Workshop on Categorical Methods in Algebra, Geometry and Mathematical Physics
Satellite to the StreetFest conference in honour of Ross Street's sixtieth birthday
July 1821 2005, Australian National University, Canberra
Workshop Abstracts

Wed, 20 July: 9:00  10:00
Higher Gauge Theory (II)
Baez, John (University of California, Riverside)Gauge theory describes how point particles transform as they trace out paths in spacetime using the formalism of connections on bundles. In higher gauge theory, we describe the parallel transport of strings as they sweep out surfaces in spacetime using ``2connections" on ``2bundles".
Ordinary gauge theory involves a groupoid in which the objects are points of spacetime and the morphisms are paths. Higher gauge theory goes further and uses a 2groupoid where the objects are points, the morphisms are paths, and the 2morphisms are ``paths of paths" which we use to describe surfaces traced out by the motion of strings. So, higher gauge theory is naturally tied to categorification.
Here we show how categorifying the concepts of bundle and connection yields the concepts of ``2bundle" and ``2connection". Just as a connection can be locally described as a Liealgebravalued 1form, a 2connection is locally described by a Liealgebravalued 1form together with a Liealgebravalued 2form. A 2connection gives wellbehaved parallel transport along paths and surfaces if a quantity called its ``fake curvature" vanishes. In this talk we sketch these ideas and discuss how 2connections on 2bundles are related to connections on nonabelian gerbes.
\smallskip
See \url{http://math.ucr.edu/home/baez/street/} for talk transparencies and links to further reading material.

Mon, 18 July: 16:40  17:20
Interpolation categories for homology theories
Biedermann, Georg (University of Western Ontario)For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions we construct a tower of interpolation categories.
These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by proving the existence of truncated versions of resolution or $E_2$model structures. Examples of functors fitting in our framework are given by every generalized homology theory represented by a ring spectrum satisfying the AdamsAtiyah condition. The constructions are closely related to the modified Adams spectral sequence and give a very conceptual approach to the associated moduli problem and obstruction theory.
As application we establish an isomorphism between certain $E(n)$local Picard groups and some Extgroups. 
Thu, 21 July: 10:10  10:50
The Galois theory of Lambdarings
Borger, James (ANU)$\Lambda$rings were introduced by Alexander Grothendieck to give a context for tackling RiemannRoch problems. It is becoming increasingly clear, however, that abstract $\Lambda$rings have a rich arithmetic nature. I will discuss what could be called the Galois theory of $\Lambda$rings and, in particular, a result with Bart de Smit that gives a complete Galoistheoretic description of the category of $\Lambda$rings that occur as the rational numbers tensor of a $\Lambda$ring that has both finite rank as an abelian group and no nonzero nilpotent elements.

