['Predicative Algebraic Set Theory'] author [{'given': 'Steve', 'family': 'Awodey'}, {'given': 'Michael A.', 'family': 'Warren'}] publication date 2005-04-12 volume 15 issue 01 page range ('1', '39') url http://www.tac.mta.ca/tac/volumes/15/1/15-01abs.html abstract In this paper the machinery and results developed in are extended to the study of constructive set theories. Specifically, we introduce two constructive set theories BCST and CST and prove that they are sound and complete with respect to models in categories with certain structure. Specifically, basic categories of classes and categories of classes are axiomatized and shown to provide models of the aforementioned set theories. Finally, models of these theories are constructed in the category of ideals. keywords algebraic set theory, categorical logic, predicativity, ideal completion, dependent type theory, Π-pretopos, small maps ams class 18B05, 18B25, 18C10, 03G30, 03E70, 03F60 dois [] DOI citations: P. Aczel and M. Rathjen, Notes on constructive set theory. Institut Mittag-Leffler (Royal Swedish Academy of Sciences) technical report number 40, 2001. Available at \texttt{http://www.ml.kva.se/preprints/archive2000-2001.php}. None S. Awodey, C. Butz, A. Simpson and T. Streicher, Relating set theories, toposes and categories of classes. In preparation, 2004. A preliminary version is available from the AST website \texttt{http://www.phil.cmu.edu/projects/ast/} as CMU technical report CMU-PHIL-146, June 2003. None S. Awodey and H. Forssell, Algebraic models of intuitionistic theories of sets and classes. Carnegie Mellon University technical report CMU-PHIL-156, July 2004. None C. Butz, Bernays-G\"{o}del type theory. \emph{Journal of Pure and Applied Algebra}, 178(1):1-23, 2003. ['Bernays–Gödel type theory'] 10.1016/s0022-4049(02)00259-1 N. Gambino, Presheaf models of constructive set theories. Submitted for publication, 2004. ['PRESHEAF MODELS FOR CONSTRUCTIVE SET THEORIES'] 10.1093/acprof:oso/9780198566519.003.0004 B. Jacobs, \emph{Categorical Logic and Type Theory}. Elsevier, Amsterdam, 1999. None P. T. Johnstone, \emph{Sketches of an Elephant} volume 2. Oxford University Press, Oxford, 2003. None A. Joyal and I. Moerdijk, \emph{Algebraic Set Theory}. Cambridge University Press, Cambridge, 1995. ['Algebraic Set Theory'] 10.1017/cbo9780511752483 S. Mac Lane and I. Moerdijk, \emph{Sheaves in Geometry and Logic}. Springer-Verlag, Berlin, 1992. None I. Moerdijk and E. Palmgren, Wellfounded trees in categories. \emph{Annals of Pure and Applied Logic}, 104:189-218, 2000. ['Wellfounded trees in categories'] 10.1016/s0168-0072(00)00012-9 I. Moerdijk and E. Palmgren, Type theories, toposes and constructive set theory: Predicative aspects of AST. \emph{Annals of Pure and Applied Logic}, 114:155-201, 2002. ['Type theories, toposes and constructive set theory: predicative aspects of AST'] 10.1016/s0168-0072(01)00079-3 I. Rummelhoff, \emph{Algebraic Set Theory}. PhD thesis, University of Oslo, 2004. Forthcoming. None D. Scott, Lambda calculus: some models, some philosophy. In J. Barwise, H. J. Keisler and K. Kunen, editors, \emph{The Kleene Symposium}, pages 381-421. North-Holland, Amsterdam, 1980. ['Lambda Calculus: Some Models, Some Philosophy'] 10.1016/s0049-237x(08)71262-x D. Scott, Category-theoretic models for intuitionistic set theory. Manuscript slides of a talk given at Carnegie Mellon University (additions made in September 1998), December 1985. None A. Simpson, Constructive set theories and their category-theoretic models. Draft, 2004. ['CONSTRUCTIVE SET THEORIES AND THEIR CATEGORY-THEORETIC MODELS'] 10.1093/acprof:oso/9780198566519.003.0003 A. K. Simpson, Elementary axioms for categories of