


Category Theory
Org: Robin Cockett (Calgary) [PDF]
 MICHAEL BARR, McGill University, Montreal, QC H3A 2K6
The category of Mackey spaces is *autonomous
[PDF] 
A standard theorem says that any locally convex topological vector
space has a finer topology, its Mackey topology with the same
set of continuous linear functionals and that is the finest possible
topology with that property. If E and F are two such spaces,
topologize the space Hom(E,F) of continuous linear transformations
E® F with the weak topology induced by the algebraic tensor
product EÄF¢ and then let [E,F] denote the associated
Mackey topology. Let F^{*} denote the dual F¢ topologized by the
Mackey topology on the weak dual and let EÄF=[E,F^{*}]^{*} (whose
underlying vector space is the algebraic tensor product). Then for
any Mackey spaces E, F, and G,
 [EÄF,G] @ [E,[F,G]]
 E @ E^{*} ^{*}
 [E,F] @ (EÄF^{*})^{*}
which is summarized by saying that the category of Mackey spaces and
continuous linear transformations is *autonomous.
This category is equivalent to the category of weakly topologized
locally convex topological vector spaces (which have the coarsest
possible topology for their set of continuous linear functionals)
which is therefore also *autonomous. They are also equivalent to
the chu category of vector spaces (which will be explained).
 MARTA BUNGE, McGill University, Dept. of Mathematics and Statistics,
805 Sherbrooke St. West, Montreal, QC H3A 2K6
Michael coverings are comprehensive
[PDF] 
The motivational example for the comprehension scheme (Lawvere '68)
came from proof theory. An example with categories as types (Gray
'69, StreetWalters '73) exhibited comprehension as the familiar
Grothendieck construction of a discrete opfibration associated with a
covariant functor F : B ® Sets on a small category B.
We introduce the setting of an "extensive 2doctrine" (E2D) in which
to state the comprehension scheme in a 2categorical setting. This
involves a 2category T "of types" and, for each object X of
T, a category E(X) of "extensive quantities of type X" with a
terminal object 1_{X}, and a "pushforward operation" E(f) :E(Y) ® E(X) for each 1cell f : Y ® X in T. For each
object X of T, we have a 2functor B_{X} : (T,X) ® E(X)
that assigns, to each 1cell f : Y ® X, the extensive
quantity of type X given by E(f)(1_{Y}). We say that the E2D
satisfies the comprehension scheme if for each X, the 2functor
B_{X} has a fully faithful right 2adjoint {}_{B} : E(X) ® (T,X), called comprehension. A 1cell f : Y ® X is called
Edense if the canonical map E(f)(1_{Y}) ® 1_{X} is an isomorphism,
and it is called an Ecovering if the unit f ® {B_{X} (f)}_{X} is an
isomorphism. It follows that every 1cell f : Y ® X admits
a unique (up to iso) factorization into an Edense 1cell Y ®Z, followed by an Ecovering 1cell Z ® X. This is called
the "Ecomprehensive factorization" of f.
The purpose of this talk is:
(1) to remark that the (pure, Fox complete spread)
factorization (BungeFunk '96) is indeed comprehensive for an E2D
with T the 2category of locally connected (Grothendieck) toposes
and E(X) the category of Lawvere distributions on X, and
(2) to prove that the (hyperpure, Michael Complete spread)
factorization (BungeFunk 2005) is comprehensive for an E2D with T
the 2category of all (Grothendieck) toposes and E(X) a category of
what we call "0distributions", or distributions with values in
0dimensional locales.
This is joint work with J. Funk.
 JOHN FOUNTAIN, University of York, York YO10 5DD, UK
Proper covers of left ample monoids
[PDF] 
The relation R^{*} is defined on a monoid M by the rule
that aR^{*} b if and only if for all x,y Î M,
If the set of idempotents E(M) of M is a commutative submonoid of
M and every R^{*}class contains an idempotent, M is
said to be left adequate. In such a monoid each
R^{*}class contains a unique idempotent and the idempotent
in the R^{*}class of an element a is denoted by
a^{f}. A left adequate monoid M is left ample if ae = (ae)^{f} a for all e Î E(M) and a Î M.
