Categories in Algebra, Geometry and Mathematical Physics
Conference in honour of Ross Street's sixtieth birthday
July 1116 2005, Macquarie University, Sydney
Conference Abstracts

Fri, 15 July: 9:00  10:00
Higher Gauge Theory (I)
Baez, John (University of California, Riverside)Gauge theory describes the parallel transport of point particles using the formalism of connections on bundles. In both string theory and loop quantum gravity, point particles are replaced by 1dimensional extended objects: paths or loops in space.
This suggests that we seek some kind of `higher gauge theory' that describes the parallel transport as we move a path along a path of paths. To find the right mathematical formalism for this, it seems we must `categorify' concepts from differential geometry, replacing smooth manifolds by smooth categories, Lie groups by Lie 2groups, Lie algebras by Lie 2algebras, bundles by 2bundles, sheaves by stacks or gerbes, and so on.
In this talk we provide an overview of work so far on this topic, with an emphasis on open problems and some tantalizing relationships between Lie 2groups and central extensions of loop groups.
\smallskip
See \url{http://math.ucr.edu/home/baez/street/} for talk transparencies and links to further reading material.

Thu, 14 July: 17:00  18:00
The Mysteries of Counting: Euler Characteristic Versus Homotopy Cardinality
Baez, John (University of California, Riverside)We all know what it means for a set to have 6 elements, but what sort of thing has $1$ elements, or $5/2$ ?
These questions have nice answers. The Euler characteristic of a space is a generalization of cardinality that admits negative integer values, while the homotopy cardinality is a generalization that admits positive real values. These concepts shed new light on basic mathematics. For example, the space of finite sets turns out to have homotopy cardinality $e$, and this explains the key properties of the exponential function. Euler characteristic and homotopy cardinality share many properties, but it's hard to tell if they are the same, because there are very few spaces for which both are welldefined. However, in many cases where one is welldefined, the other may be computed by dubious manipulations involving divergent series  and the two then agree!
\par\noindent
The challenge of unifying them remains open.
\smallskip
See \url{http://math.ucr.edu/home/baez/counting/} for talk transparencies and links to further reading material.

Tue, 12 July: 10:30  11:10
Iterated wreath product of the simplex category and iterated loop spaces
Berger, Clemens (Université de Nice  Sophia Antipolis)Certain small categories may serve to establish an explicit link between topological data and algebraic data. In this vein, Graeme Segal constructed infinite loop spaces out of special presheaves on $\Gamma$, while Bob Thomason studied the analogous construction of simple loop spaces out of special presheaves on the simplex category $\Delta$. The forgetful functor which associates to an infinite loop space the underlying simple loop space is induced by a canonical functor from $\Delta$ to $\Gamma$.
I shall interpolate between these two constructions and construct $n$fold loop spaces out of special presheaves on the $n$fold wreath product of $\Delta$. Like above, there is a canonical functor from the $n$fold wreath product of $\Delta$ to $\Gamma$ corresponding to the obvious forgetful functor. Moreover, the inductive limit over $n$ of the $n$fold wreath products of $\Delta$ is isomorphic to André Joyal's cell category $\Theta$, which is one way of connecting iterated loop spaces to higher categorical structures. The underlying combinatorics are intimately related to the multitude of trees as described by Michael Batanin and Ross Street.

Sat, 16 July: 9:45  10:25
Using Segal categories to understand simplicial monoids and simplicial categories
Bergner, Julie (University of Notre Dame)Much as Segal groupoids (and many other constructions) have been used to study simplicial or topological groups, we use Segal categories to better understand simplicial monoids. Specifically, there is a Quillen equivalence of model categories between a model structure on simplicial monoids and a ``reduced Segal category" model structure, a result which makes extensive use of algebraic theories. Using the more general notion of a multisorted algebraic theory the analogous result holds for simplicial categories with a fixed set of objects and Segal categories with the same set in degree zero. We can then apply the ideas behind this proof to prove that there is a Quillen equivalence between a model structure on the category of all small simplicial categories and a Segal category model structure.

Thu, 14 July: 16:10  16:50
Intertwiners
Berrick, A.J. (National University of Singapore)Intertwiners can be discussed for categories in general, but here the main area of application is to monoids. For the monoid of $n\times n$ matrices over a commutative ring, this has important ramifications for the algebraic Ktheory of the ring.

