Category Theory Octoberfest
Concordia University, Montreal
Saturday - Sunday, October 4 - 5, 2008

M Barr
Title: Duality of Z-groups
Abstract: A Z-group is a group that can be embedded in a cartesian power of Z and a topological Z-group is one that can be embedded topologically and algebraically in a power of Z. If D and C denote the categories of Z-groups and topological Z-groups, resp., then Hom(-,Z): CD is adjoint on the left to Hom(-,Z): DC. According to a well-known theorem of Lambek & Rattray, the fixed categories for each composite are equivalent (dual in this case, since the functors are contravariant). We identify the fixed groups of C as the ones that are a kernel of a map between powers of Z and the fixed groups of D as those that have a pure embedding (meaning the cokernel is torsion-free) into a power of Z.
A Joyal
Title: On sifted colimits and bicompact objects
Abstract: We show that a functor between cocomplete categories preserves sifted colimits iff it preserves directed colimits and reflexive coequalisers. This is in fact a special case of a more general theorem: a functor between cocomplete quasi-categories preserves sifted colimits iff it preserves directed colimits and colimits of simplicial objects. We obtain a new characterisation of varieties of (homotopy) algebras.

V Harnik
Title: Placed Composition in Higher Dimensional Categories
Abstract: Given a collection of cells in a higher dimensional category, we have two formal languages in which we can describe other cells that can be obtained from them with the help of composition operations. One, based on the ordinary composition operations, can describe all such cells. The other, based on the so called placed composition operations, seems preferable in some respects, but can describe only some such cells. When made into free mathematical structures, these languages become computads and multitopic sets and can be compared to each other.
The subject is relevant to the problem of defining the concept of weak higher dimensional category.
This is joint work with Michael Makkai and Marek Zawadowski.

C Hermida
Title: Coherence for Lax and Pseudo-algebras revisited: Universality
Abstract: The coherence theorem for pseudo-algebras is shown here as freely forcing (strict) algebras to be transportable under equivalence. The precise formulation involves (free) relative 2-fibrations: the forgetful 2-functor from pseudo-algebras to the base is the free Equ-fibration over the forgetful 2-functor for strict algebras. We exhibit the corresponding result for lax algebras, involving adjoint retracts instead of equivalences, making explicit the prominent role of maps in this context.
  1. C. Hermida, Some properties of Fib as a fibred 2-category, JPAA (1999) 59(1):1-41.
  2. C. Hermida, From coherent structures to universal properties, JPAA (2001) 165(1):7-61.
  3. S. Lack, Homotopy-theoretic aspects of 2-monads, arXiv:math.CT/0607646 (2006).

P Hofstra
Title: From Poset to Quantifier.
Abstract: We introduce a category of based posets, which carries a monoidal structure. The monoid objects in this category correspond to notions of quantification. This viewpoint allows concise conceptual proofs of the fact that various fibrational constructions, such as the Dialectica and the Girard constructions, are monadic. We'll also explain how this gives a categorical account of Skolemization.

J Kennison
Title: Spectra for Symbolic Dynamics

T Kusalik
Title: The Continuum Hypothesis in Topos Theory and Algebraic Set Theory
Abstract: In 1972, Lawvere and Tierney adapted Cohen's "forcing" proof of the independence of the Continuum Hypothesis to a topos-theoretic framework. The essence of this proof is to construct, from a given Boolean topos satisfying the Axiom of Choice, a new topos which will satisfy the same as well as the Continuum Hypothesis. I will discuss a generalization of this proof to the case in which the given topos is not necessarily Boolean, and some of the new considerations which have to be dealt with once the Boolean assumption is dropped. I will also discuss how this proof generalizes to the case of Algebraic Set Theory (developed by Joyal, Moerdijk, Awodey and others), in which the "mathematical universes" in question are no longer topoi, but are categories of classes. While the intuitionistic relative consistency result that I prove is in some sense actually weaker than Lawvere and Tierney's original result, consistency results for stronger intuitionistic systems follow from my result as a corollary, but not from Lawvere and Tierney's.

