30 dois [] DOI citations: Bunge, M. \emph{An application of descent to a classification theorem for toposes}. Mathematical Proceedings of the Cambridge Philosophical Society, 107(1), 59–79. (1990). ['An application of descent to a classification theorem for toposes'] 10.1017/s0305004100068365 Johnstone, P.T. \emph{Sketches of an elephant: A topos theory compendium}. Vols 1, 2, Oxford Logic Guides \textbf{43}, \textbf{44}, Oxford Science Publications, 2002. None Manuell, G. \emph{Presenting quotient locales}, arXiV2207.05116. July (2022). None Plewe, T. \emph{Localic triquotient maps are effective descent maps} Mathematical Proceedings of the Cambridge Philosophical Society 122(01):17 - 43 (1997) ['Localic triquotient maps are effective descent maps'] 10.1017/s0305004196001648 Townsend, C.F. \emph{An axiomatic account of weak triquotient assignments in locale theory}; early draft. www.christophertownsend.org. ['An axiomatic account of weak triquotient assignments in locale theory'] 10.1016/j.jpaa.2009.07.005 Townsend, C.F. \emph{A categorical account of the Hofmann-Mislove theorem} Math. Proc. Camb. Philos. Soc. 139, No.3, 441-455 (2005). ['A categorical account of the Hofmann–Mislove theorem'] 10.1017/s0305004105008844 Townsend, C.F. \emph{An axiomatic account of weak triquotient assignments in locale theory}, Journal of Pure and Applied Algebra, Volume 214, Issue 6, 2010, Pages 729-739. ['An axiomatic account of weak triquotient assignments in locale theory'] 10.1016/j.jpaa.2009.07.005 Townsend, C.F. \emph{Aspects of slice stability in Locale Theory} Georgian Mathematical Journal. Vol. 19, Issue \textbf{2}, (2012) 317-374. ['Aspects of slice stability in locale theory'] 10.1515/gmj-2012-0011 Townsend, C.F. \emph{Stability of Properties of Locales Under Groups}. Appl Categor Struct Vol. 25 Issue \textbf{3} (2017), 363-380. ['Stability of Properties of Locales Under Groups'] 10.1007/s10485-016-9430-6 Townsend, C.F. \emph{Double power monad preserving adjunctions are Frobenius.}. Theory and Applications of Categories, Vol. 33, No. 17, 2018, pp. 476-491 None Vickers, S.J. and Townsend, C.F. \emph{A Universal Characterization of the Double Power Locale}. Theo.Comp. Sci. 316 (2004), 297-321. \end{thebibliography} \end{document} ['A universal characterization of the double powerlocale'] 10.1016/j.tcs.2004.01.034 -------------- ['Birkhoff subfibrations of the codomain fibration'] author [{'given': 'A. S.', 'family': 'Cigoli'}, {'given': 'S.', 'family': 'Mantovani'}] publication date 2024-04-25 volume 40 issue 10 page range ('301', '323') url http://www.tac.mta.ca/tac/volumes/40/10/40-10abs.html abstract Slice categories of a semi-abelian category have a regular epireflection to their subcategories of internal Mal'tsev algebras. These are Birkhoff reflections, hence admissible with respect to regular epis in the sense of Janelidze's categorical Galois theory. We prove that when is moreover peri-abelian, these reflections form an admissible Galois structure for a larger class of morphisms, called proquotients. Starting from a careful investigation of the previous situation, we prove that all regular epireflective subfibrations in () of the codomain fibration of can be constructed from a reflective subcategory _0 of whose unit morphisms have characteristic kernel. The fibres of such reflective subfibrations are admissible with respect to proquotients precisely when _0 is a Birkhoff subcategory of . keywords categorical Galois theory; reflective subfibration; characteristic subobject; peri-abelian category ams class 18D30; 18E13; 18E50 dois [] DOI citations: F.~Borceux and D.~Bourn, \emph{Mal'cev, protomodular, homological and semi-abelian categories}, Math. Appl., vol. 566, Kluwer Acad. Publ., 2004. ['Mal’cev, Protomodular, Homological and Semi-Abelian Categories'] 10.1007/978-1-4020-1962-3 D.~Bourn, Normalization equivalence, kernel equivalence and affine categories, Springer LNM 1488, 1991, 43--62. ['Normalization equivalence, kernel equivalence and affine categories'] 10.1007/bfb0084212 D.~Bourn, 3 $\times$ 3 Lemma and protomodularity, \emph{J.~Algebra} 236 (2001) 778--795. ['3×3 Lemma and Protomodularity'] 10.1006/jabr.2000.8526 D.~Bourn, The cohomological comparison arising from the associated abelian object, preprint arXiv:1001.0905v2. None A.~S.~Cigoli, J.~Gray, and T.~Van der Linden, On the normality of Higgins commutators, \emph{J.~Pure Appl.~Algebra} 219 (2015) 897--912. ['On the normality of Higgins commutators'] 10.1016/j.jpaa.2014.05.025 A.~S.~Cigoli and A.~Montoli, Characteristic subobjects in semi-abelian categories, \emph{Theory Appl.~Categ.} 30 (2015) 206--228. None T.~Everaert, M.~Gran, Monotone-light factorization systems and torsion theories, \emph{Bull.~Sci.~Math.} 137 (2013) 996--1006. ['Monotone-light factorisation systems and torsion theories'] 10.1016/j.bulsci.2013.02.004 T.~Everaert, M.~Gran, and T.~Van der linden, Higher Hopf formulae for homology via Galois Theory, \emph{Adv.~Math.} 217 (2008) 2231--2267. ['Higher Hopf formulae for homology via Galois Theory'] 10.1016/j.aim.2007.11.001 M.~Gran, Central extensions and internal groupoids in Mal'tsev categories, \emph{J.~Pure Appl.~Algebra} 155 (2001) 139--166. ['Central extensions and internal groupoids in Maltsev categories'] 10.1016/s0022-4049(99)00092-4 M.~Gran and J.~Scherer, Conditional flatness, fibrewise localizations, and admissible reflections, \emph{J.~Austr.~Math.~Soc.} (2023). ['CONDITIONAL FLATNESS, FIBERWISE LOCALIZATIONS, AND ADMISSIBLE REFLECTIONS'] 10.1017/s1446788723000046 J.~Gray and T.