Tue, 12 July: 14:00  15:00
Derived categories of toric varieties
Bondal, Alexei (Steklov Institute, Moscow)We shall use a version of the Frobenius morphism to give a description of
derived categories of coherent sheaves on some toric varieties in terms of
constructible sheaves on real tori. We shall construct full exceptional
collections of line bundles for these varieties. We will briefly outline how our picture relates to mirror symmetry for toric varieties. 
Mon, 11 July: 10:30  11:20
On semidirect products and the representability of actions.
Borceux, Francis (Université catholique de Louvain)\thanks{This talk is a selection of results coming from recent joint works with Dominique Bourn, MariaManuel Clementino, George Janelidze and Max Kelly.}
%
I consider a category $\cal V$ with semidirect products and I write $\mathsf{Act}(G,X)$ for the set of actions of the object $G$ on the object $X$, in the sense of the theory of semidirect products in $\cal V$. I investigate the representability of the functor $\mathsf{Act}(,X)$. This is a very strong property.
Examples of categories with semidirect products are the semiabelian categories, but also the categories of topological models of a semiabelian theory. And among the nonalgebraic semiabelian categories, we have the dual of the category of pointed objects in a topos.
When $\cal V$ is the category of groups, the functor $\mathsf{Act}(,X)$ is represented by the group $\mathsf{Aut}(X)$ of automorphisms of $X$. When $\cal V$ is the category of Lie algebras, this functor is represented by the Lie algebra $\mathsf{Der}(X)$ of derivations of $X$. When $\cal V$ is the category of Boolean rings, or of commutative von Neumann regular rings, the functor is represented by the ring $\mathsf{End}(X)$ of $X$linear endomorphisms of $X$, and so on.
A representability criterion for $\mathsf{Act}(,X)$ is established in the case where $\cal V$ is semiabelian, locally presentable, with finite limits commuting with filtered colimits. This contains all categories of models of a semiabelian theory in a Grothendieck topos. For such categories, the representability of $\mathsf{Act}(,X)$ reduces to the preservation of binary coproducts. A very simple necessary condition and a very simple sufficient condition can be given in terms of amalgamation properties. The precise form of the more involved ``if and only if'' amalgamation property is related to a new notion of ``normalization of a morphism''.
The study of normal coverings in the category of pointed objects of a topos $\cal E$ gives as corollary  when these coverings exist  the representability of the functor $\mathsf{Act}(,X)$ for every object $X$ of the dual of the category of pointed objects of $\cal E$. This is the case for a wide class of toposes, which contains all boolean toposes and all toposes of presheaves.
Finally the case of topological models of a semiabelian theory is investigated and transfer results from the $\mathsf{Set}$ case to the topological case are established. 
Wed, 13 July: 9:00  9:40
Dbranes and generalized geometry
Bouwknegt, Peter (Australian National University)In this talk I will review how the category of $D$branes in topological
$A$ and $B$ models can be understood in the context of Hitchin's generalized geometry.

