




Category Theory / Théorie des catégories (R. Paré, Organizer)
 MICHAEL BARR, McGill University, Montreal, Quebec H3A 2K6
Absolute homology

Call two maps, f,g: C ® C¢, of chain complexes absolutely
homologous if for any additive functor F, the induced Ff and Fg
are homologous (induce the same map on homology). It is known that the
identity is absolutely homologous to 0 iff it is homotopic to 0
and tempting to conjecture that f and g are absolutely homologous
iff they are homotopic. This conjecture is false, but there is an
equational characterization of absolute homology. I also characterize
left absolute and right absolute (in which F is quantified over left
or right exact functors).
 RICK BLUTE, University of Ottawa, Ottawa
Constructing *autonomous categories

We begin by considering a new example of a *autonomous category due to
T. Ehrhard, called the category of ``finiteness spaces''. This is at
once a generalization of the category of sets and relations, and the
category of coherence spaces. One essentially adds a notion of
orthogonality to the usual category of sets and relations, and only
considers morphisms respecting that orthogonality. The resulting
category carries the full structure of a Seely model.
This idea of adding a notion of orthogonality to monoidal categories
seems quite general, and should lead to a number of new models of
linear logic. We consider several notions of such orthogonality,
including one applied to a category of formal distributions, first
defined by the author and P. Panangaden.
 M. BUNGE, Department of Mathematics and
Statistics, McGill University, Montréal, Québec H3A 2K6
A Van Kampen theorem for Grothendieck toposes

This work is part of a larger program of developing some aspects of
algebraic topology and functional analysis for Grothendieck toposes
regarded as generalized topological spaces. The principal motivation
for this particular paper was to provide an interpretation, in
topostheoretic terms, of (the calculation of) knot groups via branched
coverings, as advocated by R. H. Fox (1957) in topology. To this end,
we prove that the 2category of Grothendieck toposes (by which we
mean toposes bounded over a given base topos S), is extensive in
an appropriate (bicategorical) sense, thus generalizing the wellknown
extensivity (A. Carboni, S. Lack and R. C. F. Walters, 1993) of the
category of topological spaces. This is then coupled with a
bicategorical version of a (Seifert)Van Kampen theorem of R. Brown and
G. Janelidze (1997) involving descent, that we first prove and then
apply to various fibrations of quantities (intensive, extensive, and of
mixed type) on Grothendieck toposes. In the case of the fibration of
covering projections over a topos (or of locally constant objects in a
topos), the passage to a Van Kampen theorem for group(oid)s, as in
topology, is effected by means of the construction (M. Bunge 1991) of
the (Grothendieck, or ``coverings'') fundamental groupoid of an
unpointed locally connected topos bounded over an arbitrary base toposd
S.
(This is joint work with Stephen Lack, University of Sydney,
Australia.)
 R. COCKETT, University of Calgary, Calgary, Alberta
Games as polarized categorical proof theory

Categories of games are usually employed as the basis for providing a
semantics for a free category carrying a specified structure. They are
particularly attractive in this regard as their maps are combinatoric
objects. This means that, in simple cases at least, the inequality of
maps can be concretely exhibited.
Of course, any category has a combinatoric nature when its maps have a
concrete normal form. What is interesting is that often a game theory
(with a concrete normal form) can be extracted from a standard proof
theory (which does not have a nice normal form) by ``polarizing'' the
propositions and forcing the proofs to be viewed as interactions
between the two polarities.
The talk will discuss some examples of this and the polarized
categorical semantics which results.
 PIERRE DAMPHOUSSE, Universite Francois Rabelais, Faculte des sciences, Parc de
Grandmont, 37200 Tours, France
Groupe des automorphismes des la categorie des
Gensembles / The automorphisms of the category of Gsets

Nous décrivons à équivalence naturelle près le groupe des
automorphismes de la catégorie des Gensembles (et des applications
Ginvariantes) pour n'importe que groupe donné G, et nous
utilisons cette description pour obtenir une action naturelle de S_{3}
sur les cartes cellulaires.
We describe the group of automorphisms (up to natural equivalence)
of the category of Gsets (and Ginvariant maps) for any given
group G, and we use this description to describe a natural action
of S_{3} on cellular decompositions of a surface.
 R. DAWSON, St. Mary's University, Halifax, Nova Scotia B3H 3C3
Adjoining adjoints