~Van der Linden, Peri-abelian categories and the universal central extension condition, \emph{J.~Pure Appl.~Algebra} 219 (2015) 2506--2520. ['Peri-abelian categories and the universal central extension condition'] 10.1016/j.jpaa.2014.09.013 P.~J.~Higgins, Groups with multiple operators, \emph{Proc.~London Math.~Soc.} (3) 6 (1956) 366--416. ['Groups with Multiple Operators'] 10.1112/plms/s3-6.3.366 S.~A.~Huq, Commutator, nilpotency, and solvability in categories, \emph{Quart.~J.~Math.} 19 (1968) 363--389. ['COMMUTATOR, NILPOTENCY, AND SOLVABILITY IN CATEGORIES'] 10.1093/qmath/19.1.363 G.~Janelidze, Pure Galois theory in categories, \emph{J.~Algebra} 132 (1990) 270--286. ['Pure Galois theory in categories'] 10.1016/0021-8693(90)90130-g G.~Janelidze and G.~M.~Kelly, Galois theory and a general notion of central extension, \emph{J.~Pure Appl.~Algebra} 97 (1994) 135--161. ['Galois theory and a general notion of central extension'] 10.1016/0022-4049(94)90057-4 G.~Janelidze, L.~M\'arki, and W.~Tholen, Semi-abelian categories, \emph{J.~Pure Appl.~Algebra} 168 (2002) 367--386. ['Semi-abelian categories'] 10.1016/s0022-4049(01)00103-7 J.~Lewin, On Schreier varieties of linear algebras, \emph{Trans.~Amer.~Math.~Soc.} 132 (1968) 553--562. ['On Schreier varieties of linear algebras'] 10.1090/s0002-9947-1968-0224663-5 S.~Mantovani, The Ursini commutator as normalized Smith-Pedicchio commutator, \emph{Theory Appl.~Categ.} 27 (2012) 174--188. None S.~Mantovani and G.~Metere, Normalities and commutators, \emph{J.~Algebra} 324 (2010) 2568--2588. ['Normalities and commutators'] 10.1016/j.jalgebra.2010.07.043 M.~C.~Pedicchio, A categorical approach to commutator theory, \emph{J.~Algebra} 177 (1995) 647--657. ['A Categorical Approach to Commutator Theory'] 10.1006/jabr.1995.1321 T.~Streicher, Fibred categories \`a la Jean B\'enabou, arXiv:1801.02927. None J.~H.~C.~Whitehead, On adding relations to homotopy groups, \emph{Ann.~of Math.} 42 (1941), 409--428. \end{thebibliography} \end{document} ['On Adding Relations to Homotopy Groups'] 10.2307/1968907 -------------- ['Integration of 1-forms and connections'] author [{'given': 'Anders', 'family': 'Kock'}] publication date 2024-05-02 volume 40 issue 11 page range ('324', '336') url http://www.tac.mta.ca/tac/volumes/40/11/40-11abs.html abstract We present a geometric/combinatorial version of the theorem that a flat torsion-free affine connection on a manifold locally may be integrated into an affine structure. keywords Flat and torsion free affine connection. Synthetic differential geometry ams class 53B05, 51K10, 20N10 dois [] DOI citations: L.\ Auslander and L.\ Markus, Holonomy of Flat Affinely Connected Manifolds, Annals of Math.\ 62 (1955), 139-151. ['Holonomy of Flat Affinely Connected Manifolds'] 10.2307/2007104 Filip B\'{a}r, Affine connections and second order affine structures, Cahiers de Top. et G\'{e}om.\ Diff.\ Cat.\ 63 (2022), 35-58. None R.\ Blute, G.\ Cruttwell and R.\ Lucyshyn-Wright, Affine geometric spaces in tangent categories, Theory and Applications of Categories 34 (2019), 405-437. None E.\ Dubuc, Sur les mod\`{e}les de la g\'{e}om.\ diff.\ synth\'{e}tique, Cahiers de Top. et G\'{e}om.\ Diff.\ 20 (1979), 231-279. None E.\ Dubuc and A.\ Kock, Column symmetric polynomials, Cahiers de Top. et G\'{e}om.\ Diff.\ Cat.\ 50 (2019), 241-254. None C.\ Ehresmann, Sur les connexions d'ordre superieur, Atti del V Congresso dell' Unione Matematica Italiana, Pavia-Torino 1956. None A.\ Kock, {\em Synthetic Differential Geometry}, London Math.\ Soc.\ Lecture Notes Series no.\ 51, Cambridge Univ.\ Press 1981 (2nd ed.\ London Math.\ Soc.\ Lecture Notes Series no.\ 333, Cambridge Univ.\ Press 2006). None A.\ Kock, A combinatorial theory of connections, in {\em Mathematical Applications of Category Theory}, ed. J.\ Gray, A.M.S.\ Contemporary Mathematics 30 (1983), 132-144. ['A combinatorial theory of connections'] 10.1090/conm/030/749772 A.\ Kock, Lie group valued integration in well-adapted % toposes, Bull.\ Austral.\ Math.\ Soc.\ 34 (1986), 395-410. % ['Lie group valued integration in well-adapted toposes'] 10.1017/s0004972700010285 A.\ Kock, Geometric construction of the Levi-Civita parallelism, Theory and Applications of Categories 4 (1998), 195-207. None A.\ Kock, Connections and path connections in groupoids, Aarhus Math.\ Preprint Series 2006 No.\ 10. None A.\ Kock, Affine Connections and Midpoint Formation, in {\em Discrete Geometry for Computer Imagery}, 15th IAPR International Conference, DGCI 2009, Springer Lecture Notes in Computer Science 5810 (2009), 13-21. ['Affine Connections, and Midpoint Formation'] 10.1007/978-3-642-04397-0_2 A.\ Kock, {\em Synthetic Geometry of Manifolds}, Cambridge Tracts in Mathematics no.\ 180, Cambridge Univ.\ Press 2010. None A.\ Kock, Affine combinations in affine schemes, Cahiers de Top. et G\'{e}om.\ Diff.\ Cat.\ 58 (2017), 115-130. None M.\ Lawson, Generalised Heaps as Affine structures, in Hollings and Lawson: {\em Wagner's Theory of Generalised Heaps}, Springer 2017). ['Wagner’s Theory of Generalised Heaps'] 10.1007/978-3-319-63621-4 H.\ Pr\"{u}fer, Theorie der Abelschen Gruppen I, Math.\ Z.\ 20 (1924), 165-187. None J.\ Virsik, On the holonomity of higher order connections, Cahiers de Top. et G\'{e}om.\ Diff.\ Cat.\ 12 (1971), 197-212. \end{thebibliography} \small \noindent Diagrams were made with Paul Taylor's package. None -------------- ['Duality for positive opetopes and positive zoom complexes'] author [{'given': 'Marek', 'family': 'Zawadowski'}] publication date 2024-05-07 volume 40 issue 12 page range ('337', '370') url http://www.tac.mta.ca/tac/volumes/40/12/40-12abs.html abstract We show that the (positive) zoom complexes, with fairly natural morphisms, form a dual category to the category of positive opetopes with contraction epimorphisms. We also show how this duality can be slightly extended to positive opetopic cardinals.