Thu, 14 July: 15:20  16:00
Invariants of symmetric monoidal 3categories
Breen, Lawrence (Université Paris 13)In this talk, I will discuss commutativity conditions
for monoidal 2 and 3categories, and describe some invariants
arising from associated geometric structures. 
Fri, 15 July: 14:00  14:40
An intrinsic characterization of branched coverings
Bunge, Marta (McGill University, Montreal)\thanks{This is joint work with Jonathan Funk.}
%
In my lecture I will present a result within the rich theory of singular maps of toposes known as complete spreads~\bibref{1}. A geometric morphism with locally connected domain is a spread if it is zerodimensional, and it is complete if cogerms of components converge. There is an equivalence~\bibref{2} between complete spreads over any $S$bounded topos and Lawvere $S$valued distributions, for any base topos $S$. This equivalence is part of an adjunction of the comprehensive type which is analogous to the comprehensive factorization of StreetWalters~\bibref{3}, but which is relative to the symmetric monad~\bibref{4}.
In topology, complete spreads were introduced in~\bibref{5} in order to deal with branched coverings in a topological rather than a combinatorial manner. A geometric morphism over a topos $E$ (with locally connected domain) is a branched covering if it is the spread completion of a locally constant covering of a slice topos $E/U$ for some pure subobject $U$ of $1$ in $E$. This definition  given independently in~\bibref{6} and~\bibref{7}, is inspired by that given in~\bibref{5} for topological spaces.
We now seek to give an intrinsic characterization of branched coverings as a subclass of the complete spreads. To this end, we introduce some new notions in what follows. A monomorphism in a topos is said to be {\em pure} if the constant object $2$ has the sheaf property with respect to every pullback of the monomorphism. We call {\em purely skeletal} any geometric morphism whose inverse image functor preserves pure monomorphisms. The notion of a {\em purely locally constant covering} is obtained from that of a locally constant covering~\bibref{8} by allowing the splitting object to have a pure (but not necessarily global) support. \\
\begin{theorem}
A geometric morphism (with locally connected domain) is a branched covering iff it satisfies the following three conditions:
\begin{enumerate}[(i)]
\item it is a complete spread
\item it is a purely locally constant covering
\item it is purely skeletal.
\end{enumerate}
\end{theorem}
Examples showing the independence between the three notions appearing in the characterization above will be given.
\begin{references}
\bibitem M. Bunge and J. Funk, {\em Toposes, distributions, and complete spread maps}, Book. Submitted February 2005.
\bibitem M. Bunge and J. Funk, {\em Spreads and the symmetric topos}, J. Pure Appl. Alg. 113 (1996) 128.
\bibitem R. Street and R.F.C. Walters, {\em The comprehensive factorization of a functor}, Bull. Amer. Math. Soc. 79 (1973) 936941.
\bibitem M.Bunge and A. Carboni, {\em The symmetric topos}, J. Pure Appl. Alg. 105 (1995) 233249.
\bibitem R. H. Fox, {\em Covering spaces with singularities}, in Algebraic Geometry and Topology : A Symposium in Honor of S. Lefschetz, Princeton University Press, 1957, 243257.
\bibitem M. Bunge and S. B. Niefield, {\em Exponentiability and single universes}, J. Pure Appl. Alg. 148 (2000) 217250.
\bibitem J. Funk, {\em On branched covers in topos theory}, Theory and Applications of Categories 7 (2000) 122.
\bibitem M. Bunge and S. Lack, {\em Van Kampen theorems in toposes}, Advances in Mathematics 179 (2003) 291317.
\end{references}

Mon, 11 July: 16:30  17:20
Homotopy theory of small functors over large categories
Chorny, Boris (University of Western Ontario)We consider the category of small simplicial functors from a large indexing category to the category of simplicial sets. If the indexing category is cocomplete we construct a model category structure on this object with weak equivalences and fibrations being objectwise. We discuss some applications of this model category.
If the indexing category is $\cal S^{\mathrm{op}}$, then we identify, through the Quillen equivalence, the category of small simplicial contravariant functors from simplicial sets to simplicial sets with the category of maps of spaces equipped with the equivariant model structure. 
Thu, 14 July: 14:00  15:00
Batanin weak higher groupoids and homotopy types
Cisinski, DenisCharles (Université Paris 13)The operadic approach of M.\ Batanin and R.\ Street to higher category theory in terms of trees allows us to define in a precise way the notion of weak higher category and of weak higher groupoid (higher category in which the $n$arrows are invertible up to higher homotopies).
Moreover, Batanin constructed a functor from topological spaces to higher groupoids and conjectured that with a suitable notion of weak equivalences between weak higher groupoids, this should define an equivalence of categories between the homotopy category of CWcomplexes and the homotopy category of weak higher groupoids.
Using constructions and results of C.\ Berger on the homotopy theory of (variants of) Joyal cellular sets, it is possible to show a first result in that direction: the homotopy type of any topological space can be reconstructed from its associated weak higher groupoid.

Fri, 15 July: 15:45  16:25
Loop Groups and Lie 2Algebras
Crans, Alissa (Loyola Marymount University)A major result in differential geometry is Ado's theorem, which states that every finitedimensional Lie algebra comes from some Lie group. We seek to prove a higherdimensional analog of this statement for Lie 2groups and Lie 2algebras, which are categorified versions of Lie groups and Lie algebras where we have replaced the associative law and Jacobi identity, respectively, by natural isomorphisms called the `associator' and the `Jacobiator'. While this problem remains open, we describe progress toward a solution by examining the Lie 2algebras $\mathfrak{g}_k$, each having the simple Lie algebra $\mathfrak{g}$ as its Lie algebra of objects, but with a Jacobiator proportional to a real number $k$ and built from the Killing form. It seems that except for $k=0$, there does not exist a Lie 2group whose Lie 2algebra is isomorphic to $\mathfrak{g}_k$. However, we can construct for integral values of $k$ an infinitedimensional Lie 2group whose Lie 2algebra is equivalent to $\mathfrak{g}_k$. Moreover, these Lie 2groups are closely related to the KacMoody central extensions of loop groups!