The wellknown Span construction adds right adjoints to a category,
freely except for the Beck condition. If right adjoints are added
completely freely a different construction arises which we will call
P_{2}. A special case of this, for the category 2, was computed by
Schanuel and Street in 1986. This talk presents recent work by Robert
Pare, Dorette Pronk, and the speaker, in which the the general
construction is studied.
P_{2}(C) is a 2category whose objects are those of C, whose
arrows are zigzags of arrows of C, and whose 2cells are
represented by certain diagrams that we call ``fences''. Fences do not
obey the middlefour property, but appropriate equivalence classes of
them do; these equivalence classes are the actual 2cells. Various
basic properties will be presented; in particular, it is shown that the
equivalence relation on fences may be undecidable.
 J. DUSKIN, Department of Mathematics, S.U.N.Y. at Buffalo,
New York 142602900, USA
Nerves of bicategories and SimpsonTamsamani weak 2categories

In [Duskin 2002] the author gave a detailed definition of a
simplicial set Ner(B) which completely encoded the
structure of a weak 2category B in the commonly
accepted sense of a bicategory as defined in [Bénabou 1967].
Ner(B) was a subcomplex of its 2Coskeleton and was
isomorphic to its 3Coskeleton, what we called a 2dimensional
Postnikov complex, and satisfied certain very weak ``Kanlike'' horn
filling properties provided that that certain 2 and 1simplex
faces which occurred in the horns were members of a subset of
(abstractly) invertible 2 and 1simplices. We proved there
that those 2dimensional Postnikov complexes in which (1) all
degenerate 2simplices are invertible and in which (2) all horns
(x_{2}[( x_{12})  (¬ )]x_{1} [( x_{01})  (¬ )]x_{0}) Î Ù_{2}^{1}(X_{·}) have an an invertible 2simplex
filler, are exactly the nerves of bicategories. (Those in which
every 2 and 1simplex is invertible are exactly the nerves of
bigroupoids, i.e., bicategories in which every 2cell is an
isomorphism and every 1cell is an equivalence).
In this short talk we will compare this result with the double
simplicial set characterization theorem which is the specialization of
the multisimplicial approach to weak ncategories of
[Simpson 1997] and [Tamsamani 1995] as described in
[Leinster 2002]. In Ner(B), each of the sets of simplices
forms the set of objects of a natural category structure for which the
face and degeneracy maps become functorial and the 0simplices become
the objects of a discrete category. The corresponding canonical Segal
maps all admit right (in our orientation) adjoint sections and become
equivalences provided one restricts in each dimension to the full
subcategories whose nsimplex objects all have isomorphisms as their
2cell interiors. The resulting bisimplicial set is a weak
2category in the sense of SimpsonTamsamani. Conversely, given a
bisimplicial set C_{·} which is a weak 2category in the
sense of SimpsonTamsamani, one can construct a 2dimensional
Postnikov complex in which the objects of the category C_{2} @ C_{1}×_{C0} C_{1} form the set of invertible 2simplices and makes it
the nerve of a bicategory. Thus the two definitions are effectively
equivalent. In contrast to what the author asserted in the introduction
to [Duskin 2002], the SimpsonTamsamani definition does
not force the resulting bicategory to have all of its 2cells
isomorphisms. The assertion was the result of an mistaken
interpretation the SimpsonTamsamani definition.
References
 [Bénabou 1967]
 J. Bénabou, Introduction to
Bicategories. Reports of the Midwest Category Seminar, Lecture Notes
in Math. 47(1967), 177.
 [Duskin 2002]
 J. Duskin, Simplicial
Matrices and the Nerves of Weak nCategories I: Nerves of Bicategories.
Theory and Applications of Categories (10) 9(2002), 198308.
 [Leinster 2002]
 Tom Leinster, A Survey of
Definitions of nCategories, Theory and Applications of Categories (1)
10(2002), 170.
 [Simpson 1997]
 Carlos Simpson, Effective
generalized SeifertVan Kampen: how to calculate
WX. xxx.lanl.gov/qalg/9710011 v2, 19 Oct 1997.
 [Tamsamani 1995]
 Z. Tamsamani, Sur des
Notions de nCatégorie et nGroupoid NonStrict via des Ensembles
MultiSimpliciaux. xxx.lanl.gov/alggeom/95102006 v2. 15 Dec 1995.
 P. FREYD, University of Pennsylvania, Philadelphia, Pennsylvania,
USA
Algebraic real analysis