[It is with deep sadness that we inform you of the passing of Dr. Marek Zawadowski, on March 3, 2024, shortly after submitting the final version of this paper. The guest editors.] keywords positive opetope, constellation, positive zoom complex, duality ams class 18N30, 18A50, 18F70 dois [] DOI citations: J. Baez, J. Dolan, {Higher-dimensional algebra III: n-Categories and the algebra of opetopes}. Advances in Math. 135 (1998), pp. 145-206. ['Higher-Dimensional Algebra III.n-Categories and the Algebra of Opetopes'] 10.1006/aima.1997.1695 C. Berger, {A Cellular Nerve for Higher Categories}. Adv. in Mathematics 169, (2002), pp. 118-175. ['A Cellular Nerve for Higher Categories'] 10.1006/aima.2001.2056 A. Burroni, {Higher-dimensional word problems with applications to equational logic}, Theoretical Computer Science 115 (1993), no. 1, pp. 43–62. ['Higher-dimensional word problems with applications to equational logic'] 10.1016/0304-3975(93)90054-w E. Cheng, {The Category of Opetopes and the Category of Opetopic Sets}, Theory and Applications of Categories, Vol. 11, No. 16, 2003, pp. 353–374. None P-L. Curien, C. Ho Thanh, S. Mimram, {\em Syntactic approaches to opetopes}, arXiv:1903.05848 [math.CT], 2019. None M. Fiore, P. Saville, {List Objects with Algebraic Structure}, In Proceedings of the 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017), No. 16, pages 1-18, 2017. None C. Hermida, M. Makkai, J. Power, {On weak higher dimensional categories, I} Parts 1,2,3, J. Pure and Applied Alg. 153 (2000), pp. 221-246, 157 (2001), pp. 247-277, 166 (2002), pp. 83-104. ['On weak higher-dimensional categories I – 2'] 10.1016/s0022-4049(00)00129-8 C. Ho Thanh, {Opetopes. Syntactic and Algebraic Aspects.}. Doctoral thesis at the University of Paris (Denis Diderot) (2020), pp. 1-324. None J. Kock, A. Joyal, M. Batanin, J-F. Mascari, {Polynomial Functors and opetopes}, Adv. Math. 224 (2010) pp. 2690-2737. ['Polynomial functors and opetopes'] 10.1016/j.aim.2010.02.012 Tom Leinster, {Higher Operads, Higher Categories}, London Mathematical Society Lecture Note Series 298, Cambridge University Press, Cambridge 2004. Preprint available as \href{http://arxiv.org/abs/math/0305049}{arXiv:math/0305049v1} [math.CT]. ['Higher Operads, Higher Categories'] 10.1017/cbo9780511525896 M. Makkai, M. Zawadowski, {Disks and duality}, Theory and Applications of Categories, Vol. 8, 2001, No. 7, pp 114-243. None D. Oury, {On the duality between trees and disks}, Theory and Applications of Categories, Vol. 24, 2010, No. 16, pp 418-450. None T. Palm, {Dendrotopic sets}, Hopf algebras, and semiabelian categories, Fields Inst. Commun. vol. 43 (2004), 411-461 AMS, Providence, RI. ['Dendrotopic sets'] 10.1090/fic/043/20 R. Steiner, {Opetopes and chain complexes}, Theory and Applications of Categories, Vol. 26, 2012, No. 19, pp 501-519. None S. Szawiel, M. Zawadowski, {The web monoid and opetopic sets}, J. of Pure and Applied Algebra 217 (2013), pp. 1105–1140. ['The web monoid and opetopic sets'] 10.1016/j.jpaa.2012.09.030 M. Zawadowski, {On positive opetopes, positive opetopic cardinals and positive opetopic sets}. arXiv:0708.2658v2 [math.CT], (2023) pp. 1-88. None M. Zawadowski, {On ordered face structures and many-to-one computads}. ArXiv:0708.2659 [math.CT], (2007), pp. 1-95. None M. Zawadowski, {Lax Monoidal Fibrations}, in Models, Logics, and Higher-Dimensional Categories: A Tribute to the Work of Mihály Makkai (B. Hart, et al. , editors) (CRM Proceedings 53, 2011), pp. 341-424. ['Lax monoidal fibrations'] 10.1090/crmp/053/17 M. Zawadowski, {Positive Opetopes with Contractions form a Test Category}, ArXiv:1712.06033 [math.CT], (2017). \endrefs \end{document} None -------------- ['A comonad for Grothendieck fibrations'] author [{'given': 'Jacopo', 'family': 'Emmenegger'}, {'given': 'Luca', 'family': 'Mesiti'}, {'given': 'Giuseppe', 'family': 'Rosolini'}, {'given': 'Thomas', 'family': 'Streicher'}] publication date 2024-05-16 volume 40 issue 13 page range ('371', '389') url http://www.tac.mta.ca/tac/volumes/40/13/40-13abs.html abstract We prove that cloven Grothendieck fibrations over a fixed base B are the pseudo-coalgebras for a lax idempotent 2-comonad on Cat/B. We show this via an original observation that the known colax idempotent 2-monad for fibrations over a fixed base has a right 2-adjoint. As an important consequence, we obtain an original cofree construction of a fibration on a functor. We also give a new, conceptual proof of the fact that the forgetful 2-functor from split fibrations to cloven fibrations over a fixed base has both a left 2-adjoint and a right 2-adjoint, in terms of coherence phenomena of strictification of pseudo-(co)algebras. The 2-monad for fibrations yields the left splitting and the 2-comonad yields the right splitting. Moreover, we show that the constructions induced by these coherence theorems recover Giraud's explicit constructions of the left and the right splittings. keywords Grothendieck fibration, lax idempotent monad ams class 18N45, 03G30, 18N10 dois [] DOI citations: M.~Bunge. \newblock Stack completions and {M}orita equivalence for categories in a topos. \newblock \emph{Cah.\ Topol.\ G\'eom.\ Diff\'er.\ Cat\'eg.}, 20\penalty0 (4):\penalty0 401--436, 1979. None M.~Bunge and C.