classes. \emph{Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science}, pages 77-85, 1999. ['Elementary axioms for categories of classes'] 10.1109/lics.1999.782592 M. A. Warren, Predicative categories of classes. Master's thesis, Carnegie Mellon University, 2004. \endrefs %\bibliographystyle{abbrv} %\bibliography{mlogic} %% %% END DOCUMENT %% \end{document} None -------------- ['Reflective Kleislisubcategories of the category of Eilenberg-Moore algebras for factorization monads'] author [{'given': 'Marcelo', 'family': 'Fiore'}, {'given': 'Matías', 'family': 'Menni'}] publication date 2005-04-18 volume 15 issue 02 page range ('40', '65') url http://www.tac.mta.ca/tac/volumes/15/2/15-02abs.html abstract It is well known that for any monad, the associated Kleisli category is embedded in the category of Eilenberg-Moore algebras as the free ones. We discovered some interesting examples in which this embedding is reflective; that is, it has a left adjoint. To understand this phenomenon we introduce and study a class of monads arising from factorization systems, and thereby termed factorization monads. For them we show that under some simple conditions on the factorization system the free algebras are a full reflective subcategory of the algebras. We provide various examples of this situation of a combinatorial nature. keywords factorization systems, monads, Kleisli categories, Schanuel topos, Joyal species, combinatorial structures, power series ams class 18A25, 18A40, 18C20, 05A10 dois [] DOI citations: [1] \newblock F.~Bergeron. \newblock Une combinatoire du pl\'ethysme. \newblock {\em Journal of Combinatorial Theory (Series~A)}, 46:291--305, 1987. 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None [6] \newblock M.~Fiore. \newblock Mathematical models of computational and combinatorial structures. \newblock Invited address for \emph{Foundations of Software Science and Computation Structures (\mbox{FOSSACS}~2005)}, volume 3441 of \emph{Lecture Notes in Computer Science}, pages~25--46. \newblock Springer-Verlag, 2005. ['Mathematical Models of Computational and Combinatorial Structures'] 10.1007/978-3-540-31982-5_2 [7] \newblock P.~J.~Freyd and G.~M.~Kelly. \newblock Categories of continuous functors {I}. \newblock {\em Journal of Pure and Applied Algebra}, 2:169--191, 1972. \newblock (Erratum in {\em Journal of Pure and Applied Algebra}, 4:121, 1974.) ['Categories of continuous functors, I'] 10.1016/0022-4049(72)90001-1 [8] \newblock P.~T.~Johnstone. \newblock A topos-theorist looks at dilators. \newblock {\em Journal of Pure and Applied Algebra}, 58(3):235--249, 1989. ['A topos-theorist looks at dilators'] 10.1016/0022-4049(89)90039-x [9] \newblock P.~T.~Johnstone. \newblock {\em Sketches of an elephant: {A} topos theory compendium}, volumes 43-44 of {\em Oxford Logic Guides}. \newblock Oxford University Press, 2002. None [10] \newblock A.~Joyal. \newblock Une theorie combinatoire des s\'eries formelles. \newblock {\em Advances in Mathematics}, 42:1--82, 1981. ['Une théorie combinatoire des séries formelles'] 10.1016/0001-8708(81)90052-9 [11] \newblock A.~Joyal. \newblock Foncteurs analytiques et espec\`es de structures. \newblock In G.~Labelle and P.~Leroux, editors, {\em Combinatoire \'enum\'erative}, volume 1234 of {\em Lecture Notes in Mathematics}, pages 126--159. Springer-Verlag, 1986. ['Combinatoire énumérative'] 10.1007/bfb0072503 [12] \newblock G.~M.~Kelly. \newblock {\em Basic Concepts of Enriched Category Theory}. \newblock Number~64 in LMS Lecture Notes. Cambridge University Press, 1982. None [13] \newblock G.~M.~Kelly. \newblock A survey of totality for enriched and ordinary categories. \newblock {\em Cahiers de Top.~et G\'eom.~Diff.~Cat\'egoriques}, 27:109--132, 1986. None [14] \newblock S.