The coalgebraic definition of the real closed interval gives rise,
somewhat paradoxically, to an algebraic definition: the theory of
``interval algebras'' is given by a constant, a pair of unary
operations, a binary operation and a finite set of equations; every
nontrivial interval algebra has a simple quotient; the full category
of simple interval algebras has a terminator, to wit, the real closed
interval. Since every model of any consistent extension of the
algebraic theory of the closed interval will therefore have a map back
to the closed interval a ``completeness theorem'' of remarkable power
is obtained. (The von Neumann integral for any compact group, as an
example, can be obtained as a consequence.) And the complete theory of
all continuous selfmaps of the closed interval can be given an
entirely syntactical characterization. (That complete theory includes a
good part of differential calculus.)
 JONATHON FUNK, University of Saskatchewan
More results on branched covers in topos theory

The application of branched covers to braid group orderings in [3] has
led to a continued investigation of branched covers in topos theory [1,
2]. Lawvere's topos distributions, pure geometric morphisms, and
complete spread geometric morphisms provide a framework by which we may
examine branched covers. One aspect that we consider is a comparison
of the notions of pure and complete spread (or closure) relative to
different base toposes. A fact about this comparison can be
paraphrased as follows: an object of a locally connected topos
E over another topos S is Eclosed
in its Sclosure. We comment on some consequences of this
fact.
References
 [1]
 M. Bunge and S. Niefield, Exponentiability and
single universes. J. Pure Appl. Algebra 148(2000), 217250.
 [2]
 J. Funk, On branched covers in topos theory.
Theory Appl. Categ. (1) 7(2000), 122.
 [3]
 , The Hurwitz action and braid group
orderings. Theory Appl. Categ. (7) 9(2001), 121150.
 D. GARRAWAY, Colby College, Waterville, Maine, USA
Generalized supremumenriched categries and their sheaves

In this talk we explore taxons enriched in the monoidal category of
complete lattices and the associated laxsemifunctors and
laxtransformations. We first show that for Q a
suptaxon the semicategory lax_{inf}(Q^{co},rel)
is equivalent to the semicategory of bimodules on Q An
immediate corollary then is that the category of Qvalued
sets is equivalent to the category
map(tax_{inf}(Q^{co},rel) ). From this it
follows that for H, a complete Heyting algebra, the
category of sheaves on H is equivalent to the category
map(tax_{inf}(H^{co},rel)). Thus a sheaf
on a complete Heyting algebra is a structure preserving semifunctor.
 A. JOYAL, UQAM
Kan complexes and quasicategories

A quasicategory is a simplicial set satisfying the restricted Kan
conditions of Boardman and Vogt. A quasicategory X has an associated
homotopy category hX. We show that X is a Kan complex iff hX is a
groupoid. The result plays a an important role in the theory of
quasicategories. We apply it to study initial objects in a
quasicategory. We introduce limits and colimits of diagrams in a
quasicategory.
 F. LINTON, Wesleyan University, Middletown, Connecticut 06459, U.S.A
On associativity for tensor products in varieties

One may introduce multilinear maps in the setting of any concrete
category, and speak of tensor products of sequences of objects there as
universal solutions to the problem of ``linearizing'' multilinear
maps. Classical experience (abelian groups, modules over fixed
commutative rings, pointed sets, etc.) leads to the expectation
that iterated binary tensor products should associate, and therefore
serve as sequential tensor products. For varieties that are entropic,
by which is meant that each defining operation is a homomorphism in the
variety, this expectation is indeed fulfilled. That impediments arise
in general, though, is seen already in the variety of (associative)
semigroups. There is, however, an astonishing plethora of examples of
varieties in which, despite the failure of entropy, associativity
nonetheless prevails. What little we know regarding the success or
failure of associativity in absence of entropy will be shared here.
 J. MACDONALD, University of British Columbia, Vancouver, British
Columbia V6T 1Z2
Embedding of algebras

