Seminars of the CENTRE de RECHERCHE en THEORIE des CATEGORIES CATEGORY THEORY RESEARCH CENTER C ---------> R | / | | / | | / | | / | v / v T ---------> C 18 Jan 2000 2:30 - 4:00 M Barr Acyclic Models This is the first in a series, not every week, but irregularly throughout the term. Subsequent talks are planned for Jan 25, Feb 29 (and later ones will be announced when dates have been arranged). 25 Jan 2000 2:30 - 4:00 M Barr Acyclic Models 1 Feb 2000 2:30 - 4:00 M Makkai Simplicial sets, weak omega groupoids, and homotopy 8 Feb 2000 2:30 - 4:00 S Crans Extended teisi in _Ab_ 15 Feb 2000 2:30 - 4:00 M Makkai Simplicial sets, weak omega groupoids, and homotopy II 29 Feb 2000 2:30 - 4:00 M Barr Acyclic Models 7 March 2000 2:30 - 4:00 Michael Barr Acyclic Models Tuesday, 14 March 2000 2:00 - 3:30 Richard Squire Subfunctors and submonads of the powerset monad 4:00 - 5:00 R.A.G. Seely Sigma Pi logic Tuesday, 21 March 2000 2:00 - 3:30 Marta Bunge Stone locales : the dominance-entire connection 4:00 - 5:00 Michael Barr Acyclic Models Tuesday, 28 March 2000 2:30 - 4:00 Alex Simpson A universal characterisation of the interval ABSTRACT: I shall discuss a universal characterisation of the closed interval, which makes sense in the general setting of a category with finite products. (Joint work with Martin Escardo, University of St Andrews, Scotland.) Tuesday, 4 April 2000 2:00 - 3:30 M Makkai Protocategories: the link between Joyal's theta-categories and multitopic categories 4:00 - 5:00 Michael Barr Acyclic Models Tuesday, 11 April 2000 2:00 - 3:30 Sjoerd Crans The shuffle pasting, I 4:00 - 5:00 Michael Barr Acyclic Models Tuesday, 18 April 2000 2:00 - 3:30 Sjoerd Crans The shuffle pasting, II 4:00 - 5:00 Michael Makkai On Quillen model structures and FOLDS. Tuesday, 25 April 2000 2:15 - 3:15 Michael Barr Acyclic Models 4:00 - 5:00 Richard Squire Singly generated algebras without the natural numbers Tuesday, 2 May 2000 2:00 - 3:30 Mihaly Makkai Protocategories: a link between Joyal's theta categories and multitopic categories (continued) 4:00 - 5:00 Richard Squire Singly generated algebras without the natural numbers II Tuesday, 9 May 2000 3:30 - 5:00 Sjoerd Crans The shuffle pasting, III Tuesday, 16 May 2000 3:30 - 5:00 Carsten Butz Quine's New Foundations `is' higher-order arithmetic Tuesday, 23 May 2000 3:30 - 5:00 Michael Barr Acyclic Models Tuesday, 30 May 2000 3:30 - 5:00 Michael Barr Acyclic Models Tuesday, 6 June 2000 TBA Tuesday, 13 June 2000 2:00 - 3:30 Sjoerd Crans The shuffle pasting, IV 4:00 - 5:00 Michael Barr Acyclic models Tuesday, 20 June 2000 2:30 - 4:00 M Makkai A new theory of pasting Tuesday, 27 June 2000 2:15 - 3:15 M Makkai A new theory of pasting II 4:00 - 5:00 Michael Barr Acyclic models Tuesday, 22 August (sic) 2000 2:30 - 4:00 S Crans The shuffle pasting, V (This will probably be Sjoerd's last seminar as a McGill post-doc.) Tuesday, 12 September 2000 2:30 - 4:00 J Lambek Injective hulls of partially ordered monoids Tuesday, 19 September 2000 2:30 - 4:00 M Barr Mackey topologies and injectives Abstract: Let C be a full subcateory of topological abelian groups (TAGs) that is complete and contains the circle group T (1-torus). A character is a continuous homomorphism to T. Denote the (discrete) group of characters of A by A^. It is clear that for any group A there is a weakest topology \sigma A that has the same character group as A, namely the topology induced by the natural embedding of A into T^{A^}. Depending on the nature of C, there may or may not be a strongest such topology (among topologies that belong to C). If there is and it is functorial, we denote the functor by \tau and say that C admits Mackey coreflections. Assume that every object of C has enough characters to separate points (called DS). Then C has Mackey reflections iff T is injective (called DE) in C. The "if" direction is sort of known although not quite stated explicitly in the main