Tue, 19 July: 16:15  16:55
Moore normalization and DoldKan theorem for semiabelian categories
Bourn, Dominique (Université du Littoral)The usual DoldKan theorem asserts that, if $\mathbb A$ is an abelian category, the Moore normalization functor $N : \mathrm{Simpl}\,\mathbb A \rightarrow \mathrm{Ch}\,\mathbb A$ from simplicial objects in $\mathbb A$ to chain complexes in $\mathbb A$ is an equivalence of categories. This normalization construction $N : \mathrm{Simpl}\,\mathbb C \rightarrow \mathrm{Ch}\,\mathbb C$ is clearly possible as soon as the category $\mathbb C$ is pointed and finitely complete. Certainly, it is far from being an equivalence
It is proved here that when the category $\mathbb C$ is semiabelian (such as, for instance, the category $\mathrm{Gp}$ of groups or the category $\mathrm{Rg}$ of rings) this functor $N$ is necessarily monadic. This must be thought of as the semiabelian formulation of the DoldKan theorem. The fact that the category $\mathbb C$ is semiabelian is a quasinecessary condition for this monadicity theorem. Indeed, assuming that the category $\mathbb C$ is pointed (which is necessary to define the functor $N$), the existence of the left adjoint is equivalent to the existence of coproducts in $\mathbb C$, and the fact that $N$ is conservative is equivalent to the fact that the category $\mathbb C$ is protomodular. 
Thu, 21 July: 11:50  12:30
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. 
Thu, 21 July: 16:40  17:20
The periodic table of $n$categories:\\ lowdimensional results
Cheng, Eugenia (University of Chicago)We examine the periodic table of weak $n$categories for the lowdimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to commutative monoids, and degenerate bicategories to monoidal categories; however, to understand the situation fully we should examine the totalities of such structures. Categories naturally form a 2category {\bfseries Cat}, so we can take the full sub2category of this whose 0cells are the degenerate categories. On the other hand monoids naturally form a category, but we can regard this as a discrete 2category to make the comparison. We show that this construction does not yield a biequivalence; to get an equivalence we must ignore the natural transformations and consider only the {\it category} of degenerate categories.
A similar situation occurs for degenerate bicategories. The tricategory of such does not yield an equivalence with monoidal categories; we must consider only the categories of such structures.
For doubly degenerate bicategories the situation is more subtle. The tricategory of such is not naturally triequivalent to the category of commutative monoids (regarded as a tricategory). However, in this case considering just the categories does not give an equivalence either; to get an equivalence we must consider the {\it bicategory} of doubly degenerate bicategories.
We conclude with some remarks about how the above cases might generalise for degenerate, doubly degenerate and triply degenerate tricategories, and for $n$fold degenerate $n$categories.

Thu, 21 July: 14:30  15:30
Local test categories and equivariant homotopy theory
Cisinski, DenisCharles (Université Paris 13)Grothendieck's theory of local test categories is an attempt to give a general answer to the following question: when does the category of presheaves on a small category model the homotopy category of CWcomplexes? The main well known example is the category of simplicial sets. Grothendieck conjectured that for any test category $A$ (that is a small category satisfying some nice combinatorial axioms) the category of presheaves on $A$ has a model structure Quillen equivalent with the one on simplicial sets, and that has now been proved. The relative version of this theory (also initiated by Grothendieck) gives rise to a lot of Quillen model structures on presheaf categories closely related to classical homotopy theory (or localized versions of it).
In this talk, I shall discuss how we can recover the equivariant homotopy theory (for a given presheaf of groups) from this view point. I shall also try to show how some ``higher Galois theory" yoga shows up naturally in this setting. 
Tue, 19 July: 17:00  17:40
On Lawverecompleteness for lax algebras
Clementino, Maria Manuel (Universidade de Coimbra)The notion of reflexive and transitive $(T,V)$algebra  or $(T,V)$category , for a symmetric monoidalclosed category $V$ and a lax monad $T$ in the category of $V$matrices, has been recently introduced and studied \bibref{2,3}. It comprises $V$categories, when $T$ is the identity monad, and Barr's relational algebras \bibref{1}, when $V=2$. Although it was this latter instance that raised the subject  having in mind that several interesting results of convergence structures could be generalized to other settings  the former one gives a new insight to these structures. Indeed, a considerable amount of knowledge of $V$categories can be interpreted in the general setting of $(T,V)$categories.
In this talk we will concentrate on Lawvere's notion of (Cauchy)complete $V$category. We will deal with a (possible) generalization of this concept, explore some examples and results.
\begin{references}
\bibitem M.\ Barr, {\em Relational algebras}, in: Springer Lecture Notes in Math.\ 137 (1970), 3955.
\bibitem M.M.\ Clementino and D.\ Hofmann, {\em Topological features of lax algebras}, Appl.\ Categ.\ Struct.\ 11 (2003), 267286.
\bibitem M.M.\ Clementino and W.\ Tholen, {\em Metric, Topology and Multicategory  a Common Approach}, J.\ Pure Appl.\ Algebra 179 (2003), 1347.
\bibitem F.W.\ Lawvere, {\em Metric spaces, generalized logic, and closed categories}, Rend.\ Sem.\ Mat.\ Fis.\ Milano 43 (1973), 135166.
\end{references}