Thus a right cancellative monoid is left ample; here R^{*}
is the universal relation. Every inverse monoid is left ample.
A left ample monoid is proper if the intersection of the
minimum left cancellative congruence and R^{*} is trivial.
The structure of proper left ample monoids can be described in terms
of right cancellative monoids and commutative monoids of idempotents.
Moreover, any left ample monoid M has a proper cover, that
is, a proper left ample monoid P together with a homomorphism from
P onto M which restricts to an isomorphism from E(P) onto
E(M). We consider how such covers can be constructed.
 PETER FREYD, Pennsylvania University
*autonomous structures on old categories
[PDF] 
Some anciently studied categories turn out to have *autonomous
structures not previously noted; for example, the category of finitely
generated additive groupvalued functors from finitely generated
abelian groups. This is, in fact, the free abelian category on one
object generator. As all free structures on one generator it has a
monoidal structure (it may be identified as composition of functors).
It is neither the tensor product nor the "par" but lies between
them.
 JONATHON FUNK, University of the West Indies, Cave Hill Campus
Semigroups and toposes
[PDF] 
We shall present a strictly semigroup description of the classifying
topos B(G) [2] of an inverse semigroup G. A
left *semigroup is a semigroup S together with an assignment
s® s^{*} satisfying:
(i) (s^{*})^{*} = s,
(ii) ss^{*}s = s, and
(iii) (s^{*} st)^{*} = (st)^{*} s,
for all s,t Î S. A morphism of left *semigroups is a function
h : S® T such that
(i) h(s^{*}) = h(s)^{*}, and
(ii) h(st) = h(s)h(s^{*} st).
Such a morphism h is said to be étale if every equation
t=h(f)t in T, where f is a strong idempotent (f=f^{*} f) of S,
can be lifted uniquely to an equation s=fs in S, meaning
h(s)=t.
Proposition 1
B(G) is equivalent to the category of étale morphisms of
left *semigroups over the inverse semigroup G.
We shall also present a strictly topos description of Eunitary
inverse semigroups [3]. A ØØseparated object
of a topos is one that is separated for the ØØtopology in the
topos [1]. (F is ØØseparated iff the diagonal
subobject F\rightarrowtail F×F is equal to its double
negation.)
Proposition 2
An inverse semigroup G is Eunitary iff the object d :G® E of B(G) is ØØseparated, where E=
idempotents of G, and d(t) = t^{*} t.
References
 [1]

P. T. Johnstone,
Sketches of an Elephant: A Topos Theory Compendium.
Clarendon Press, Oxford, 2002.
 [2]

A. Kock and I. Moerdijk,
Presentations of étendues.
Cahiers Topologie Géom. Différentielle Catég. (2)
32(1991), 145164.
 [3]

Mark V. Lawson,
Inverse Semigroups: The Theory of Partial Symmetries.
World Scientific Publishing Co., Singapore, 1998.
 NICOLA GAMBINO, University of Quebec at Montreal
Presheaf and sheaf models for constructive set theory
[PDF] 
I will present presheaf and sheaf models for Constructive
ZermeloFrankel set theory (CZF), analogous to the ones defined by
Dana Scott for Intuitionistic ZermeloFrankel set theory (IZF).
 PIETER HOFSTRA, Calgary

 ROBIN HOUSTON, University of Manchester, Oxford Road, Manchester M13 9PL
When coherence comes for free
[PDF] 
A monoidal category is equipped with associativity and unit
isomorphisms, subject to coherence conditions. Under a
commonlysatisfied hypothesis ("tensor generation"), these coherence
conditions are redundantin the sense that the existence of
arbitrary associativity and unit isomorphisms implies the existence of
a coherent collection of them. The same principle extends to the
symmetric case.
I shall explain the hypothesis, and describe how (when the hypothesis
is satisfied) coherent associativity, symmetry, and unit isomorphisms
may be constructed from arbitrary ones.