Sat, 16 July: 11:30  12:10
$n$fold operads in iterated monoidal categories
Forcey, Stefan (Tennessee State University, Nashville)This presentation explains how the iterated generalization of
braidings and symmetries on a monoidal category allows a highly
structured generalization of the operads that live in one. After a
brief recounting of the meaning of iterated monoidal category, I
introduce a new family tree of simple combinatorial (and pictorial!)
examples. Then I define $n$fold operads and their algebras, and give
examples which correspond to the just mentioned categories. Finally
there is a discussion of the way that these operads might be helpful in
the description of higherdimensional weakly enriched iterated monoidal
categories.

Tue, 12 July: 15:20  16:00
Conformal field theory and Frobenius algebras in modular tensor categories
Fuchs, Jürgen (Karlstads Universitet, Sweden)A class of Frobenius algebras in modular tensor categories provides the right tools for proving the existence of twodimensional rational conformal quantum field theories. For an algebra $A$ in this class, the construction used in the proof provides in particular a dictionary between algebraic structures and physical concepts, part of which looks as follows:\\[1ex]
\newcommand{\PBS}[1]{\let\temp=\\#1\let\\=\temp}
\begin{tabular}{>{\PBS\raggedright\hspace{0pt}}m{13em}@{\quad}p{1em}@{\quad}>{\PBS\raggedright\hspace{0pt}}p{10.5em}}
category of left $A$modules &= &boundary conditions \\[.5ex]
category of $A$bimodules &= &defect lines \\[.5ex]
bimodule morphisms between
certain induced bimodules &= &bulk fields \\[.5ex]
Picard group of the
bimodule category &= &internal symmetries
\end{tabular}\\[1ex]
Moritaequivalent algebras yield the same conformal field theory; the Brauer group of the modular tensor category describes the subset of modular invariant partition functions that arise from an automorphism of the fusion rules. 
Thu, 14 July: 9:00  10:00
Lie theory for $L_\infty$algebras
Getzler, Ezra (Northwestern University, Evanston)The Deligne groupoid is a homotopy functor from nilpotent differential graded Lie algebras concentrated in positive degree to groupoids. We generalize this to a functor from nilpotent $L_\infty$algebras concentrated in degrees $(n, \infty)$ to $n$groupoids, or rather, to their nerves. On restriction to abelian differential graded Lie algebras, i.e. chain complexes, our functor is the DoldKan functor. The construction uses methods from rational homotopy theory, and gives a generalization of the CampbellHausdorff formula.
If time permits, we will discuss the generalization to the nonnilpotent case. 
Thu, 14 July: 10:30  11:10
Gerbes of chiral differential operators
Gorbunov, Vasiliy (Kentucky University)In this talk we will introduce the gerbe attached to a compact algebraic or complex manifold whose characteristic class is the first Pontriagin class of the manifold. This gerbe is closely related to the socalled `half twisted sigma model' on a manifold introduced by Witten in the late 80s. We mention some applications of this construction to topology.