We look at some general conditions for determining when the unit of an
adjunction is monic when the generated monad is defined on a category
of algebras. Among other possible applications these conditions will be
shown to provide a common approach to some classic embedding theorems
in algebra.
 M. MAKKAI, McGill University, Montreal, Quebec H3A 2K6
Comparing definitions of weak ncategory for low values of n

The main topic is the comparison of the Gordon/Power/Street definition
of tricategory with the multitopic 3category (Hermida/Makkai/Power
and Makkai) and with the 3dimensional theta category (Joyal). The
2dimensional case, which is wellunderstood, will be discussed for
the purposes of introduction. Partial results in other dimensions will
be mentioned.
 S. NIEFIELD, Union College
Homotopy pullbacks, lax pullbacks, and exponentiability

We consider a unified approach to homotopy pullbacks, lax pullbacks,
and pseudopullbacks via the notion of an Epullback, where E is an
object of a category E with finite limits. It turns out
that Epullbacks of objects over B exist in E if and
only if the exponentials B^{E} exist in E.
The goal of the talk is to discuss Eexponentiable morphisms of
E, i.e., morphisms for which a certain Epullback
functor has a right adjoint. In particular, we will consider the
relationship between Eexponentiability and ordinary exponentiability
of morphisms of E, and apply these results to the
categories of topological spaces, small categories, and Grothendieck
toposes.
 DORETTE PRONK, Department of Mathematics and Statistics, Dalhousie University,
Halifax, Nova Scotia B3H 3J5
Free extensions of double categories

The syntax of double categories is more complicated than the syntax for
categories. Free extensions, being the ``polynomials'' of the double
category, provide a valuable way of studying this. In this talk I
discuss joint work with Robert Dawson and Robert Pare on free
extensions of double categories by cells and arrows. I will give
conditions under which there are normal forms for cells and arrows in
the resulting double category. I will also show that if those
conditions are not satisfied, equality for cells in the new double
category may be undecidable. An application of this relating to the
work by Brown and Mosa on symmetric double categories and connections
will be discussed.
 ROBERT ROSEBRUGH, Mount Allison
Updating views universally

A new treatment of view updates for the `sketch data model' (a semantic
data model based on certain mixed sketches) will be presented. A
database state is a model M of such a sketch E. Database
systems require support for views, since view support allows
application programs to continue to operate unchanged when the
underlying system is modified. Fully supporting views depends on a
solution to the view update problem: when can a view update be
propagated to the full database? We define a view to be a sketch
morphism K :V ® E and a view
database N to be the result of substitution of a model M along
K. We say that a view database is updatable if it enjoys a
universal property for substitution along K. We explain why this
property gives a complete and correct treatment of view updatability.
and determines a solution to the view update problem. We also show
that, in various contexts, updatability can be guaranteed independently
of the view database.
This is joint work with Michael Johnson.
 PHIL SCOTT, University of Ottawa, Ottawa, Ontario
Yoneda techniques and decision problems

The word problem for arrows in free cartesian and bicartesian closed
categories is intimately related to decision problems in typed lambda
calculi. Solving the latter involves intricate prooftheoretic
techniques (related to socalled operational semantics in computer
science). In the last decade, modeltheoretic ``reductionfree
normalization'' has been developed, which is both strikingly faster in
computing normal forms and is not based on rewriting theory at all
(Berger & Schwichtenberg, LICS 1991). Such normalization techniques
have been increasingly studied by categorists, starting with
Altenkirch, Hofmann, & Streicher (CTCS, 1995). I will report on some
recent works (e.g. Cubric, Dybjer, Scott (MSCS, 1998),
Altenkirch, Dybjer, Hofmann, Scott (LICS, 2001)) which use Yoneda and
sheaf methods to solve decision problems for free (bi)ccc's. The
methods resemble JoyalGordenPowerStreet coherence techniques, and
may be interpreted as a kind of ``reverse Lambek program'': instead of
using proof theory to solve coherence problems (a la Lambek), we use
techniques from categorical coherence theory to solve decision problems
in proof theory. If time permits, we discuss applications to other
logical systems and categorical frameworks for reductionfree
normalization via generalized logical relations.
 R. SEELY, Department of Mathematics, McGill University, Montreal,
Quebec H3A 2K6
Poly bicategories of modules