 ANDRÉ JOYAL, Université du Québec à Montréal
Beyond Category Theory
[PDF] 
The theory of quasicategories is extending both category theory and
homotopy theory. We shall discuss the similarities and the
differences between these theories. We shall discuss the somewhat
surprising fact that a general groupoid can be treated as an
equivalence relation. In particular, a group defines an equivalence
relation on a point; the quotient is the classifying space of the
group.
 STEVE LACK, University of Western Sydney, Locked Bag 1797, Penrith South
DC NSW 1797, Australia
Partial maps in higherdimensional categories
[PDF] 
Given a category C with a suitable class M of morphisms, one
obtains a corresponding notion of partial map, where the M's provide
the possible domains of definition. Typically the M's will be
monomorphisms.
If C has higherdimensional structure (for example if it is a
bicategory or tricategory) then new phenomena occur; among other
things, one might expect to interpret the condition that the M's be
monomorphisms in a relaxed way suitable for a bicategory or
tricategory.
In this talk I will focus on the case where C is the (2)category of
categories and where C is a certain category of bicategories. In
particular I will describe how "partial morphisms of bicategories"
are the same thing as the "2sided enrichments" of Kelly, Labella,
Schmitt, and Street.
 TOM LEINSTER, University of Glasgow, University Gardens, Glasgow G12 8QW,
UK
The Thompson groups
[PDF] 
In the 1960s, Richard Thompson (and, independently, Freyd and Heller)
discovered three groups, F, T and V, with several remarkable
properties. F, in particular, turns out to be one of those
structures that appears unexpectedly in many diverse parts of
mathematics. It also has a very natural and simple categorical
description: it is the symmetry group of the `generic idempotent
object'. I will explain what this means, how it differs from Freyd
and Heller's earlier description, and how it belongs to the large
family of existing descriptions of free categories with structure.
Joint work with Marcelo Fiore.
 DORETTE PRONK, Dalhousie University, Halifax, NS B3H 3J5
Adjoining Cycles of Adjoint Arrows
[PDF] 
The P_{2} construction [2] freely adds right adjoints to all arrows
(or a suitably chosen subset of the arrows) of a category. When one
considers this construction, one may wonder what one needs to do in
order to obtain a 2category in which every arrow has both a left and
a right adjoint. As was shown in [1], such arrows come in cycles,
either an infinite cycle or a finite cycle. In this talk we will
present a construction which freely adds specified cycles of adjoints
to classes of arrows in a category that is freely generated on a
graph. (This is our first step toward such a construction for
arbitrary categories and 2categories.) As a result we obtain
families of nontrivial examples of categories containing cycles of
arrows which are both left and right adjoints.
This is joint work with Robert Dawson and Robert Paré.
References
 [1]

P. I. Booth,
Sequences of adjoint functors.
Arch. Math. 23(1972), 489493.
 [2]

R. J. M. Dawson, R. Paré and D. A. Pronk,
Adjoining adjoints.
Adv. in Math. 178(2003), 99140.
 PEDRO RESENDE, Instituto Superior Técnico, Av. Rovisco Pais, 1049001
Lisboa, Portugal
Applications of quantale theory to groupoids and inverse
semigroups
[PDF] 
Quantales are simple algebraic structures which can be found very
often, explicitly or less so, in mathematics. They have properties
that make them analogous to rings, and similarly to rings the richer
aspects of the theory only become available when we restrict to
quantales satisfying special conditions.
In this talk we shall mainly address a class of quantales that is both
easy to describe and closely related to groupoids and inverse
semigroups, namely the socalled inverse quantale frames, which form a
category that is equivalent to the category of complete and infinitely
distributive inverse semigroups. Partly because of this equivalence,
quantales turn out to be good mediating objects for the purpose of
constructing étale groupoids from inverse semigroups (for instance
the germ groupoid of a pseudogroup, or Paterson's universal groupoid
of an inverse semigroup).
We shall examine the threefold interplay between quantales, groupoids
and inverse semigroups, and some of its known or conjectured
consequences as regards one or more of the following topics: more
general semigroups (such as guarded semigroups); more general
groupoids (such as open groupoids); generalizations of groupoid
cohomology; the structure of groupoid C^{*}algebras.