Fri, 15 July: 11:20  12:00
Associativity Problems in SemiAbelian Categories
Janelidze, George (Cape Town University and Tbilisi Mathematical Institute)A pointed category with finite coproducts is called semiabelian \bibref{11} if it is Barr exact and Bourn protomodular. The main result in \bibref{11} asserts that a category is semiabelian if and only it satisfies all of the socalled old axioms. Here ``all old axioms" in fact means ``all firstorder exactness axioms, invented before Barr exactness, that hold in all varieties of $\omega$groups (= groups with multiple operators in the sense of P.\ Higgins \bibref{7})". However, on the one hand there are semiabelian varieties of universal algebras far away from $\omega$groups (see \bibref{4}), and on the other hand there are useful stronger axioms that do not necessarily hold in all semiabelian categories but hold in all important varieties of $\omega$groups, including groups, rings, (modules or) algebras over rings, and Lie algebras. The aim of this talk is to discuss these axioms and their relationship with associativity in concrete algebraic cases; a number of open problems will be stated  hoping that their solution will lead to the discovery of a convincing associativity axiom. The discussion will refer to the following papers:
\begin{references}
\bibitem F.\ Borceux, G.\ Janelidze, and G.\ M.\ Kelly, {\em Internal object actions}, Commentationes Mathematicae Universitatis Carolinae, accepted
\bibitem D.\ Bourn, {\em Commutator theory in strongly protomodular categories}, Univ.\ Littoral Preprint 208, 2004
\bibitem D.\ Bourn and G.\ Janelidze, {\em Extensions with abelian kernels in protomodular categories}, Georgian Math.\ Journal 11, 4, 2004, 645654
\bibitem D.\ Bourn and G.\ Janelidze, {\em Characterization of protomodular varieties of universal algebras}, Theory Applications of Categories 11, No 6, 2003, 143147
\bibitem M.\ Gerstenhaber, {\em A categorical setting for the Baer extension theory}, Proceedings of Symposia in Pure Mathematics 17, American Math.\ Soc., Providence, RI, 1970, 5064
\bibitem M.\ Gran, {\em Applications of categorical Galois theory in universal algebra}, Fields Institute Communications 43, 2004, 243280
\bibitem P.\ J.\ Higgins, {\em Groups with multiple operators}, Proc.\ London Math.\ Soc.\ (3) 6 1956, 366416
\bibitem S.\ A.\ Huq, {\em Commutator, nilpotency and solvability in categories}, Quart.\ J.\ Math.\ Oxford (2) 19, 1968, 363389
\bibitem G.\ Janelidze, {\em Internal crossed modules}, Georgian Math.\ Journal 10, 1, 2003, 99114
\bibitem G.\ Janelidze and L.\ Márki, {\em Radicals of rings and pullbacks}, Journal of Pure and Applied Algebra 97, 1994, 2936
\bibitem G.\ Janelidze, L.\ Márki, and W.\ Tholen, {\em Semiabelian categories}, Journal of Pure and Applied Algebra 168, 202, 367386
\bibitem G.\ Janelidze and M.\ C.\ Pedicchio, {\em Pseudogroupoids and commutators}, Theory and Applications of Categories 8, No 15, 2001, 405456
\bibitem S.\ Mac Lane, {\em Duality for groups}, Bulletin of American Math.\ Soc.\ 56, 1950, 485516.
\bibitem L.\ Márki and R.\ Wiegandt, {\em Remarks on radicals in categories}, Lecture Notes in Mathematics 962, Springer, 1982, 190196
\bibitem M.\ C.\ Pedicchio, {\em A categorical approach to commutator theory}, Journal of Algebra 177, 1995, 647657
\bibitem G.\ Orzech, {\em Obstruction theory in algebraic categories I}, Journal of Pure and Applied Algebra 2 1972, 287314
\end{references}