~Hermida. \newblock Pseudomonadicity and 2-stack completions. \newblock In \emph{Models, logics, and higher-dimensional categories}, volume~53 of \emph{CRM Proc. Lecture Notes}, pages 29--54. Amer. Math. Soc., Providence, RI, 2011. \newblock \doi{10.1090/crmp/053/02}. None M.~Bunge and R.~Par\'{e}. \newblock Stacks and equivalence of indexed categories. \newblock \emph{Cah.\ Topol.\ G\'eom.\ Diff\'er.\ Cat\'eg.}, 20\penalty0 (4):\penalty0 373--399, 1979. None F.~Conduch\'{e}. \newblock Au sujet de l'existence d'adjoints \`a droite aux foncteurs ``image r\'{e}ciproque'' dans la cat\'{e}gorie des cat\'{e}gories. \newblock \emph{C. R. Acad. Sci. Paris S\'{e}r. A-B}, 275:\penalty0 A891--A894, 1972. None S.~Eilenberg and J.C. Moore. \newblock {Adjoint functors and triples}. \newblock \emph{Illinois J. Math.}, 9\penalty0 (3):\penalty0 381 -- 398, 1965. \newblock \doi{10.1215/ijm/1256068141}. ['Adjoint functors and triples'] 10.1215/ijm/1256068141 N.~Gambino and J.~Kock. \newblock Polynomial functors and polynomial monads. \newblock \emph{Math. Proc. Cambridge Philos. Soc.}, 154\penalty0 (1):\penalty0 153--192, 2013. \newblock \doi{10.1017/S0305004112000394}. ['Polynomial functors and polynomial monads'] 10.1017/s0305004112000394 J.~Giraud. \newblock M\'{e}thode de la descente. \newblock \emph{Bull. Soc. Math. France M\'{e}m.}, 2:\penalty0 viii+150, 1964. ['Méthode de la descente'] 10.24033/msmf.2 J.~Giraud. \newblock \emph{Cohomologie non ab{\'e}lienne}, volume 179 of \emph{Grundlehren der mathematischen Wissenschaften}. \newblock Springer Berlin, Heidelberg, 1971. \newblock \doi{10.1007/978-3-662-62103-5}. ['Cohomologie non abélienne'] 10.1007/978-3-662-62103-5 J.~W. Gray. \newblock Fibred and cofibred categories. \newblock In \emph{Proc. {C}onf. {C}ategorical {A}lgebra ({L}a {J}olla, {C}alif., 1965)}, pages 21--83. Springer, New York, 1966. ['Fibred and Cofibred Categories'] 10.1007/978-3-642-99902-4_2 N.~Johnson and D.~Yau. \newblock \emph{2-dimensional categories}. \newblock Oxford University Press, Oxford, 2021. \newblock \doi{10.1093/oso/9780198871378.001.0001}. ['2-Dimensional Categories'] 10.1093/oso/9780198871378.001.0001 K.~Kapulkin and P.~LeFanu Lumsdaine. \newblock The simplicial model of univalent foundations (after voevodsky). \newblock \emph{J. Eur. Math. Soc.}, 23:\penalty0 2071–2126, 2021. \newblock \doi{10.4171/JEMS/1050}. ['The simplicial model of Univalent Foundations (after Voevodsky)'] 10.4171/jems/1050 G.~M. Kelly and R.~Street. \newblock Review of the elements of {$2$}-categories. \newblock In \emph{Category {S}eminar ({P}roc. {S}em., {S}ydney, 1972/1973)}, volume Vol. 420 of \emph{Lecture Notes in Math.}, pages 75--103. Springer, Berlin-New York, 1974. ['Category Seminar'] 10.1007/bfb0063096 A.~Kock. \newblock \emph{Limit monads in categories}. \newblock PhD thesis, Univ. Chicago, 1967. None A.~Kock. \newblock Monads for which structures are adjoint to units. \newblock \emph{J. Pure Appl. Algebra}, 104\penalty0 (1):\penalty0 41--59, 1995. \newblock \doi{10.1016/0022-4049(94)00111-U}. ['Monads for which structures are adjoint to units'] 10.1016/0022-4049(94)00111-u S.~Lack. \newblock Codescent objects and coherence. \newblock \emph{J. Pure Appl. Algebra}, 175\penalty0 (1):\penalty0 223--241, 2002. \newblock \doi{10.1016/S0022-4049(02)00136-6}. ['Codescent objects and coherence'] 10.1016/s0022-4049(02)00136-6 A.~D. Lauda. \newblock Frobenius algebras and ambidextrous adjunctions. \newblock \emph{Theory Appl. Categ.}, 16\penalty0 (4):\penalty0 84--122, 2006. None A.~J. Power, G.~L. Cattani, and G.~Winskel. \newblock A representation result for free cocompletions. \newblock \emph{J. Pure Appl. Algebra}, 151\penalty0 (3):\penalty0 273--286, 2000. \newblock \doi{10.1016/S0022-4049(99)00063-8}. ['A representation result for free cocompletions'] 10.1016/s0022-4049(99)00063-8 A.J. Power. \newblock A general coherence result. \newblock \emph{J. Pure Appl. Algebra}, 57\penalty0 (2):\penalty0 165--173, 1989. \newblock \doi{10.1016/0022-4049(89)90113-8}. ['A general coherence result'] 10.1016/0022-4049(89)90113-8 R.~Street. \newblock Fibrations and {Y}oneda's lemma in a {$2$}-category. \newblock In \emph{Category {S}eminar ({P}roc. {S}em., {S}ydney, 1972/1973)}, Lecture Notes in Math., Vol. 420, pages 104--133. Springer, Berlin, 1974. ["Fibrations and Yoneda's lemma in a 2-category"] 10.1007/bfb0063102 R.~Street. \newblock Fibrations in bicategories. \newblock \emph{Cah.\ Topol.\ G\'eom.\ Diff\'er.\ Cat\'eg.}, 21\penalty0 (2):\penalty0 111--160, 1980. None R.~Street. \newblock Conduch{\'e} functors. \newblock Manuscript, 1986. \newblock Available in extended form at \href{http://www.math.mq.edu.au/~street/Pow.fun.pdf} {{\tt www.math.mq.edu.au/{\~{ }}street/Pow.fun.pdf}} (accessed on 3/12/2023). None R.~Street and D.~Verity. \newblock The comprehensive factorization and torsors. \newblock \emph{Theory Appl. Categ.}, 23\penalty0 (3):\penalty0 42--75, 2010. None T.~Streicher. \newblock Fibered {C}ategories {\`a} la {J}ean {B\'enabou}. \newblock Manuscript, 2022. \newblock Available at \href{http://www2.mathematik.tu-darmstadt.de/~streicher/FIBR/FiBo.pdf} {{\tt www2.mathematik.tu-darmstadt.de/{\~{ }}streicher/FIBR/FiBo.pdf}} (accessed on 3/12/2023). None M.~Weber. \newblock Yoneda structures from 2-toposes. \newblock \emph{Appl. Categ. Structures}, 15\penalty0 (4):\penalty0 373--399, 2007. \newblock \doi{10.1007/s10485-007-9079-2}. ['Yoneda Structures from 2-toposes'] 10.1007/s10485-007-9079-2 V.~Z{\"o}berlein. \newblock Doctrines on 2-categories. \newblock \emph{Math. Z.