 PHIL SCOTT, University of Ottawa
Geometry of Interaction and the Dynamics of Proofs
[PDF] 
Girard's Geometry of Interaction (GoI) program develops a mathematical
modelling of the dynamics of cutelimination in prooftheory.
Girard's work (19881995, 2004) is stated in the language of
operator algebras. He gave a novel modelling of proofs, interpreting
cuts via feedback in an intrinsic theory of types, data and
algorithms. However, as emphasized by Hyland and Abramsky, there are
deep connections of GoI with the recent theory of traced monoidal
categories of JoyalStreetVerity. Indeed, traces lead to new
insights into Girard's Execution Formula, a kind of power series
representing an invariant of cutelimination. Recently, in a series
of papers, E. Haghverdi and I have reexamined the categorical
foundations of GoI. For example, we develop a typed version,
Multiobject GoI (MGoI), which includes all previous as well as several
new models. MGoI depends on a new theory of partial traces, trace
classes and an abstract theory of orthogonality (related to work of
Hyland and Schalk). I shall survey some of this recent work, along
with Soundness and Completeness Theorems for GoI semantics. If time
permits, we also explore some of the new directions in GoI.
 ROBERT SEELY, McGill University, 805 Sherbrooke St. W, Montreal QC,
H3A 2K6
Differential Categories
[PDF] 
We introduce the notion of a differential category: a
(semi)additive symmetric monoidal category with a comonad (a
"coalgebra modality") and a differential combinator, satisfying a
number of coherence conditions. In such a category, one should regard
the base maps as "linear", and the coKleisli maps as "smooth"
(infinitely differentiable). Although such categories do not
necessarily arise from models of linear logic, one should think of
this as replacing the usual dichotomy of linear vs. stable
maps established for coherence spaces.
To illustrate this approach, we give a number of examples, the most
important of which, a monad S_{¥} on the category of vector
spaces, with a canonical differential combinator, fully captures the
usual notion of derivatives of smooth maps. Our models are somewhat
more general than are allowed by other approaches (such as Ehrhard's
and Regnier's, which inspired our work). For example, differential
categories are monoidal categories, rather than monoidal closed or
*autonomous categories. This allows us to capture various
"standard models" of differentiation which are notably not closed.
Second, we relax the condition that the comonad be a "storage"
modality in the usual sense of linear logic, again so as to allow the
standard models which do not necessarily give rise to a full storage
modality. However, when the comonad is a storage modality, we can
describe an extension of the notion of differential category which
captures the notnecessarilyclosed fragment of EhrhardRegnier's
differential lcalculus.
Joint work by R. Blute, J. R. B. Cockett, and R. A. G. Seely.
 ALEX SIMPSON, University of Edinburgh, United Kingdom
Applications of Algebraic Set Theory
[PDF] 
Algebraic set theory considers the categorytheoretic structure
implicit in the notion of smallness arising from the set/class size
distinction of firstorder set theory. Hitherto, it has mainly
aroused foundational interest as a reorganization of models of (many
variants of) set theory, focusing on their algebraic structure.
However, by providing a natural environment for carrying out internal
arguments involving large structures and size distinctions, algebraic
set theory should also be a theory with applications. In this talk, I
shall attempt to explore some of the possible directions such
applications might take.
 BEN STEINBERG, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario
K1S 5B6
Mobius inversion, groupoids and inverse semigroup algebras
[PDF] 
Using Mobius inversion, we give an explicit isomorphism between the
algebra of a finite inverse semigroup and the algebra of its
underlying groupoid (which is in turn isomorphic to a direct sum of
matrix algebras over the local groups). From this one obtains a
description of the irreducible representations and a character sum
formula for calculating intertwining numbers. Applications include
explicit decompositions of tensor and exterior powers of
representations of partial permutation inverse semigroups and
calculation of the eigenvalues with multiplicities for random walks on
finite triangularizable semigroups.
 PAUL TAYLOR, University of Manchester, UK
Computable Real Analysis without Set Theory or Turing Machines
[PDF] 
The many schools of computable or constructive analysis accept without
question the received notion of set with structure. They rein in the
wild behaviour of settheoretic functions using the double bridle of
topology and recursion theory, adding encodings of explicit numerical
representations to the epsilons and deltas of metrical analysis.