Mon, 11 July: 14:45  15:25
BiHeyting toposes and essential covers
Johnstone, Peter (Cambridge University)Recent work of Francis Borceux and Dominique Bourn has highlighted the
importance of (Grothendieck) toposes in which subobject lattices are
coHeyting algebras as well as Heyting algebras. Clearly, if a topos admits
an essential surjection from a Boolean topos, then it has this property.
We investigate the extent to which the converse holds. 
Mon, 11 July: 9:00  10:00
The theory of quasicategories (I)
Joyal, André (Université du Québec à Montréal)We shall describe a few salient points of the theory of quasicategories, stressing the similarities and the differences with category theory. For example, the theories of limits and Kan extensions are formally identical. The quasicategory HOT is the primary example of an $\infty$topos.
We introduce the notion of an absolutely exact quasicategory, in which the equivalence relations are general groupoids. The notion may capture one of the main difference between HOT and the category of sets. The theorem of Kan on the equivalence between the homotopy category of pointed connected spaces and the category of group objects in TOP has the following generalisation: in any exact quasicategory the full quasicategory of pointed connected objects is equivalent to the quasicategory of group objects.
Absolutely exact quasicategories abound. The quasicategory of models of any algebraic theory is absolutely exact. Here an algebraic theory is defined to be a quasicategory with finite products. Models of algebraic theories can be taken iteratively. A group object in the quasicategory of group objects is a braided group object or a 2fold loop space, etc.
There is also a notion of category object in any left exact quasicategory. A category object is said to be reduced if its groupoid of isomorphisms is trivial (this is related to the notion of complete Segal space introduced by Charles Rezk). Every category object in HOT can be reduced (more generally, every category object of an absolutely exact quasicategory). The quasicategory of reduced categories in HOT is equivalent to QCAT, the quasicategory of small quasicategories. More generally, there is a notion of reduced $n$category object for every $n$. The quasicategory of reduced $n$categories in HOT is equivalent to the quasicategory of quasi$n$categories properly defined. $\Theta$sets can be used to model quasi$n$categories. 
Tue, 12 July: 9:00  10:00
Noncommutative Fourier transform, Chen's iterated integrals and higherdimensional holonomy~(I)
Kapranov, Michael (Yale University)We set up a framework for a noncommutative version of
the Fourier transform which relates functions of noncommuting variables
and ordinary functions on the space of unparametrized paths. It is
based on Chen's analogs of exponential functions that are
generating functions of his iterated integrals. Then we explain how to
extend this correspondence to represent higherdimensional membranes
by elements of a certain differential graded algebra $A$. This is
related to the concept of holonomy of gerbes that attracted
a lot of attention recently. We will also give the interpretation
of higher gerbe holonomy in terms of Chen's iterated integrals
of forms of higher degree with coefficients in Liealgebraic
analogs of crossed modules and crossed complexes. If one views higher
holonomy as a ``pasting integral" then 2dimensional associativities
translate into vanishing of some brackets in the structure dgLie
algebra which is automatic in the crossed module case but has to be
imposed in general. 
Sat, 16 July: 10:45  11:25
Mirror Symmetry for manifolds of general type
Katzarkov, Ludmil (University of California, Irvine)In this talk we will demonstrate how ideas from category theory
can be used to approach problems from low dimensional topology. 
Mon, 11 July: 11:30  12:00
Ross Street and the early days of Category Theory in Australia
Kelly, Max (Sydney University) 
Sat, 16 July: 9:00  9:40
The affine TemperleyLieb category  applications to representation theory, operator algebras and physics
Lehrer, Gus (Sydney University)I shall explain how to construct a category with objects whose
endomorphisms form algebras called the affine TemperleyLieb algebras.
They may be analysed in the spirit of cellular algebras, and the
homomomorphisms between their functorially defined cell modules are
natural transformations. Arising from this are explicit computations of
discriminants of canonical forms, and other information about the
modules. If time permits I shall indicate some applications to affine
Hecke algebras, Jones' planar algebras and the study of phase changes in
physics.

Mon, 11 July: 15:45  16:25
A general theory of selfsimilarity
Leinster, Tom (Glasgow University)I will present the beginnings of a general theory of selfsimilarity.
In principle this concerns selfsimilar anythings, but I will focus on
the topological case.
Consider a selfsimilar object $X$. Typically, $X$ looks like several
copies of itself glued to several copies of another object $Y$, and $Y$
looks like several copies of itself glued to several copies of $X$or the
same kind of thing with a larger family $X_1, X_2, \ldots$ of objects.
Now observe:
%
\begin{enumerate}
\item This is a system of simultaneous equations in which the usual
algebraic operations have been replaced by gluing. The gluing
instructions can be regarded as `higherdimensional formulas'.
\item Often such a system has a canonical solution $X_1, X_2, \ldots$
In this case, the $X_i$s  which may be highly complex fractal
spaces  are determined by a system of algebraic equations.
\end{enumerate}
%
There is a clean and simple formalism for such systems of
simultaneous equations. An explicit criterion determines whether
a system has a canonical solution; if so, an explicit construction
gives it.
Many wellknown selfsimilar spaces arise as examples. One of the
simplest is the real interval $[0, 1]$; indeed, the starting point of the
theory was Freyd's characterization of $[0, 1]$ as a terminal coalgebra. 
Wed, 13 July: 9:45  10:25
Generalized bialgebras and triples of operads
Loday, JeanLouis (Université de Strasbourg)The classical PoincaréBirkhoffWitt theorem and CartierMilnorMoore theorem can be viewed as a rigidity result on cocommutative Hopf algebras intertwining the operads Com, As, and Lie of commutative algebras, associative algebras and Lie algebras respectively. Recently, in joint work with M. Ronco, we proved a similar rigidity result intertwining the operads As, 2as, and $B$infinity, where 2as is the operad of algebras with 2 associative operations. We will present a general program aimed at discovering rigidity results for triples of operads $(C,A,P)$, where $C$ (resp.\ $A$, resp.\ $P$) is the operad governing the coalgebra structure (resp.\ the algebra strucutre, resp.\ the primitives).