Tue, 19 July: 11:30  12:10
Differential and Smooth Categories
Cockett, Robin (University of Calgary)\thanks{This is joint work with Richard Blute and Robert Seely, both
of whom join me in wishing Ross Happy Birthday!}
%
Recently T. Ehrhard and L. Regnier introduced
the ``differential $\lambda$calculus" as an abstract
syntax for differentiation. Ehrhard also introduced
semantic settings which modeled the calculus
(e.g. Finiteness and Koethe spaces).
This talk will address the categorical semantics of
differentiation in the spirit of the above. However,
we shall take the opportunity to generalize those ideas
so that standard models of differentiable functions
(from first year calculus) are included. One effect
of this generalization is to remove the necessity for
higher order constructs (often absent from simpler
models): so it seems appropriate to miss out the
$\lambda$ and say that these categories model the
``differential calculus".
A differential category is an additive category with a
coalgebra modality and a differential combinator.
A coalgebra modality is a comonad in which each cofree
coalgebra of the modality is a comonoid. The differentiable
or smooth functions are the maps in the
coKleisli category. The coKleisli category of a
differentiable category is a smooth category. We
shall show how, under suitable conditions, one can
recover an underlying differential category from
a smooth category.

Mon, 18 July: 14:30  15:30
Open/closed modular operads
Getzler, Ezra (Northwestern University, Evanston)Modular operads are a generalization of cyclic operads in which
composition is along graphs which are not necessarily trees. We may
reinterpret graphs as oriented surfaces with configurations of closed
simple curves. In this talk, we explain how the theory of open/closed
modular operads emerges if we consider oriented surfaces with
antiinvolutions which preserve the configuration of curves. 
Wed, 20 July: 16:15  16:55
Descent on 2fibrations and strongly 2regular 2categories
Hermida, Claudio (Instituto Superior Técnico, Lisbon)We consider pseudodescent in the context of 2fibrations.
A 2category of descent data is associated to a 3truncated
simplicial object in the base 2category.
A morphism $q$ in the base induces (via commaobjects and pullbacks)
an internal category whose truncated simplicial nerve induces in
turn the 2category of descent data for $q$. When the 2fibration admits direct
images, we provide the analogues of the BeckBénabouRoubaud
theorem, identifying the 2category of descent data with that of
pseudoalgebras for the pseudomonad $q^{*}\Sigma_{q}$.
We introduce a notion of \textit{strong 2regularity\/} for a 2category ${\mathcal R}$, so that its basic 2fibration of internal fibrations
${\mathit cod}:{\mathsf{Fib}}({\mathcal R})\rightarrow{\mathcal R}$ admits
direct images. In this context, we show that \textit{essentiallysurjectiveonobjects\/} morphisms,
defined by a certain lax colimit, are of effective descent by means of a Beckstyle pseudomonadicity theorem.

Thu, 21 July: 9:00  10:00
The theory of quasicategories (II), a perspective
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. 
Mon, 18 July: 9:00  10:00
Noncommutative Fourier transform, Chen's iterated integrals and higherdimensional holonomy~(II)
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. 
Mon, 18 July: 11:50  12:30
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. 
Thu, 21 July: 11:00  11:40
Weak units and homotopy 3types.
Kock, Joachim (Universitat Autònoma de Barcelona)\thanks{Joint work with A.\ Joyal.}
It is shown that every braided
monoidal category arises as $\mathrm{End}(I)$ for a weak unit $I$ in an
otherwise completely strict monoidal $2$category. This implies a
version of Simpson's weakunit conjecture in dimension $3$, namely
that oneobject $3$groupoids that are strict in all respects, except
that the object has only weak identity arrows, can model all simply
connected homotopy $3$types. The proof has a clear conceptual
content and relies on a geometrical argument with string diagrams
and configurations spaces.