}, 148:\penalty0 267--279, 1976. \newblock \doi{10.1007/BF01214522}. \end{thebibliography} \newpage None -------------- ['The oplax limit of an enriched category'] author [{'given': 'Soichiro', 'family': 'Fujii'}, {'given': 'Stephen', 'family': 'Lack'}] publication date 2024-05-17 volume 40 issue 14 page range ('390', '412') url http://www.tac.mta.ca/tac/volumes/40/14/40-14abs.html abstract We show that 2-categories of the form ℬ𝐂𝐚𝐭 are closed under slicing, provided that we allow ℬ to range over bicategories (rather than, say, monoidal categories). That is, for any ℬ-category 𝕏, we define a bicategory ℬ/𝕏 such that ℬ𝐂𝐚𝐭/𝕏≅ (ℬ/𝕏)𝐂𝐚𝐭. The bicategory ℬ/𝕏 is characterized as the oplax limit of 𝕏, regarded as a lax functor from a chaotic category to ℬ, in the 2-category 𝐁𝐈𝐂𝐀𝐓 of bicategories, lax functors and icons. We prove this conceptually, through limit-preservation properties of the 2-functor 𝐁𝐈𝐂𝐀𝐓→ 2𝐂𝐀𝐓 which maps each bicategory ℬ to the 2-category ℬ𝐂𝐚𝐭. When ℬ satisfies a mild local completeness condition, we also show that the isomorphism ℬ𝐂𝐚𝐭/𝕏≅ (ℬ/𝕏)𝐂𝐚𝐭 restricts to a correspondence between fibrations in ℬ𝐂𝐚𝐭 over 𝕏 on the one hand, and ℬ/𝕏-categories admitting certain powers on the other. keywords Enriched categories, bicategories ams class 18D20, 18N10 dois [] DOI citations: Igor Bakovi{\'c}. \newblock Fibrations of bicategories. \newblock Preprint available at \url{http://www.irb.hr/korisnici/ibakovic/groth2fib.pdf}. None Renato Betti, Aurelio Carboni, Ross Street, and Robert Walters. \newblock Variation through enrichment. \newblock {\em J. Pure Appl. Algebra}, 29(2):109--127, 1983. ['Variation through enrichment'] 10.1016/0022-4049(83)90100-7 Jean B{\'{e}}nabou. \newblock Introduction to bicategories. \newblock In {\em Reports of the {M}idwest {C}ategory {S}eminar}, pages 1--77. Springer, Berlin, 1967. ['Introduction to bicategories'] 10.1007/bfb0074299 Mitchell Buckley. \newblock Fibred 2-categories and bicategories. \newblock {\em J. Pure Appl. Algebra}, 218(6):1034--1074, 2014. ['Fibred 2-categories and bicategories'] 10.1016/j.jpaa.2013.11.002 Richard Garner. \newblock Topological functors as total categories. \newblock {\em Theory Appl. Categ.}, 29(15):406--422, 2014. None R.~Gordon and A.~J. Power. \newblock Enrichment through variation. \newblock {\em J. Pure Appl. Algebra}, 120(2):167--185, 1997. ['Enrichment through variation'] 10.1016/s0022-4049(97)00070-4 John~W. Gray. \newblock {\em Formal category theory: adjointness for {$2$}-categories}. \newblock Lecture Notes in Mathematics, Vol. 391. Springer-Verlag, Berlin-New York, 1974. ['Formal Category Theory: Adjointness for 2-Categories'] 10.1007/bfb0061280 Richard Garner and Michael Shulman. \newblock Enriched categories as a free cocompletion. \newblock {\em Adv. Math.}, 289:1--94, 2016. ['Enriched categories as a free cocompletion'] 10.1016/j.aim.2015.11.012 Claudio Hermida. \newblock Some properties of {${\bf Fib}$} as a fibred {$2$}-category. \newblock {\em J. Pure Appl. Algebra}, 134(1):83--109, 1999. ['Some properties of Fib as a fibred 2-category'] 10.1016/s0022-4049(97)00129-1 Stephen Lack. \newblock Limits for lax morphisms. \newblock {\em Appl. Categ. Structures}, 13(3):189--203, 2005. ['Limits for Lax Morphisms'] 10.1007/s10485-005-2958-5 Stephen Lack. \newblock Icons. \newblock {\em Appl. Categ. Structures}, 18(3):289--307, 2010. ['Icons'] 10.1007/s10485-008-9136-5 Stephen Lack and Michael Shulman. \newblock Enhanced 2-categories and limits for lax morphisms. \newblock {\em Adv. Math.}, 229(1):294--356, 2012. ['Enhanced 2-categories and limits for lax morphisms'] 10.1016/j.aim.2011.08.014 Ross Street. \newblock Fibrations and {Y}oneda's lemma in a {$2$}-category. \newblock In {\em Category {S}eminar ({P}roc. {S}em., {S}ydney, 1972/1973)}, Lecture Notes in Math., Vol. 420, pages 104--133. Springer, Berlin, 1974. ["Fibrations and Yoneda's lemma in a 2-category"] 10.1007/bfb0063102 Ross Street. \newblock Enriched categories and cohomology. \newblock {\em Repr. Theory Appl. Categ.}, (14):1--18, 2005. \newblock Reprinted from Quaestiones Math. {{\bf{6}}} (1983), no. 1-3, 265--283, with new commentary by the author. ['ENRICHED CATEGORIES AND COHOMOLOGY'] 10.1080/16073606.1983.9632304 Mark Weber. \newblock Yoneda structures from 2-toposes. \newblock {\em Appl. Categ. Structures}, 15(3):259--323, 2007. \end{thebibliography} \newpage \end{document} ['Yoneda Structures from 2-toposes'] 10.1007/s10485-007-9079-2 -------------- ["Bi-directional models of `Radically Synthetic' Differential Geometry"] author [{'given': 'Matías', 'family': 'Menni'}] publication date 2024-05-23 volume 40 issue 15 page range ('413', '429') url http://www.tac.mta.ca/tac/volumes/40/15/40-15abs.html abstract The radically synthetic foundation for smooth geometry formulated in postulates a space T with the property that it has a unique point and, out of the monoid T^T of endomorphisms, it extracts a submonoid R which, in many cases, is the (commutative) multiplication of a rig structure. The rig R is said to be bi-directional if its subobject of invertible elements has two connected components. In this case, R may be equipped with a pre-order compatible with the rig structure. We adjust the construction of `well-adapted' models of Synthetic Differential Geometry in order to build the first pre-cohesive toposes with a bi-directional R. We also show that, in one of these pre-cohesive variants, the pre-order on R, derived radically synthetically from bi-directionality, coincides with that defined in the original model. keywords Axiomatic Cohesion, (Radically) Synthetic Differential Geometry ams class 58A03, 18B25, 18F10, 03G30 dois [] DOI citations: M.