Rory Lucyshyn-Wright
Title: Domains as Algebras of a Lax Monad: Towards an Integrated Lax-Algebraic Presentation of Domain-Theoretic Topology
Abstract: Seal's description of topological spaces as lax algebras of the filter monad (which follows Barr's for the ultrafilter monad) yields a particularly effective interplay of order-theoretic, topological, and algebraic concepts. The presence of the continuous lattices as the algebras of this monad suggests the possibility of a presentation which would effectively integrate all three aspects of these objects. Moreover, their role in this setting suggests a possible connection with domain-theoretic topology and even a possibility of applying lax-algebraic techniques to domain theory. Indeed, very recent work to be presented now shows that the topological spaces described by the strict algebras of a monotone-relational extension of this monad are those which satisfy a familiar and fundamental domain-theoretic approximation property. Consequently, the sober spaces among these are in fact precisely the continuous dcpos under the Scott topology. This not only strongly confirms the above intuitions and opens a door to lax-algebraic study of domain-theoretic topology, it also frames continuity in general dcpos and its natural topological generalization as a manifestation of the same essential algebraic nature possessed by continuous lattices. That we may isolate such a property and recover a description of a well-known class of domain-theoretic objects in this manner is compelling evidence of the flexibility, suitability, and elegance of this particular lax-algebraic framework as an integrated means of studying the confluence of order and topology in connection with algebraic concepts.

G Lukács
Title: A cartesian closed category that might be useful for higher-type computation
Abstract: A map f:XY between Hausdorff spaces is said to be k-continuous if the restriction f|K of f to every compact subspace K of X is continuous. The space X is a k-space if every k-continuous function from X to a Hausdorff space is continuous. The category kHaus of Hausdorff k-spaces and their continuous maps has two useful properties: It is cartesian closed, and it is a coreflective subcategory of Haus, the category of Hausdorff spaces and their continuous maps.
A T1 space X is zero-dimensional if it admits a base consisting of clopen sets, or equivalently, if X embeds into a power of 2={0,1}. Recently, Schroeder showed that the exponential NNN (when calculated in kHaus) is not Tychonoff, and thus it is not zero-dimensional.
In this talk, a cartesian closed category consisting of zero-dimensional spaces, which contains an entire hierarchy N, NN, NNN,…, is presented. It is shown that this hierarchy is equivalent to the Kleene-Kreisel one.
  1. Martín Escardó and Reinhold Heckmann. Topologies on spaces of continuous functions. In Proceedings of the 16th Summer Conference on General Topology and its Applications (New York), volume 26, pages 545-564, 2001/02.
  2. M.H. Escardo. Exhaustible sets in higher-type computation. Log. Methods Comput. Sci., 4(3), 2008.
  3. G. Lukács. A convenient subcategory of Tych, Appl. Categ. Structures 12 (2004), no. 4, 369-377.
  4. M. Schroeder. The sequetial topology on NNN is not regular, preprint.

J Morton
Title: 2-Vector Spaces and Finite Groupoids
Abstract: In this talk I will describe an explicit construction of a weak 2-functor from a bicategory of spans of groupoids into Kapranov-Voevodsky 2-vector spaces. Time permitting I will discuss some applications to topology.

S Niefield
Title: Par-valued lax functors and exponentiability
Abstract: We know that the category of Set-valued functors on a small category B is a topos. Replacing Set by a bicategory B whose objects are sets, we consider the category Lax(B,B) of B-valued lax functors on B with map-valued oplax transformations as morphism. Since this category is rarely even cartesian closed (e.g., Lax(B,Span) is equivalent to Cat/B), the usual exponentiability questions arise. In this talk, we consider the case where B is the partially-ordered 2-category Par whose morphisms are partial maps. In particular, we characterize exponentiable objects of Lax(B,Par), by establishing its equivalence with a subcategory of Cat/B. Moreover, since the latter is itself a slice category, a simple translation gives a characterization of exponentiable morphisms, as well.