Fri, 15 July: 14:45  15:25
Triangulated Derivators
Maltsiniotis, Georges (Université Paris 7)The aim of the theory of derivators is to introduce the ``good'' framework for homotopical and homological algebra. Triangulated derivators are more particularly concerned with homological algebra. Soon after the introduction of triangulated categories by Verdier, it was observed that this notion was insufficient to take hold of all the structures carried by the derived category of an abelian one. In particular the derived category with its structure of triangulated category does not satisfy a universal property. Moreover homotopical limits or colimits in a triangulated category are defined only up to a non unique isomorphism. In particular the cone is not functorial.
In his thesis Bernhard Keller proved a universal property of the derived category of an abelian, or even an exact, category by considering ``towers of triangulated categories''. Instead of looking only to the derived
category of the abelian one, he considers the derived categories of the categories of cubical diagrams. More generally, fix for example a Grothendieck category $\mathcal{A}$. For every small category $I$, the category of presheaves on $I$ with values in $\mathcal{A}$ is still a Grothendieck category, and one can consider its derived category $\mathbf{D}(I)$. The derived category of $\mathcal{A}$ is just $\mathbf{D}(e)$ ($e$ being the final category). For every functor $u:I\to J$ between small categories the inverse image of presheaves induces a functor $u^*:\mathbf{D}(J)\to\mathbf{D}(I)$, and every natural transformation $\alpha:u\to v$ induces a natural transformation $\alpha^*:v^*\to u^*$. This defines a contravariant $2$functor from the $2$category $\mathcal{C}at$ of small categories to the $2$category $\mathcal{C}AT$ of all categories. The idea of Grothendieck is that it is this $2$functor, considered as a structure on the derived category $\mathbf{D}(e)$, that carries all ``the triangulated structure'' of the derived category.
A triangulated derivator is a 2functor $\mathbf{D}$ defined on a full $2$subcategory $\mathcal{D}iag$ of $\mathcal{C}at$ (satisfying some stability properties) with values in $\mathcal{C}AT$ satisfying a list of axioms. Some of these axioms are similar to those defining the notion of ``homotopy theory'' of Alex Heller. The most important is the one saying that for every functor $u:I\to J$ in $\mathcal{D}iag$, the functor $u^*:\mathbf{D}(J)\to\mathbf{D}(I)$ has a left adjoint $u^{}_!$ and a right adjoint $u^{}_*$. Some other axioms are close to those considered by Jens Franke in his study of systems of triangulated diagram categories. The basic theorem of the theory of triangulated derivators is that for every category $I$ in $\mathcal{D}iag$, $\mathbf{D}(I)$ is a triangulated category, and that for every functor $u:I\to J$ in $\mathcal{D}iag$, $u^*$ defines a triangulated functor. Bernhard Keller has proved that to every exact category $\mathcal{E}$ one can associate a triangulated derivator $\mathbf{D}_{\mathcal{E}}$ defined on finite direct categories such that $\mathbf{D}_{\mathcal{E}}(e)$ is the derived category of $\mathcal{E}$.
One can define a Ktheory of triangulated derivators. It is conjectured that if $\mathcal{E}$ is an exact category, the Ktheory of the triangulated derivator $\mathbf{D}_{\mathcal{E}}$ coincides with the one of the exact category $\mathcal{E}$, and that the Ktheory of triangulated derivators satisfies some additivity and localization theorems. Amnon Neeman has a plan for proving the additivity conjecture. 
Wed, 13 July: 11:30  12:10
On modular categories and group actions on braided tensor categories
Mueger, Michael (Radboud University, Nijmegen)Modular categories as defined by Turaev some ten years ago are a fairly special but also extremely interesting class of braided tensor categories with connections to quantum and loop groups, number theory, and topological and conformal quantum field theory. After recalling the definition and some more or less wellknown results I will discuss two open conjectures concerning modular categories: the congruence subgroup conjecture and the modular embedding conjecture.
I will then discuss actions of discrete groups on (more general) braided tensor categories and their bearing on the conjectures mentioned above. This is work in progress and I will summarize the state of affairs.