~Artin, A.~Grothendieck, and J.~L. Verdier. \newblock {\em Th\'eorie de topos et cohomologie \'etale des sch\'emas}, volume 269-270 of {\em Lecture notes in mathematics}. \newblock Springer-Verlag, 1972. \newblock (SGA4). ['Théorie des Topos et Cohomologie Etale des Schémas'] 10.1007/bfb0081551 J.~L. {Bell}. \newblock {\em {A primer of infinitesimal analysis.}} \newblock Cambridge: Cambridge University Press, 1998. None M.~Bunge, F.~Gago, and A.~M. San Luis~Fern{\'a}ndez. \newblock {\em Synthetic differential topology}, volume 448 of {\em Lond. Math. Soc. Lect. Note Ser.} \newblock Cambridge: Cambridge University Press, 2018. ['Synthetic Differential Topology'] 10.1017/9781108553490 J.~R.~B. Cockett and G.~S.~H. Cruttwell. \newblock Differential structure, tangent structure, and {SDG}. \newblock {\em Appl. Categ. Struct.}, 22(2):331--417, 2014. ['Differential Structure, Tangent Structure, and SDG'] 10.1007/s10485-013-9312-0 M.~Demazure and P.~Gabriel. \newblock {\em Groupes alg\'ebriques. {T}ome {I}: {G}\'eom\'etrie alg\'ebrique, g\'en\'eralit\'es, groupes commutatifs}. \newblock Masson \& Cie, \'Editeur, Paris, 1970. \newblock Avec un appendice {\it Corps de classes local} par Michiel Hazewinkel. None E.~J. {Dubuc}. \newblock {Sur les modeles de la g\'eom\'etrie diff\'erentielle synthetique.} \newblock {\em {Cah. Topologie G\'eom. Diff\'er. Cat\'egoriques}}, 20:231--279, 1979. None P.~T. Johnstone. \newblock Remarks on punctual local connectedness. \newblock {\em Theory Appl. Categ.}, 25:51--63, 2011. None A.~Kock. \newblock A simple axiomatics for differentiation. \newblock {\em Math. Scand.}, 40:183--193, 1977. ['A simple axiomatics for differentiation.'] 10.7146/math.scand.a-11687 A.~{Kock}. \newblock {\em {Synthetic differential geometry. 2nd ed.}} \newblock Cambridge: Cambridge University Press, 2nd ed. edition, 2006. None A.~Kock. \newblock {\em Synthetic geometry of manifolds}, volume 180 of {\em Camb. Tracts Math.} \newblock Cambridge: Cambridge University Press, 2010. None R.~{Lavendhomme}. \newblock {\em {Basic concepts of synthetic differential geometry.}} \newblock Dordrecht: Kluwer Academic Publishers, 1996. ['Basic Concepts of Synthetic Differential Geometry'] 10.1007/978-1-4757-4588-7 F.~W. Lawvere. \newblock Outline of synthetic differential geometry. \newblock Notes of the February 1998 talks in the Buffalo Geometry Seminar. With corrections (Nov. 1998). Available from Lawvere's webpage. None F.~W. Lawvere. \newblock {Categorical dynamics.} \newblock {\em Var. Publ. Ser., Aarhus Univ.}, 30:1--28, 1979. None F.~W. Lawvere. \newblock Categories of spaces may not be generalized spaces as exemplified by directed graphs. \newblock {\em Repr. Theory Appl. Categ.}, 9:1--7, 2005. \newblock Reprinted from Rev. Colombiana Mat. {20} (1986), no. 3-4, 179--185. None F.~W. Lawvere. \newblock Axiomatic cohesion. \newblock {\em Theory Appl. Categ.}, 19:41--49, 2007. None F.~W. Lawvere. \newblock Euler's continuum functorially vindicated. \newblock In {\em Logic, Mathematics, Philosophy: Vintage Enthusiasms}, volume~75 of {\em The Western Ontario Series in Philosophy of Science}, pages 249--254. Springer Science+Bussiness Media B. V., 2011. ['Euler’s Continuum Functorially Vindicated'] 10.1007/978-94-007-0214-1_13 M.~Menni. \newblock Sufficient cohesion over atomic toposes. \newblock {\em Cah. Topol. G\'eom. Diff\'er. Cat\'eg.}, 55(2):113--149, 2014. None M.~Menni. \newblock {A Basis Theorem for 2-rigs and Rig Geometry}. \newblock {\em {Cah. Topol. G\'eom. Diff\'er. Cat\'eg.}}, 62(4):451--490, 2021. None M.~{Menni}. \newblock {The hyperconnected maps that are local}. \newblock {\em {J. Pure Appl. Algebra}}, 225(5):15, 2021. \newblock Id/No 106596. ['The hyperconnected maps that are local'] 10.1016/j.jpaa.2020.106596 F.~Marmolejo and M.~Menni. \newblock {Level $\epsilon$}. \newblock {\em {Cah. Topol. G\'eom. Diff\'er. Cat\'eg.}}, 60(4):450--477, 2019. None I.~Moerdijk and G.~E. Reyes. \newblock {\em Models for smooth infinitesimal analysis}. \newblock Springer-Verlag, New York, 1991. ['Models for Smooth Infinitesimal Analysis'] 10.1007/978-1-4757-4143-8 J.~Rosick{\'y}. \newblock Abstract tangent functors. \newblock Diagrammes 12, 1984. None D.~Yetter. \newblock {On right adjoints to exponential functors.} \newblock {\em J. Pure Appl. Algebra}, 45:287--304, 1987. \end{thebibliography} \end{document} ['On right adjoints to exponential functors'] 10.1016/0022-4049(87)90077-6 -------------- ['KZ-pseudomonads and Kan injectivity'] author [{'given': 'Ivan Di', 'family': 'Liberti'}, {'given': 'Gabriele', 'family': 'Lobbia'}, {'given': 'Lurdes', 'family': 'Sousa'}] publication date 2024-05-24 volume 40 issue 16 page range ('430', '478') url http://www.tac.mta.ca/tac/volumes/40/16/40-16abs.html abstract We introduce the notion of Kan injectivity in 2-categories and study its properties. For an adequate 2-category , we show that every set of morphisms induces a KZ-pseudomonad on whose 2-category of pseudoalgebras is the locally full sub-2-category of all objects (left) Kan injective with respect to and morphisms preserving Kan extensions. The main ingredient is the construction of a (pseudo)chain whose appropriate “convergence" is ensured by a small object argument. keywords 2-category, Kan injectivity, KZ-pseudomonad, small object argument ams class 18D70, 18D65, 18N10, 18N15 dois [] DOI citations: Distributive laws for pseudomonads {II} F.