~{Mac Lane}. \newblock {\em Categories for the Working Mathematician}. \newblock Graduate Texts in Mathematics. Springer-Verlag, 1971. ['Categories for the Working Mathematician'] 10.1007/978-1-4612-9839-7 [15] \newblock S.~{Mac Lane} and I.~Moerdijk. \newblock {\em Sheaves in Geometry and Logic: {A} First Introduction to Topos Theory}. \newblock Universitext. Springer-Verlag, 1992. ['Sheaves in Geometry and Logic'] 10.1007/978-1-4612-0927-0 [16] \newblock M.~M{\'e}ndez. \newblock Species on digraphs. \newblock {\em Advances in Mathematics}, 123(2):243--275, 1996. ['Species on Digraphs'] 10.1006/aima.1996.0073 [17] \newblock M.~M\'endez and O.~Nava. \newblock Colored species, c-monoids and plethysm, {I}. \newblock {\em Journal of Combinatorial Theory (Series~A)}, 64:102--129, 1993. ['Colored species, c-monoids, and plethysm, I'] 10.1016/0097-3165(93)90090-u [18] \newblock M.~Menni. \newblock About {$\text{\font\cyrss=wncyss10\cyrss I}$}-quantifiers. \newblock {\em Applied Categorical Structures}, 11(5):421--445, 2003. None [19] \newblock M.~Menni. \newblock Symmetric monoidal completions and the exponential principle among labeled combinatorial structures. \newblock {\em Theory and Applications of Categories}, 11:397--419, 2003. None [20] \newblock J.~Myhill. \newblock Recursive equivalence types and combinatorial functions. \newblock {\em Bull.~Amer. Math.~Soc.}, 64:373--376, 1958. 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['The Family Approach to Total Cocompleteness and Toposes'] 10.2307/1999291 [24] \newblock R.~Street and R.~Walters. \newblock Yoneda structures on 2-categories. \newblock {\em Journal of Algebra}, 50:350--379, 1978. \endrefs \end{document} %%% Local Variables: ['Yoneda structures on 2-categories'] 10.1016/0021-8693(78)90160-6 -------------- ['Model structures for homotopy ofinternal categories'] author [{'given': 'T.', 'family': 'Everaert'}, {'given': 'R.W.', 'family': 'Kieboom'}, {'given': 'T. Van der', 'family': 'Linden'}] publication date 2005-06-23 volume 15 issue 03 page range ('66', '94') url http://www.tac.mta.ca/tac/volumes/15/3/15-03abs.html abstract The aim of this paper is to describe Quillen model category structures on the category of internal categories and functors in a given finitely complete category . Several non-equivalent notions of internal equivalence exist; to capture these notions, the model structures are defined relative to a given Grothendieck topology on . Under mild conditions on , the regular epimorphism topology determines a model structure where is the class of weak equivalences of internal categories (in the sense of Bunge and Paré). For a Grothendieck topos we get a structure that, though different from Joyal and Tierney's, has an equivalent homotopy category. In case is semi-abelian, these weak equivalences turn out to be homology isomorphisms, and the model structure on induces a notion of homotopy of internal crossed modules. In case is the category of groups and homomorphisms, it reduces to the case of crossed modules of groups. The trivial topology on a category determines a model structure on where is the class of strong equivalences (homotopy equivalences), the class of internal functors with the homotopy lifting property, and the class of functors with the homotopy extension property. As a special case, the “folk” Quillen model category structure on the category = of small categories is recovered. keywords internal category, Quillen model category, homotopy, homology ams class Primary 18G55 18G50 18D35; Secondary 20J05 18G25 18G30 dois [] DOI citations: \emph{Th{\'e}orie des topos et cohomologie {\'e}tale des sch{\'e}mas. {T}ome 1: Th{\'e}orie des topos}, Lecture notes in mathematics, vol. 269, Springer, 1972, S{\'e}minaire de G{\'e}ometrie Alg{\'e}brique du Bois-Marie 1963-1964 (SGA4), 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 J.