Wed, 13 July: 10:45  11:25
Bimodules over operads: encoding deep structure of morphisms
Scott, Jonathan (École Polytechnique Fédèrale de Lausanne)\thanks{Joint work with K.\ Hess (EPFL) and P.E.\ Parent (U. Ottawa).}
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An operad $P$ determines a category of $P$algebras and their morphisms. The algebras and morphisms are linked, in the sense that in order to alter the morphisms, one must alter $P$, changing the algebras in the process.
We will show how $P$bimodules can be used to give more freedom in describing morphisms. Let $R$ be a $P$bimodule. If $R$ is a $P$cooperad, then the category of $P$algebras and ``$R$relative" morphisms is a full subcategory of the Kleisli category for the triple associated to $R$. We apply this idea to unravel the algebraic structure in morphisms of bar/cobar constructions.
As a concrete example, we construct a bimodule over the associative operad of chain complexes that yields the category of associative algebras and ``strongly homotopymultiplicative" morphisms.

Fri, 15 July: 10:30  11:10
Loop Groups and Categorified Geometry
Stevenson, Danny (University of California, Riverside)\thanks{Joint work with John Baez, Alissa Crans and Urs Schreiber.}
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I will describe aspects of this joint work on the construction of a 2group associated to the KacMoody central extension of the loop group. I will explain how this construction can be understood in terms of `categorified bundles', i.e.\ gerbes and 2bundles. Finally I will show how the 2group we construct is related to the group $\mathrm{String}(n)$ which plays an important role in the work of Stolz and Teichner on elliptic objects. 
Fri, 15 July: 16:30  17:20
Torsion theories in seminormal categories
Tholen, Walter (York University)\thanks{Joint work with M. M. Clementino and D. Dikranjan.}
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The notion of semiabelian category combines two powerful algebraic properties, Barrexactness and Bournprotomodularity, and provides, among other things, an appropriate axiomatic setting for homological algebra of noncommutative structures. The category of topological groups fails to be semiabelian since equivalence relations fail to be effective, but it is still Barrregular and Bournprotomodular. Clementino and Borceux called such pointed categories `homological', and more recently Bourn and Gran showed that they provide a good environment for the study of torsion theories.
In this talk we shall present a generalized and relativized notion of semiabelian category, which captures certain properties of categories such as the category of topological groups that are lost when we view them just as homological categories. Hence we give a general setting that allows for a smooth treatment of torsion theories, which we characterize in terms of radicals and closure operators. Examples well beyond the realm of semiabelian or even homological categories are also presented.

Tue, 12 July: 11:20  12:00
Swiss cheese as the categorical ether
Voronov, Alexander (University of Minnesota)We show that the Swisscheese operad may serve as the ``$n$categorical ether'', bookkeeping the composition structure for morphisms of different levels.

Mon, 11 July: 14:00  14:40
Wellsupported compact closed categories of processes
Walters, Robert F.C. (Università dell'Insubria)In \bibref{1,3} the notion of wellsupported compact closed categories was introduced  symmetric monoidal categories for which every object has a separable algebra structure.
More recently it was shown in \bibref{2} that the category whose arrows are cospans of $A$labelled graphs between finite sets is the generic symmetric monoidal category with a separable algebra which has an $A$family of actions. Such a category we regard as being a simple example of a category of processes with sequential operations.
In this lecture we would like to describe other examples, both syntactic and semantic, with behaviour functors between them. We will describe also parallel operations and distributive laws on such categories of processes.
\begin{references}
\bibitem A.\ Carboni, {\em Matrices, relations and group representations}, J.\ Algebra, 138:497529, 1991.
\bibitem R.\ Rosebrugh, N.\ Sabadini, R.F.C.\ Walters, {\em Symmetric separable algebras in monoidal categories and $\mathrm{Cospan(Graph)}$}, Abstracts of the International Category Theory Conference, CT'04, Vancouver 2004.
\bibitem R.F.C.\ Walters, {\em The tensor product of matrices}, Lecture, International Conference on Category Theory, LouvainlaNeuve, 1987.
\end{references}

Thu, 14 July: 11:20  12:00
Categorical Deformation Theory, Vassiliev Theory and Other Topological Invariants
Yetter, David (Kansas State University)We discuss the relationship between the deformation theory of monoidal categories, monoidal functors and monoidal natural transformations, both in relation to the classical algebraic deformation theory of Gerstenhaber and to its application to Vassiliev's finitetype invariants of classical knots and links.
Extensions to other topological applications in both three and four dimensions will be discussed.