~Marmolejo ['Distributive laws for pseudomonads II'] 10.1016/j.jpaa.2004.04.008 Doctrines on 2-categories Z{\"o}berlein, V. None Distributive Laws For Pseudomonads F.~Marmolejo ['Distributive laws for pseudomonads II'] 10.1016/j.jpaa.2004.04.008 Formal Category Theory: Adjointness for 2-Categories J.~W.~Gray ['Formal Category Theory: Adjointness for 2-Categories'] 10.1007/bfb0061280 Singular coverings of toposes Bunge, M. and Funk, J. None Bi-accessible and bipresentable 2-categories Di Liberti, I. and Osmond, A. ['Bi-accessible and Bipresentable 2-Categories'] 10.1007/s10485-024-09794-9 Order-preserving reflectors and injectivity Carvalho, M. and Sousa, L. ['Order-preserving reflectors and injectivity'] 10.1016/j.topol.2010.12.016 Categorical topology Herrlich, H. ['Categorical topology'] 10.1016/0016-660x(71)90105-x Kan extensions and lax idempotent pseudomonads Marmolejo, F. and Wood, R.~J. None Monads for which structures are adjoint to units A.~Kock \doi{https://doi.org/10.1016/0022-4049(94)00111-U} ['Monads for which structures are adjoint to units'] 10.1016/0022-4049(94)00111-u Locally Presentable and Accessible Categories Ad{\'a}mek , J. and Rosický, J. \doi{10.1017/CBO9780511600579} None Enriched weakness Lack, S. and Rosick{\`y}, J. None Doctrines in categorical logic Kock, A. and Reyes, G.~E.~ ['Doctrines in Categorical Logic'] 10.1016/s0049-237x(08)71104-2 Kan injectivity in order-enriched categories Ad{\'a}mek, J. and Sousa, L. and Velebil, J. ['Kan injectivity in order-enriched categories'] 10.1017/s0960129514000024 Structures defined by finite limits in the enriched context, I Kelly, Gregory M None Limits indexed by category-valued 2-functors Street, Ross ['Limits indexed by category-valued 2-functors'] 10.1016/0022-4049(76)90013-x Lax orthogonal factorisation systems Clementino, Maria Manuel and Franco, Ignacio Lopez None Algebraic weak factorisation systems I: Accessible AWFS Bourke, John and Garner, Richard ['Algebraic weak factorisation systems I: Accessible AWFS'] 10.1016/j.jpaa.2015.06.002 Accessibility and presentability in 2-categories Ivan {Di Liberti} and Fosco Loregian \doi{https://doi.org/10.1016/j.jpaa.2022.107155} ['Accessibility and presentability in 2-categories'] 10.1016/j.jpaa.2022.107155 Distributive laws via admissibility Walker, C. ['Distributive Laws via Admissibility'] 10.1007/s10485-019-09567-9 Pointwise extensions and sketches in bicategories Street, R. None Categories for the working mathematician Mac Lane, S. ['Categories for the Working Mathematician'] 10.1007/978-1-4612-9839-7 Category theory in context Riehl, E. None A logic of orthogonality Ad{\'a}mek, J. and H{\'e}bert, M. and Sousa, L. None Categories of continuous functors, {I} Freyd, {P.~J.} and Kelly, {G.~M.} ['Categories of continuous functors, I'] 10.1016/0022-4049(72)90001-1 Completions of categories: Seminar lectures given 1966 in Z{\"u}rich Lambek, Joachim ['Completions of Categories'] 10.1007/bfb0077265 On limit-preserving functors Kennison, JF ['On limit-preserving functors'] 10.1215/ijm/1256053963 Continuous categories and exponentiable toposes Peter Johnstone and André Joyal \doi{https://doi.org/10.1016/0022-4049(82)90083-4} ['Continuous categories and exponentiable toposes'] 10.1016/0022-4049(82)90083-4 A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on G.~M.~Kelly \doi{10.1017/s0004972700006353} ['A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on'] 10.1017/s0004972700006353 Sketches of an Elephant: A Topos Theory Compendium: 1 Volume Set Peter T. Johnstone ['Sketches of an Elephant A Topos Theory Compendium'] 10.1093/oso/9780198515982.001.0001 Universal abstract elementary classes and locally multipresentable categories Michael Lieberman and Ji{\v{r}}{\'{\i}} Rosick{\'{y}} and Sebastien Vasey \doi{10.1090/proc/14326} ['Universal abstract elementary classes and locally multipresentable categories'] 10.1090/proc/14326 Theories, Sites, Toposes Olivia Caramello ['Theories, Sites, Toposes'] 10.1093/oso/9780198758914.001.0001 Aspects of topoi P. Freyd ['Aspects of topoi'] 10.1017/s0004972700044828 Lex colimits R.~Garner and S.~Lack \doi{https://doi.org/10.1016/j.jpaa.2012.01.003} ['Lex colimits'] 10.1016/j.jpaa.2012.01.003 Approximate Injectivity Rosick{\'y}, J. and Tholen, W. \doi{10.1007/s10485-017-9510-2} ['Approximate Injectivity'] 10.1007/s10485-017-9510-2 Locally presentable and accessible categories Ad\'{a}mek, J. and Rosick\'{y}, J. \doi{10.1017/CBO9780511600579} None On a bicomma object condition for {KZ}-doctrines Bunge, M. and Funk, J. ['On a bicomma object condition for KZ-doctrines'] 10.1016/s0022-4049(98)00108-x Conspectus of variable categories R.~Street ['Conspectus of variable categories'] 10.1016/0022-4049(81)90021-9 Kan-injectivity of locales and spaces Carvalho, M. and Sousa, L. ['On Kan-injectivity of Locales and Spaces'] 10.1007/s10485-015-9413-z Aspects of algebraic algebras D.~Hofmann and L.