~Ad{\'a}mek, H.~Herrlich, J.~Rosick{\'y}, and W.~Tholen, \emph{Weak factorization systems and topological functors}, Appl. Categ. Struct. \textbf{10} (2002), 237--249. 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['Homotopies of small categories'] 10.4064/fm-114-3-209-217 M.~Gran, \emph{Internal categories in {M}al'cev categories}, J. Pure Appl. Algebra \textbf{143} (1999), 221--229. ['Internal categories in Mal’cev categories'] 10.1016/s0022-4049(98)00112-1 M.~Gran and J.~Rosick{\'y}, \emph{Special reflexive graphs in modular varieties}, Alg. Univ. \textbf{52} (2004), 89--102. None J.~W. Gray, \emph{Fibred and cofibred categories}, Proceedings of the Conference on Categorical Algebra, La Jolla 1965 (S.~Eilenberg, D.~K. Harrison, S.~MacLane, and H.~R{\"o}hrl, eds.), Springer, 1966, pp.~21--83. ['Proceedings of the Conference on Categorical Algebra'] 10.1007/978-3-642-99902-4 G.~Janelidze, \emph{Internal crossed modules}, Georgian Math. J. \textbf{10} (2003), no.~1, 99--114. ['Internal Crossed Modules'] 10.1515/gmj.2003.99 G.~Janelidze and G.~M. Kelly, \emph{Galois theory and a general notion of central extension}, J. Pure Appl. Algebra \textbf{97} (1994), 135--161. 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Parent, and A.~Tonks, \emph{A model structure {\`a} la {T}homason on {{\bf 2-Cat}}}, {\tt math.AT/0411154}, 2005. \end{thebibliography} \end{document} None -------------- ['The shape of a category up to directed homotopy'] author [{'given': 'Marco', 'family': 'Grandis'}] publication date 2005-06-23 volume 15 issue 04 page range ('95', '146') url http://www.tac.mta.ca/tac/volumes/15/4/15-04abs.html abstract This work is a contribution to a recent field, Directed Algebraic Topology. Categories which appear as fundamental categories of `directed structures', e.g. ordered topological spaces, have to be studied up to appropriate notions of directed homotopy equivalence, which are more general than ordinary equivalence of categories. Here we introduce past and future equivalences of categories—sort of symmetric versions of an adjunction—and use them and their combinations to get `directed models' of a category; in the simplest case, these are the join of the least full reflective and the least full coreflective subcategory. keywords homotopy theory, adjunctions, reflective subcategories, directed algebraic topology, fundamental category, concurrent processes ams class 55Pxx, 18A40, 68Q85 dois [] DOI citations: A. Bauer, L. Birkedal and D.S. Scott, Equilogical spaces, Theoretical Computer Science \b315 (2004), 35--59. ['Equilogical spaces'] 10.1016/j.tcs.2003.11.012 F. Borceux and M. Korostenski, Open localizations, J. Pure Appl. Algebra \b74 (1991), 229--238. ['Open localizations'] 10.1016/0022-4049(91)90113-g A.C. Ehresmann, Localization of universal problems. Local colimits, Appl. Categ. Structures \b10 (2002), 157--172. ['Localization of Universal Problems. Local Colimits'] 10.1023/a:1014342114336 L. Fajstrup, E. Goubault and M. Raussen, Algebraic topology and concurrency, Preprint 1999. None L. Fajstrup, M. Raussen, E. Goubault and E. Haucourt, Components of the fundamental category, Appl. Categ. Structures \b12 (2004), 81--108. ['Components of the Fundamental Category'] 10.1023/b:apcs.0000013812.75342.de P. Gaucher and E. Goubault, Topological deformation of higher dimensional automata, Preprint 2001. \newline \texttt{http://arXiv.org/abs/math.AT/0107060} ['Topological deformation of higher dimensional automata'] 10.4310/hha.2003.v5.n2.a3 E. Goubault, Geometry and concurrency: a user's guide, in: Geometry and concurrency, Math. Structures Comput. Sci. \b10 (2000), no. 4, pp. 411--425. ["Geometry and concurrency: a user's guide"] 10.1017/s0960129500003133 M. Grandis, Cubical homotopical algebra and cochain algebras, Ann. Mat. Pura Appl. \b170 (1996) 147--186. ['Cubical homotopical algebra and cochain algebras'] 10.1007/bf01758987 M. Grandis, Higher fundamental functors for simplicial sets, Cah. Topol. G\'{e}om. Diff\'{e}r. Cat\'{e}g. \b42 (2001), 101--136. None M. Grandis, Directed homotopy theory, I. The fundamental category, Cah. Topol. G\'{e}om. Diff\'{e}r. Cat\'{e}g. \b44 (2003), 281--316. None M. Grandis, Directed combinatorial homology and noncommutative tori (The breaking of symmetries in algebraic topology), Math. Proc. Cambridge Philos. Soc., to appear. [Dip. Mat. Univ. Genova, Preprint 480 (2003).] None M. Grandis, Inequilogical spaces, directed homology and noncommutative geometry, Homology Homotopy Appl. \b6 (2004), 413--437. \newline \texttt{http://www.rmi.acnet.ge/hha/volumes/2004/n1a21/v6n1a21.pdf} ['Inequilogical spaces, directed homology and noncommutative geometry'] 10.4310/hha.2004.v6.n1.a21 M. Grandis and R. Par\'{e}, Adjoints for double categories, Cah. Topol. G\'{e}om. Diff. Cat\'{e}g. \b45 (2004), 193--240. None G. Janelidze and W. Tholen, Functorial factorization, well--pointedness and separability, J. Pure Appl. Algebra \b142 (1999), 99--130. ['Functorial factorization, well-pointedness and separability'] 10.1016/s0022-4049(98)00095-4 G.M. Kelly, On the ordered set of reflective subcategories, Bull. Austral. Math. Soc. \b36 (1987), 137--152. ['On the ordered set of reflective subcategories'] 10.1017/s0004972700026381 G.M. Kelly and F. W. Lawvere, On the complete lattice of essential localizations, Actes du Colloque en l'Honneur du Soixanti\`{e}me Anniversaire de R. Lavendhomme (Louvain--la-Neuve, 1989). Bull. Soc. Math. Belg. S\'{e}r. A \b41 (1989), 289--319. None F.W. Lawvere, Unity and identity of opposites in calculus and physics, The European Colloquium of Category Theory (Tours, 1994), Appl. Categ. Structures \b4 (1996), 167--174. ['Unity and identity of opposites in calculus and physics'] 10.1007/bf00122250 S. Mac Lane, Categories for the working mathematician, Springer, Berlin 1971. ['Categories for the Working Mathematician'] 10.1007/978-1-4612-9839-7 G. Rosolini, Equilogical spaces and filter spaces, Categorical studies in Italy (Perugia, 1997). Rend. Circ. Mat. Palermo (2) Suppl. No. 64, (2000), 157--175. None D. Scott, A new category? Domains, spaces and equivalence relations, Unpublished manuscript (1996). \newline \texttt{http://www.cs.cmu.edu/Groups/LTC/} \end{thebibliography} None -------------- ['Algebraic Models of Intuitionistic Theories of Sets and Classes'] author [{'given': 'S.', 'family': 'Awodey'}, {'given': 'H.', 'family': 'Forssell'}] publication date 2005-09-21 volume 15 issue 05 page range ('147', '163') url http://www.tac.mta.ca/tac/volumes/15/5/15-05abs.html abstract This paper constructs models of intuitionistic set theory in suitable categories. First, a Basic Intuitionistic Set Theory () is stated, and the categorical semantics are given. Second, we give a notion of an ideal over a category, using which one can build a model of in which a given topos occurs as the sets. And third, a sheaf model is given of a Basic Intuitionistic Class Theory conservatively extending . The paper extends the results in by introducing a new and perhaps more natural notion of ideal, and in the class theory of part three. keywords algebraic set theory, topos theory, sheaf theory ams class 18B05, 18B25, 18C10, 03G30, 03E70, 03F60 dois [] DOI citations: I. Rummelhoff, Class categories and polymorphic $\Pi_1$ types. PhD thesis, University of Oslo, forthcoming. None S. Awodey, C. Butz, A. Simpson, and T. Streicher, Relating topos theory and set theory via categories of classes. Tech. Report CMU-PHIL-146, Carnegie Mellon, 2003. Available from: {\tt www.phil.cmu.edu/projects/ast} None P. Aczel and M. Rathjen, Notes on Constructive Set Theory. Technical Report 40, Institut Mittag-Leffler (Royal Swedish Academy of Sciences), 2001. None P. T. Johnstone, Sketches of an Elephant. Clarendon Press, Oxford, 2002. ['Sketches'] 10.1093/oso/9780198515982.003.0007 A. A. Fraenkel and Y. Bar--Hillel, Foundations of Set Theory, 1st edition. North--Holland Publishing Company, Amsterdam, 1958. None C.~Butz, G{\"o}del-{B}ernays type theory. Journal of Pure and Applied Algebra, 178(1):1--23, 2003. None A.~Simpson, Elementary Axioms for Categories of Classes (Extended Abstract). In Fourteenth Annual IEEE Symposium on Logic in Computer Science: 77--85, 1999. ['Elementary axioms for categories of classes'] 10.1109/lics.1999.782592 A. Joyal and I. Moerdijk, Algebraic set theory. London Mathematical Society, Lecture Note Series 220, Cambridge University Press, 1995. \end{references*} \end{document} ['Algebraic Set Theory'] 10.1017/cbo9780511752483 -------------- ['Generic commutative separable algebras and cospans of graphs'] author [{'given': 'R.', 'family': 'Rosebrugh'}, {'given': 'N.', 'family': 'Sabadini'}, {'given': 'R. F. C.', 'family': 'Walters'}] publication date 2005-10-12 volume 15 issue 06 page range ('164', '177') url http://www.tac.mta.ca/tac/volumes/15/6/15-06abs.html abstract We show that the generic symmetric monoidal category with a commutative separable algebra which has a Σ-family of actions is the category of cospans of finite Σ-labelled graphs restricted to finite sets as objects, thus providing a syntax for automata on the alphabet Σ. We use this result to produce semantic functors for Σ-automata. keywords separable algebra, cospan category ams class 18B20, 18D10, 68Q05, 68Q85 dois [] DOI citations: L. Abrams, Two dimensional topological quantum field theories and Frobenius algebras, J. Knot Theory Ramifications, \textbf{5}, 569--587, 1996. ['TWO-DIMENSIONAL TOPOLOGICAL QUANTUM FIELD THEORIES AND FROBENIUS ALGEBRAS'] 10.1142/s0218216596000333 J. Beck, Distributive laws, in \emph{Seminar on Triples and Categorical Homology Theory}, Lecture Notes in Mathematics, \textbf{80}, pages 119--140, Springer-Verlag, 1969. ['Distributive laws'] 10.1007/bfb0083084 R. Betti, A. Carboni, R.H. Street and R.F.C. Walters, Variation through enrichment, Journal of Pure and Applied Algebra, \textbf{29}, 109--127, 1983. ['Variation through enrichment'] 10.1016/0022-4049(83)90100-7 A. Carboni, R.F.C. Walters, Cartesian Bicategories I, J. Pure Applied Algebra, \textbf{49}, 11--32, 1987. ['Cartesian bicategories I'] 10.1016/0022-4049(87)90121-6 A. Carboni, Matrices, relations and group representations, J. Algebra, \textbf{138}, 497--529, 1991. ['Matrices, relations, and group representations'] 10.1016/0021-8693(91)90057-f H.S.M. Coxeter, W.O.J. Moser, \emph{Generators and %relations for discrete groups}, 2nd edition, Springer-Verlag, 1964. % None F. DeMeyer, E. Ingraham, \emph{Separable algebra over commutative rings}, Springer Lecture Notes in Mathematics, \textbf{181}, 1971. ['Separable Algebras Over Commutative Rings'] 10.1007/bfb0061226 F. Gadducci, R. Heckel, An inductive view of graph transformation, Springer LNCS, \textbf{1376}, 223--237, 1997. ['An inductive view of graph transformation'] 10.1007/3-540-64299-4_36 F. Gadducci, R. Heckel, M. Llabres, A bi-categorical axiomatisation of concurrent graph rewriting, ENTCS, \textbf{29}, 1999. None R. Gates, On generic separable objects, {\sl Theory