Category Theory Octoberfest
Concordia University, Montreal
Saturday  Sunday, October 4  5, 2008
Abstracts
 M Barr
 Title: Duality of Zgroups
Abstract:
A Zgroup is a group that can be embedded in a cartesian power of Z
and a topological Zgroup is one that can be embedded topologically
and algebraically in a power of Z. If D and C denote the
categories of Zgroups and topological Zgroups, resp., then
Hom(,Z): C → D is adjoint on the left to
Hom(,Z): D → C.
According to a wellknown 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 torsionfree) 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 quasicategories 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 Pseudoalgebras revisited: Universality
Abstract:
The coherence theorem for pseudoalgebras is shown here as
freely forcing (strict) algebras to be transportable under
equivalence. The precise formulation involves (free) relative
2fibrations: the forgetful 2functor from pseudoalgebras to the base
is the free Equfibration over the forgetful 2functor 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.
References
 C. Hermida, Some properties of Fib as a fibred
2category, JPAA (1999) 59(1):141.
 C. Hermida, From coherent structures to universal
properties, JPAA (2001) 165(1):761.
 S. Lack, Homotopytheoretic aspects of 2monads,
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
Abstract
 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 topostheoretic 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 LucyshynWright
 Title: Domains as Algebras of a Lax Monad: Towards an Integrated
LaxAlgebraic Presentation of DomainTheoretic 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 ordertheoretic, 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 domaintheoretic topology and even a possibility of applying
laxalgebraic techniques to domain theory. Indeed, very recent work to
be presented now shows that the topological spaces described by the
strict algebras of a monotonerelational extension of this monad are
those which satisfy a familiar and fundamental domaintheoretic
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
laxalgebraic study of domaintheoretic 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 wellknown class of domaintheoretic objects in this
manner is compelling evidence of the flexibility, suitability, and
elegance of this particular laxalgebraic 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
highertype computation
Abstract:
A map f:X → Y between Hausdorff spaces is
said to be kcontinuous if the restriction
f_{K} of f to every compact subspace
K of X is continuous. The space X is a
kspace if every kcontinuous function from
X to a Hausdorff space is continuous. The category
kHaus of Hausdorff kspaces 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 T_{1} space X is zerodimensional 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 N^{NN} (when
calculated in kHaus) is not Tychonoff, and thus it is not
zerodimensional.
In this talk, a cartesian closed category consisting of zerodimensional
spaces, which contains an entire hierarchy N,
N^{N},
N^{NN},…, is presented. It
is shown that this hierarchy is equivalent to the KleeneKreisel one.
References

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 545564, 2001/02.

M.H. Escardo.
Exhaustible sets in highertype computation.
Log. Methods Comput. Sci., 4(3), 2008.

G. Lukács.
A convenient subcategory of Tych,
Appl. Categ. Structures 12 (2004), no. 4, 369377.

M. Schroeder.
The sequetial topology on
N^{NN}
is not regular, preprint.
 J Morton
 Title: 2Vector Spaces and Finite Groupoids
Abstract:
In this talk I will describe an explicit construction of a weak 2functor from a
bicategory of spans of groupoids into KapranovVoevodsky 2vector
spaces. Time permitting I will discuss some applications to topology.
 S Niefield
 Title:
Parvalued lax functors and exponentiability
Abstract: We know that the category of Setvalued 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 Bvalued lax functors on B with mapvalued 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 partiallyordered 2category 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 Vmanifolds 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: KockZöberlein monads from monads on SET
Abstract: The downset 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 EilenbergMoore algebras. In fact, such situations can be generalized
by investigating monoids in the homsets of the corresponding Kleisli
category: one can construct "structured" versions of certain monads on
SET, thus obtaining monads of KockZöberlein 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 Vcategorical 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 VCat; furthermore, once ob(VCat) has been made into a Vcategory
itself via the Gromov construction, H (and other functors) become Vfunctors. 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 MartinLof's
intensional type theory, homotopy theory and higherdimensional category
theory. In particular, we describe a new model of type theory using
strict omegagroupoids. Using this model we are able to answer a
previously open question, due to Hofmann and Streicher, regarding the
higherdimensional 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.