source. The other direction seems not have been suspected. The main tool used in the proof is (insert blare of trumpets here) the category chu(Ab,T). (Joint work with H Kleisli) Tuesday, 26 September 2000 2:30 - 4:00 M Barr On HSP completions of categories of TAGs Abstract: For a category C of TAGs, let PC, SP and HP denote the closure of C under products, subobjects, and Hausdorff quotients, resp. A category of TAGs is said to satisfy * DS (dually separated) if every group has a separating family of characters; * DE (dually embedded) if every character on a toplogically embedded subgroup can be extended to the full group; * DC (dually closed) if every closed (topologically embedded) subgroup is the intersection of the kernels of all the characters that vanish on it. Equivalently if the quotient is DS. Proposition: HSPC is closed under products, subobjects, and Hausdorff quotients. Theorem 1: Let C have finite products and satisfy DS, DE, and DC. Then so do PC, SC, and HC. Theorem 2: Let C satisfy DS, DE, and DC. Then so does the category of all groups of the form A x D, where A is in C and D is discrete. It follows from the first talk that HSPC in that case has Mackey coreflections. Examples will be given. Tuesday, 3 October 2000 2:30 - 4:00 Claudia Casadio Some applications of bilinear logic to linguistics Tuesday, 10 October 2000 2:30 - 4:00 M Hebert Injectivity and orthogonality classes: recent results Abstract: Let K be a full subcategory of a locally l-presentable category (l any infinite regular cardinal). Then * K is a l-injectivity class iff it is closed under products, l-filtered colimits and l-pure subobjects (iff it is axiomatizable by regular sentences in L(l,l)) * (l> w) K is a l-orthogonality class iff it is closed under limits and l-filtered colimits (iff it is axiomatizable by limit sentences in L(l,l)) (This is known to be false for l =w). Here l-injective (l-orthogonal) means injective (orthogonal) w.r.t. a set of morphisms between l-presentable objects. Other related results will be recalled. Details and proofs can be found in the following manuscripts/papers available on request (or on the web where linked): * [H] Hebert, M., Lambda*-injectivity + special compactness = lambda-injectivity, 1999. * [H] Hebert, M., Purity and injectivity in accessible categories, JPAA 129 (1998), 143-147. * [HR] Hebert, M., Rosicky, J., Uncountable orthogonality is a closure property, to appear in Bull. London Math. Soc. * [HAR] Hebert, M., Adamek, J., Rosicky, J., More on orthogonality in locally presentable categories, to appear in Cahiers Top. Geom. Diff. Categ. Tuesday, 17 October 2000 2:30 - 4:00 M Barr Derived functors using effacements Abstract: It is well known that Tor, for example, can be defined using flat resolutions instead of projective. However, the usual proof involves starting with a projective resolution which can be mapped to the flat resolution and then comparing them. I got curious about defining Tor using flat resolutions even in the absence of projective resolutions. For example in module categories in a topos, objects have flat resolutions but not usually projective resolutions. I actually explore this for a right exact functor T: A --> C, using what I call effaceable objects. Tuesday, 24 October 2000 2:30 - 4:00 M Barr Derived functors using effacements II Tuesday, 31 October 2000 2:30 - 4:00 M Bunge The relative pure-entire factorization for geometric morphisms Abstract: \documentclass[12pt]{article} \title{The Relative Pure-Entire Factorization for Geometric Morphisms} \author{Marta Bunge} \begin{document} \maketitle \begin{abstract} \vspace{5mm} A locale $A$ in a topos $\cal E$ is said to be a \emph{Stone locale} if it is a compact and zero-dimensional locale. An equivalent description says that $A$ is the locale of ideals of a Boolean algebra in $\cal E$. A geometric morphism ${\varphi}\colon {\cal F}\to{\cal E}$ is called \emph{entire} (respect. \emph{pure}) if $\varphi$ is localic and defined by a Stone locale (respect. if ${\varphi}_{*}(2_{\cal F}) {\cong} (2_{\cal E})$, where $2 = 1 + 1$). In [P.T.Johnstone, {\it Factorization Theorems for Geometric Morphisms II, Categorical aspects of Topology and Analysis}, Springer, {\bf LNM 915} (1982) 216-233] it is shown that every geometric morphism $\varphi$ admits a unique factorization $\varphi \cong {\psi}\cdot{\pi}$ where $\psi$ is entire and $\pi$ is pure. Suppose now that there is a base topos $\cal S$ over which the toposes are defined and that we only consider geometric morphisms ``over $\cal S$''. Also suppose that instead of $2 = 1 + 1$ in the above, we take the object $\Omega_{\cal S}$ of truth-values in the topos $\cal S$. The question that we answer here is the following: under what conditions on ${\varphi}\colon {\cal F} \to {\cal E}$ (over $\cal S$) does one obtain a relativized version of the pure-entire factorization mentioned above. In the process of answering it in [M. Bunge, J. Funk, M. Jibladze, T. Streicher, {\it Relative Stone Locales}, in preparation], we encounter several interesting versions of well-known notions, constructive versions of classically known results, and new problems. \end{abstract} \end{document} Tuesday, 7 November 2000 2:30 - 4:00 M Bunge Do free distribution algebras exist? Abstract: \documentclass[12pt]{article} \title{Do Free Distribution Algebras Exist?} \author{Marta Bunge} \begin{document} \maketitle \begin{abstract} \vspace{5mm} If $\cal S$ is an elementary topos and $\Omega_{\cal S}$ is its subobjects classifier, then the adjoint pair $F \dashv U$ given by ${\Omega_{\cal S}}^{(-)} \dashv \Omega_{\cal S}^{(-)} \colon {\cal S}^{op}\to {\cal S}$ is tripleable (R.Par\'e, {\it Colimits in Topoi}, {\sl Bulletin of the AMS} {\bf 80}(1974) 556-561). Moreover, there exists an equivalence between ${\cal S}^{op}$ and the category of \emph{complete atomic Heyting algebras} in $\cal S$ (over $\cal S$, the latter equipped withy the forgetful functor and its left adjoint -- the \emph{free} complete atomic Heyting algebra functor). In [M.Bunge, J. Funk, M. Jibladze, T. Streicher, {\it Distribution Algebras}, to appear in {\sl Advances in Mathematics}{\bf 156}(2000)] we prove a \emph{relative} version of this result with an interesting interpretation in terms of distributions and their algebraically duals. This is done by replacing $\cal S$ by a topos $\cal E$ bounded over $\cal S$, and by replacing ${\cal S}^{op}$ by the category of $\cal S$-valued distributions on $\cal E$ in the sense of [F.W. Lawvere, {\it Extensive and Intensive Quantities}, Lectures at Aarhus University Workshop, 1983]. However, we are seemingly forced to make a hypothesis on $\cal E$ as a topos over $\cal S$ for the tripleableness to hold. The tripleableness question is in fact only dependent on the existence of a left adjoint to the forgetful functor from the category of \emph{distribution algebras} in $\cal E$ to $\cal E$, that is on the existence of \emph{free distribution algebras}. The theorem holds for any topos $\cal E$ which is an essential localization of a presheaf topos, as well as when the base topos $\cal S$ is {\it Set}. The question itself as well as related matters will be discussed in this talk. \end{abstract} \end{document} Tuesday, 14 November 2000 2:30 - 4:00 M Zawadowski Model completions and propositional logics (Abstract on the web page) Tuesday, 21 November 2000 2:30 - 4:00 M Zawadowski Model completions and propositional logics II (Abstract on the web page) Tuesday, 28 November 2000 2:30 - 4:00 M Zawadowski Model completions and propositional logics III Tuesday, 5 December 2000 2:30 - 4:00 M Zawadowski Model completions and propositional logics IV COFFEE as usual after the first talk PLACE: BURNSIDE HALL 920, McGILL UNIVERSITY =================================================== (Any comments, suggestions to rags@math.mcgill.ca)