~Sousa ['Aspects of algebraic Algebras'] 10.23638/lmcs-13(3:4)2017 -------------- ['Higher coverings of racks and quandles – Part I'] author [{'given': 'Fara', 'family': 'Renaud'}] publication date 2024-05-31 volume 40 issue 17 page range ('479', '536') url http://www.tac.mta.ca/tac/volumes/40/17/40-17abs.html abstract This article is the first part of a series of three articles, in which we develop a higher covering theory of racks and quandles. This project is rooted in M. Eisermann's work on quandle coverings, and the categorical perspective brought to the subject by V. Even, who characterizes coverings as those surjections which are central, relatively to trivial quandles. We extend this work by applying the techniques from higher categorical Galois theory, in the sense of G. Janelidze, and in particular we identify meaningful higher-dimensional centrality conditions defining our higher coverings of racks and quandles. In this first article (Part I), we revisit the foundations of the covering theory of interest, we extend it to the more general context of racks and mathematically describe how to navigate between racks and quandles. We explain the algebraic ingredients at play, and reinforce the homotopical and topological interpretations of these ingredients. In particular we study and insist on the crucial role of the left adjoint of the conjugation functor between groups and racks (or quandles). We rename this functor , and explain in which sense it sends a rack to its group of homotopy classes of paths. We characterize coverings and relative centrality using , but also develop a more visual “geometrical” understanding of these conditions. We use alternative generalizable and visual proofs for the characterization of central extensions of racks and quandles. We complete the recovery of M. Eisermann's suitable constructions of weakly universal covers, and fundamental groupoids from a Galois-theoretic perspective. We sketch how to deduce M. Eisermann's detailed classification results from the fundamental theorem of categorical Galois theory. As we develop this complementary understanding of the subject, we lay down all the ideas and results which will articulate the higher-dimensional theory developed in Part II and III. keywords (covering theory of) racks and quandles, categorical Galois theories, central extensions and characterizations, centralization, fundamental groupoid, homotopy classes of paths, weakly universal covers ams class 18E50; 57K12; 08C05; 55Q05; 18A20; 18B40; 20L05 dois [] DOI citations: M.~Barr, \emph{Exact categories}, Lecture Notes in Math. 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Algebra \textbf{216} (2012), no.~8, 1823--1826. \end{thebibliography} \end{document} ['Generalising connected components'] 10.1016/j.jpaa.2012.02.020 -------------- ['The 2-localization of a model category'] author [{'given': 'Dubuc E.', 'family': 'J.'}, {'given': 'Girabel', 'family': 'J.'}] publication date 2024-06-06 volume 40 issue 18 page range ('537', '574') url http://www.tac.mta.ca/tac/volumes/40/18/40-18abs.html abstract In this paper we elaborate on a 2-categorical construction of the homotopy category of a Quillen model category. Given any category A and a class of morphisms Σ⊂A containing the identities, we construct a 2-category obtained by the addition of 2-cells determined by homotopies. A salient feature here is the use of a novel notion of cylinder introduced in . The inclusion 2-functor A has a universal property which yields the 2-localization of A at Σ provided that the arrows of Σ become equivalences in . This result together with a fibrant-cofibrant replacement is then used to obtain the 2-localization of a model category C at the weak equivalences W. The set of connected components of the hom categories yields a novel proof of Quillen's results. We follow the general lines established in , for model bicategories. keywords localization, 2-category, homotopy ams class 18N10, 18N40, 18N55 dois [] DOI citations: M.E. Descotte, E.J. Dubuc, M. Szyld, \emph{A localization of bicategories via homotopies}, Theory and Applications of Categories, Vol. 35, No. 23, pp 845-874 (2020). None M.E. Descotte, E.J. Dubuc, M. Szyld, \emph{Model bicategories and their homotopy bicategories}, Advances in Mathematics Volume 404, Part B, (2022). ['Model bicategories and their homotopy bicategories'] 10.1016/j.aim.2022.108455 W. G. Dwyer, J. Spalinski, \emph{Homotopy theories and model categories}, Handbook of Algebraic Topology (I. M. James, ed.), Elsevier Science B.V., 1995. 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Gray, \emph{Formal Category theory: Adjointness for 2-categories}, Lecture Notes in Mathematics, Vol. 391, Springer-Verlag, Berlin-New York, 1974. ['Formal Category Theory: Adjointness for 2-Categories'] 10.1007/bfb0061280 %P. Hirschhorn, \emph{Model Categories and Their localizations}, Mathematical Surveys and Monographs Volume 99 (2003). None %M. Hovey, \emph{Model categories}, Mathematical Surveys and Monographs, vol. 63, Amer. Math. Soc., Providence, RI, 1999. None %S. Mac Lane, \emph{Categories for the Working Mathematician}, Graduate Texts in Mathematics Volume 5 (1971). None %J. Milnor, \emph{The geometric realization of a semi–simplicial complex}, %Ann. of Math., vol. 65 (1957), %357–362. None D. A. Pronk, \emph{Etendues and stacks as bicategories of fractions}, Compositio Mathematica, Volume 102 (1996) no. 3, p. 243-303. None D. Quillen, \emph{Homotopical Algebra}, Springer Lecture Notes in Mathematics 43 (1967). ['Homotopical Algebra'] 10.1007/bfb0097438 E. Riehl, \emph{Higher Dimensional Categories Model Categories and Weak factorization Systems}, \url{http://www.math.jhu.edu/~eriehl/essay.pdf} None %R. W. Thomason, \emph{Cat as a Closed Model Category}, Cahiers de %Topologie et Geometrie Differentielle XXI-3. (1980), 305-324. None M. Szyld, \emph{The homotopy relation in a category with weak equivalences}, https://arxiv.org/abs/1804.04244v1 (2018). \end{thebibliography} \end{document} None -------------- --------------------- --------------------- No Tex File Found For 40/Foreword/Foreword.tex --------------------- --------------------- --------------------- --------------------- No Tex File Found For 40/Postface/Postface.tex --------------------- ---------------------