Wed, 20 July: 11:30  12:10
BatalinVilkovsky classes in graph complex
Lazarev, Andrey (Bristol University) 
Mon, 18 July: 10:10  10:50
Model categories and homotopy colimits in toric topology
Panov, Taras (Moscow State University)\thanks{Homotopytheoretical aspects of toric topology is a joint project with Nigel Ray and Rainer Vogt.}
%
Since the pioneering work of Davis and Januszkiewicz, algebraic topologists have been drawn increasingly
towards the study of spaces which arise from wellbehaved actions of the torus $T^n$. Investigations are no longer confined to the properties of Davis and Januszkiewicz's toric manifolds, but have extended to related geometrical structures, such as momentangle complexes, subspace arrangements or torus manifolds of Hattori and Masuda, as well as the homotopy types of associated spaces and their rationalisations and localisations. We refer to this enlarged field of activity as {\it toric topology}.
From the viewpoint of applications in combinatorics and commutative algebra it is important to understand the topology of loop spaces of different spaces arising in toric topology. Many of these spaces (e.g., toric manifolds, momentangle complexes and their Borel
constructions) admit a simple presentation as a colimit of a certain diagram of spaces over the face category of a simplicial complex. This opens a way to construct good algebraic and topological models for the associated loop spaces by studying the behavior of the loop and classifying space functors, and their algebraic analogues,
with respect to homotopy colimits in different algebraic and topological model categories (spaces, topological monoids, noncommutative DGAs etc.). This study leads to better understanding homotopy colimits themselves in these categories, which may be of independent interest in homotopy theory. Our main applications concern Stanley's ``Combinatorial commutative algebra",
in particular, we apply our models to effective calculation of Extcohomology of StanleyReisner face rings for several important classes of simplicial complexes, as well as to different combinatorial problems
related to the numbers of faces in triangulations (known to
combinatorial geometers as {\it fvectors} of simplicial complexes).

Wed, 20 July: 10:30  11:10
Homotopy Quantum Field Theories with background a 2type, formal maps and gerbes.
Porter, Timothy (University of Wales, Bangor)\thanks{Joint work with Vladimir Turaev.}
Homotopy quantum field theories can be used to study `$d$manifolds with background $B$'. These were introduced by Turaev in 2000 but generalise ideas of Segal on Conformal QFTs. Turaev classified HQFTs for $d = 1$ when the background was a $K(G,1)$. Brightwell and Turner examined the case when $B$ is a $K(A,2)$ and Rodrigues showed that for any dimension $d$ the HQFTs only depended on the $(d+1)$type of $B$.
There is an obvious question. If we have an algebraic model for the $2$type of $B$, can we extend Turaev and BrightwellTurner to classify all such HQFTs? VT and TP have done this in terms of the crossedmodule model for the 2type and a combinatorial notion of formal map. A slightly surprising spinoff is an interpretation of these HQFTs as classifying gerbes of a suitable type, generalising $G$torsors. 
Wed, 20 July: 15:15  15:55
Localic germ groupoids of inverse semigroups
Resende, Pedro (Instituto Superior Técnico, Lisbon)Groupoids and inverse semigroups, which cater for more general notions of symmetry than groups, have many applications in algebra and geometry, and they are related in many ways  in particular, there are several constructions of topological groupoids from inverse semigroups. Based on the correspondence, which I shall recall, between localic \'{e}tale groupoids and quantales that has been established in \bibref{1}, in this talk I shall study the groupoid of germs of a pseudogroup, showing that its construction can be extended to any inverse semigroup whose idempotents form a frame, yielding a localic groupoid whose spectrum is, in the case of, say, the pseudogroup of partial homeomorphisms of a Hausdorff space, the usual topological germ groupoid. The construction we provide can be carried over to an arbitrary topos (immediately yielding, for instance, a $G$equivariant construction via an interpretation in the topos of $G$sets), and the germ groupoid obtained is universal in the sense that, as a quantale, it is the image of an inverse semigroup by a left adjoint functor.
\begin{references}
\bibitem P. Resende, {\em Étale groupoids and their quantales}, preprint, arXiv:math/0412478.
\end{references}