Recently, with the notions of ``poly bicategory'' and ``linear
bicategory'', we generalized several bicategorical notions to a setting
where there were two (implicit or explicit) ``horizontal''
compositions. Poly bicategories have poly 2cells which may be
regarded as arrows with strings of 1cells as domains and as
codomains; the representability of such arrows by ordinary arrows with
single object domains and codomains gives the notion of linear
bicategory. A good higher dimensional structure based on lax functors
can be built over poly (and linear) bicategories, consisting of ``poly
functors'', ``poly modules'' between them, and ``poly transformations''
between poly modules: poly functors between poly bicategories P and
P¢ together with poly modules and their transformations form a module
poly bicategory provided P is representable and closed in the sense
that every 1cell has both a left and a right adjoint (in the
appropriate linear sense). Usually constructing module (bi)categories
requires some completeness and cocompleteness; in the present setting
we can identify such conditions as necessary for the representability
of the module poly bicategory, rather than for its existence.
This is joint work with J. R. B. Cockett and J. Koslowski.
 JAVAD TAVAKOLI, Saskatchewan Indian Federated College (SIFC), University of Regina,
Regina, Saskatchewan S4S 0A2
On completion of a poset in a topos

Let P be a poset in a topos E. The canonical
topology on the object of left (right) ideals of P induces a
subobject [^(P)](\checkP). If we let [`(P)] be the MacNielle completion of P, there is an
embedding [`(P)]\hookrightarrow [^(P)] which has a
left exact left adjoint. Moreover there is an isomorphism
lim_{¬} W @ [^(P)]. There is
also c:W\hookrightarrow[^(P)] in E,
which may represent ``characteristic sheaves'' in [^(P)].
Just as
E/X \rightleftarrows E(X, W) 

is an adjunction induced by the truth value 1® W,
where the set of fixed points of the adjunction is the full subcategory
Sub_{E}(X) Ì E/X. If we replace P
by the natural numbers object N, the projection ([^(N)])_{leq}®[^(N)] will also induce an
adjunction
E/X \rightleftarrows E(X, 
^ N

). 

The fixed points E ® X of this adjunction may be called
``finite filters''.
 J. WICK PELLETIER, York University, 4700 Keele St., Toronto, Ontario M3J 1P3
Spatiality of involutive quantales

This talk on based on joint work with C. J. Mulvey. We are concerned
with finding a notion of spatiality of involutive quantales which when
applied to the spectrum MaxA of a C^{*}algebra A captures our
intuitive sense of that Gelfand quantale as a space. The approach taken
in defining such a notion is motivated by that of Giles and Kummer in
their article on noncommutative spaces [Indiana Univ. Math. J.
21(1971)]. We show that X is spatial if, and only if, it has enough
points in the sense defined earlier in terms of algebraically
irreducible representations [Mulvey, Pelletier, J. Pure Appl. Algebra
159(2001)]. A consequence is that a locale is spatial as an involutive
unital quantale precisely when it yields the topology of a classical
topological space.
 R. WOOD, Dalhousie University, Halifax
2categories of algebras for double dualization 2monads

Double dualization with respect to W, the ordered set of truth
values, provides a 2monad on ord, the 2category of ordered
sets, whose 2category of algebras is that of (constructively)
completely distributive (CCD) lattices, functions which preserve all
suprema and all infima, and inequalities. However,
(,W):ord^{op}®ord is not monadic. We show
that there is a specialization of a generalization of ordered sets for
which homming into W is 2monadic and for which the
2category of algebras is the above 2category of CCD lattices. Our
methodology is quite general and applicable to similar double
dualization monads on 2categories of the form Vcat,
for suitable monoidal V. If time permits, such examples
will be examined.