D Pronk
Title: Translation Groupoids and Orbifold Homotopy Theory
Abstract: Orbifolds (originally introduced as V-manifolds by Satake) are paracompact spaces with an atlas which exhibits the local structure as the orbit space of the action of a finite group on Euclidean space. An orbifold is called representable if it can be presented as the orbit space of a manifold by the action of a compact Lie group. A large class of orbifolds, including all orbifolds for which the local groups act effectively, is known to be representable and some people have conjectured that all orbifolds are representable.
Orbifolds can be represented by smooth étale groupoids with a proper diagonal, where two such étale groupoids represent the same orbifold if and only if hey are Morita equivalent. Together with Ieke Moerdijk, I introduced a notion of maps between orbifolds that was obtained by taking the bicategory of fractions of the category of orbifold groupoids with respect to Morita equivalences. This notion of map is the right one to study orbifold homotopy theory, and allows one to consider the sheaf cohomology of an orbifold. For this reason some people have called such maps "good maps".
It is not hard to see that an orbifold is representable precisely when it can be represented by a smooth translation groupoid. In this talk I will show that good maps between representable orbifolds can be considered as spans of equivariant maps between translation groupoids. I will also discuss the exact shape of equivariant Morita equivalences.
This prepares the way to consider invariants coming from equivariant homotopy theory for orbifolds. Such invariants are orbifold invariants precisely when they are invariant under Morita equivalence. In this talk we will consider several examples of such invariants.
This is joint work with Laura Scull (UBC).

G Seal
Title: Kock-Zöberlein monads from monads on SET
Abstract: The down-set monad on ORD can be seen as a structured version of the powerset monad on SET; similarly, the open filter monad on TOP is strongly reminiscent of the filter monad on SET. These parallels are supported by the fact that corresponding monads yield the same category of Eilenberg-Moore algebras. In fact, such situations can be generalized by investigating monoids in the hom-sets of the corresponding Kleisli category: one can construct "structured" versions of certain monads on SET, thus obtaining monads of Kock-Zöberlein type.

W Tholen
Title: Towards an enriched understanding of Hausdorff and Gromov metrics
Abstract: The Hausdorff metric for subsets of a metric space has seen renewed interest in recent years, especially through the work of Gromov who used it in order to introduce a distance between compact metric spaces. By providing finitely generated groups with a metric and considering converging sequences of the emerging metric spaces he proved an important existence theorem for nilpotent subgroups of such groups. Bill Lawvere has mentioned repeatedly the need for us to better understand the V-categorical meaning of Hausdorff and Gromov distances. In this talk we will tackle this task when V is just a quantale, hoping that this case will also shed light on more general situations. To this end we will show that the Hausdorff metric gives rise to an interesting monad H on V-Cat; furthermore, once ob(V-Cat) has been made into a V-category itself via the Gromov construction, H (and other functors) become V-functors. We also give categorical characterizations of both the Hausdorff and the Gromov constructions.
Joint work with Andrei Akhvlediani and Maria Manuel Clementino

M Warren
Title: Types and groupoids
Abstract: In this talk we discuss connections between Martin-Lof's intensional type theory, homotopy theory and higher-dimensional category theory. In particular, we describe a new model of type theory using strict omega-groupoids. Using this model we are able to answer a previously open question, due to Hofmann and Streicher, regarding the higher-dimensional structure of the identity type construction in type theory.

N Yanofsky
Title: On the algorithmic informational content of categories.
Abstract: With Kolmogorov complexity theory, researchers define the algorithmic informational content of a string as the length of the shortest program/Turing machine that describes the string. After a brief review of some features of Kolmogorov complexity theory we present the rudimentary beginnings of a programming language that can be used to describe categories, functors and natural transformations. With this in hand we define the informational content of these categorical notions as the shortest such program. Some basic consequences of our definition are presented and we show that our definition is a generalization of Kolmogorov complexity theory of strings.