and Applications of Categories}, \textbf{4}, 204--248, 1998. None M. Grandis, Finite sets and symmetric simplicial sets, %Theory and Applications of Categories, 8, 244--253, 2001. % % % None A. Joyal, R.H. Street, D. Verity, Traced monoidal categories, Math. Proc. Cambridge Philosophical Soc. \textbf{119} (3), 425--446, 1996. ['Traced monoidal categories'] 10.1017/s0305004100074338 P. Katis, N. %Sabadini, R.F.C. Walters, Bicategories of %processes, Journal of Pure and Applied %Algebra, 115, 141--178, 1997. % None P. Katis, N. Sabadini, R.F.C. Walters, Span(Graph): A categorical algebra of transition systems, Proc. AMAST '97, SLNCS \textbf{1349}, 307--321, Springer Verlag, 1997. ['Span(Graph): A categorical algebra of transition systems'] 10.1007/bfb0000479 P. Katis, N. Sabadini, R.F.C. Walters, On the algebra of systems with feedback and boundary, Rendiconti del Circolo Matematico di Palermo Serie II, Suppl. \textbf{63}, 123--156, 2000. None P. Katis, N. Sabadini, R.F.C. Walters, A formalisation of the IWIM Model, in: Proc. COORDINATION 2000, (Eds.) A. Porto, G.-C. Roman, LNCS \textbf{1906}, 267--283, Springer Verlag, 2000. ['A Formalization of the IWIM Model'] 10.1007/3-540-45263-x_17 P. Katis, N. Sabadini, R.F.C. Walters, Feedback, trace and fixed-point semantics, Theoret. Informatics Appl. \textbf{36}, 181--194, 2002. ['Feedback, trace and fixed-point semantics'] 10.1051/ita:2002009 J. Kock, \emph{Frobenius algebras and 2D topological Quantum Field Theories}, Cambridge University Press, 2004. ['Frobenius algebras'] 10.1017/cbo9780511615443.005 F. W. Lawvere, Ordinal sums and equational doctrines, Springer Lecture Notes in Mathematics, \textbf{80}, 141--155, 1967. ['Ordinal sums and equational doctrines'] 10.1007/bfb0083085 S. Lack, Composing PROPs, {\sl Theory and Applications of Categories,} \textbf{13}, 147--163, 2004. None E.H. Moore, Concerning the abstract group of order $k!$ and $\frac{1}{2}k!$, Proc. London Math. Soc, \textbf{28}, 357--366, 1897. None J. Meseguer, U. Montanari, Petri Nets are Monoids, Information and Computation, \textbf{88}, 105--155, 1990. Also SRI-CSL-88-3, January 1988. ['Petri nets are monoids'] 10.1016/0890-5401(90)90013-8 R. Rosebrugh, N. Sabadini, R.F.C. Walters, Minimization and minimal realization in Span(Graph), Mathematical Structures in Computer Science, \textbf{14}, 685--714, 2004. ['Minimisation and minimal realisation in Span(Graph)'] 10.1017/s096012950400430x R. Rosebrugh, N. Sabadini, R.F.C. Walters, Symmetric separable algebras in monoidal categories and Cospan(Graph), Abstracts of the International Category Theory Conference, CT'04, Vancouver 2004. None R. Rosebrugh, R.J. Wood, Factorization Systems and Distributive Laws, J. Pure Appl. Alg. \textbf{175}, 327--353, 2002. None N. Sabadini, R.F.C. Walters, Hierarchical automata and P systems, CATS'03, ENTCS \textbf{78}, 2003. ['Hierarchical automata and P-systems'] 10.1016/s1571-0661(04)81003-x P. Sobocinski, Process congruences from reaction rules, in Luca Aceto's ``Concurrency Column'', Bulletin of the EATCS vol. \textbf{84}, 2004. None R.F.C. Walters, The tensor product of matrices, Lecture, International Conference on Category Theory, Louvain-la-Neuve, 1987. \end{thebibliography} \end{document} None -------------- ['A Galois theory with stable units for simplicial sets'] author [{'given': 'João J.', 'family': 'Xarez'}] publication date 2006-07-07 volume 15 issue 07 page range ('178', '193') url http://www.tac.mta.ca/tac/volumes/15/7/15-07abs.html abstract We recall and reformulate certain known constructions, in order to make a convenient setting for obtaining generalized monotone-light factorizations in the sense of A. Carboni, G. Janelidze, G. M. Kelly and R. Paré. This