Fundamental conceptual results such as the HeineBorel theorem can
only be saved by settheoretic tricks such as Turing tapes with
infinitely many nontrivial symbols.
It doesn't have to be like that.
When studying computable continuous functions, we should never
consider uncomputable or discontinuous ones, only to exclude them
later. By the analogy between topology and computation, we
concentrate on open subspaces. So we admit +, , ×,
¸, < , > , ¹ , Ù and Ú, but not £ , ³ ,
=, \lnot or Þ. Universal quantification captures the
HeineBorel theorem, being allowed over compact spaces.
Dedekind completeness can also be presented in a natural logical style
that is much simpler than the constructive notion of Cauchy sequence,
and also more natural for both analysis and computation.
Since open subspaces are defined as continuous functions to the
Sierpi\'nski space, rather than as subsets, they enjoy a "de Morgan"
duality with closed subspaces that is lost in intuitionistic set,
type or topos theories. Dual to " compact spaces is
$ over "overt" spaces. Classically, all spaces are overt,
whilst other constructive theories use explicit enumerations or
distance functions instead. Arguments using $ and overtness
are both dramatically simpler and formally dual to familiar ideas
about compactness.
 BENNO VAN DEN BERG, Universiteit Utrecht
On a realisability model for CZF
[PDF] 
Recently, in two unpublished papers, Streicher and Lubarsky have
(independently) put forward realisability models for CZF. In this
talk, I will show that the two models are the same and make clear how
their work can be understood in the context of algebraic set theory
(AST). I hope also to indicate how AST can be used to demonstrate the
validity of several principles in this model. (This is all part of the
speaker's PhD thesis.)
 JAAP VAN OOSTEN, Utrecht University, PO Box 80010, NL3508 TA The Netherlands
AST in realizability
[PDF] 
It is shown how David McCarty's wellknown realizability model for IZF
fits into Joyal and Moerdijk's framework of Algebraic Set Theory. We
shall remark on a way to eliminate the external ordinals from the
construction, as well as using recent work of M. Warren in order to
obtain prooftheoretic properties of the model by topostheoretic
means.
 MICHAEL WARREN, Carnegie Mellon University, 5000 Forbes Ave., Pittsburgh,
Pennsylvania 15213, USA
Coalgebras in a category of classes
[PDF] 
It is a basic result from topos theory that if E is an
elementary topos and G is a cartesian comonad on E, then
the category E_{G} of coalgebras for G is also an
elementary topos. We extend this result to the setting of algebraic
set theory by showing that if C is one of several kinds of
categories of classes and G is a cartesian comonad which preserves
small maps, then C_{G} is also a category of classes of the
same kind as C. We then turn to the consideration of
several useful corollaries. First, categories of classes are, under
suitable conditions, stable under the formation of internal
presheaves. Secondly, it follows that several of the set theories
considered in the literature on algebraic set theory possess the
disjunction and existence properties.
 RICHARD WOOD, Dalhousie University
Variation and Enrichment
[PDF] 
The parametrized 2category constructions Fib/S, for S with finite
limits, and Wcat, for W a bicategory, are further unified by
considering, for fixed W, the 2category of pseudofunctors H: A ® W which are locally discrete fibrations. This
2category is biequivalently described as a 2category whose objects
are laxfunctors W^{c} o ® mat, where mat is the bicategory
whose objects are sets and whose homcategories are given by
mat(X,A) = set^{AxX}. The biequivalence is a direct
generalization of the Grothendieck biequivalence between fibrations
and CATvalued pseudofunctors and is mediated by pulling back a
universal local discrete fibration mat_{*} ® mat. Further, the
2category is also biequivalent to the classical [^(]W)cat, where
[^(]W) is the bicategory whose objects are those of W with
[^(]W)(w,x) = set^{W(w,x)o p}. We will show how to recover the
usual variable and enriched categories within this framework.
Work with JRB Cockett and SB Niefield.