setting is used to study the existence of monotone-light factorizations both in categories of simplicial objects and in categories of internal categories. It is shown that there is a non-trivial monotone-light factorization for simplicial sets, such that the monotone-light factorization for reflexive graphs via reflexive relations is a special case of it, obtained by truncation. More generally, we will show that there exists a monotone-light factorization associated with every full subcategory Mono(F_n), n≥ 0, consisting of all simplicial sets whose unit morphisms are monic for the localization F_n:𝐒𝐞𝐭^Δ^op→𝐒𝐞𝐭^Δ^op_n, which truncates each simplicial set after the object of n-simplices. The monotone-light factorization for categories via preorders is as well derived from the proposed setting. We also show that, for regular Mal'cev categories, the reflection of internal groupoids into internal equivalence relations necessarily produces monotone-light factorizations. It turns out that all these reflections do have stable units, in the sense of C. Cassidy, M. Hébert and G. M. Kelly, giving rise to Galois theories. keywords simplicial object, simplicial set, internal category, internal preorder, regular category, Mal'cev category, descent theory, Galois theory, reflection with stable units, monotone-light factorization, Kan extension, elementary topos, geometric morphism ams class 18A32, 18A40, 18G30, 12F10, 55U10, 08B05, 18B25 dois [] DOI citations: Carboni, A., Janelidze, G., Kelly, G. M., Par\'{e}, R. \textit{On localization and stabilization for factorization systems.} App. Cat. Struct. \textbf{5} (1997) 1--58. ['On Localization and Stabilization for Factorization Systems'] 10.1023/a:1008620404444 Cassidy, C., H\'{e}bert, M., Kelly, G. M. \textit{Reflective subcategories, localizations and factorization systems.} J. Austral. Math. Soc. \textbf{38A} (1985) 287--329. None Gabriel, P., Zisman, M. \textit{Calculus of fractions and homotopy theory.} Ergebnisse der mathematik, Vol. 35. Berlin-Heidelberg-New York: Springer 1967. ['Calculus of Fractions and Homotopy Theory'] 10.1007/978-3-642-85844-4 Janelidze, G. \textit{Internal categories in Mal'cev varieties.} Preprint, York University, Toronto (1990). None Janelidze, G. \textit{Pure Galois theory in categories.} J. Algebra \textbf{132} (1990) 270--286. ['Pure Galois theory in categories'] 10.1016/0021-8693(90)90130-g Janelidze, G., Tholen, W. \textit{Functorial factorization, well-pointedness and separability.} J. Pure Appl. Algebra \textbf{142} (1999) 99--130. ['Functorial factorization, well-pointedness and separability'] 10.1016/s0022-4049(98)00095-4 Kelly, G.M. \textit{A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on.} Bull. Austral. Math. Soc. \textbf{22} (1980) 1--83. ['A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on'] 10.1017/s0004972700006353 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{Internal monotone-light factorization for categories via preorders.} Theory Appl. Categories \textbf{13} (2004) 235--251. ['The monotone-light factorization for categories via preorders'] 10.1090/fic/043/25 Xarez, J. J. \textit{The monotone-light factorization for categories via preordered and ordered sets.} PhD thesis, University of Aveiro (Portugal), 2003. ['The monotone-light factorization for categories via preorders'] 10.1090/fic/043/25 Xarez, J. J. \textit{The monotone-light factorization for categories via preorders.} in Galois theory, Hopf algebras and semiabelian Categories, 533--541, Fields Inst. Commun. \textbf{43}, Amer. Math. Soc., Providence, RI, 2004. \end{thebibliography} \end{document} ['The monotone-light factorization for categories via preorders'] 10.1090/fic/043/25 --------------