Mon, 18 July: 15:50  16:30
Some localizations of Segal spaces
Robinson, Hugh (Australian National University)The category of Segal spaces was proposed by Charles Rezk in 2000 as a
suitable candidate for a model category for homotopy theories. Quillen
functors induce morphisms in this category and the morphisms induced by
Quillen pairs are ``adjoint'' in a useful sense. Quillen's original total
derived functors are then obtained as a suitable localization of these
morphisms within the category of Segal spaces.
We also consider a construction of ``homotopy fibres'' within a pointed
homotopy theory modelled by a Segal space. Then the homotopy fibre of a
map is preserved by a similar localization which remembers only the
homotopy category plus the automorphism groups of objects. 
Mon, 18 July: 11:00  11:40
Localizations in stable homotopy theory
Rosický, Jiri (Masaryk University)\thanks{Joint work with C. Casacuberta and J. J. Gutiérrez.}
%
Localizing subcategories are very important in stable homotopy theory and the authors of \bibref{2} mention that they do not know any example of a localizing subcategory $\mathcal L$ without a localization functor, which is the same thing as $\mathcal L$ not being coreflective. We will justify their suspicion that the answer may depend on set theory by showing that, assuming Vop\v enka's principle, every localizing subcategory $\mathcal L$ of the homotopy category $\mathcal S$ of spectra is coreflective. Moreover, $\mathcal L$ is generated by a single object and, dually, every colocalizing subcategory $\mathcal C$ of $\mathcal S$ is reflective and generated by a single object. The consequence is that every localizing subcategory of $\mathcal S$ is a cohomological Bousfield class.
Our results are true for monogenic stable homotopy categories $\mathrm{Ho}(\mathcal K)$ (in the sense of \bibref{2}) more general than $\mathcal S$  we have to assume at least that $\mathcal K$ is a combinatorial model category. This means that $\mathcal K$ is locally presentable and the modelcategory structure is cofibrantly generated. Our proofs are based on techniques developed in \bibref{1} and transfered to homotopy theory in \bibref{3}.
\begin{references}
\bibitem J.\ Adámek and J.\ Rosický, {\em Locally Presentable and Accessible Categories}, Cambridge University Press 1994.
\bibitem M.\ Hovey, J.\ H.\ Palmieri and N.\ P.\ Strickland, {\em Axiomatic Stable Homotopy Theory}, Mem.\ Amer.\ Math.\ Soc.\ 610 (1997).
\bibitem C.\ Casacuberta, D.\ Scevenels and J.\ H.\ Smith, {\em Implications of largecardinal principles in homotopical localizations}, to appear in Adv.\ Math.
\end{references}

Wed, 20 July: 14:30  15:10
Topological and conformal field theory as Frobenius algebras
Runkel, Ingo (MaxPlanckInstitut, Potsdam)A twodimensional topological field theory can be defined as a functor from a category Cob of twodimensional cobordisms to the category of vector spaces. It is known that such functors are in onetoone correspondence to certain Frobenius algebras.
If one replaces the target category by a more general (modular) braided tensor category $C$, a functor from Cob to $C$ is in turn related to certain Frobenius algebras in $C$. The latter can be seen to describe twodimensional conformal, rather than topological, field theories. 
Thu, 21 July: 15:50  16:30
Orientals
Steiner, Richard (University of Glasgow)The oriented simplexes or orientals are a family of $\omega$categories associated to simplexes. They were constructed by Ross Street, who expressed the view that they were fundamental structures of nature. I will support this view by showing that the category of orientals arises in other
contexts: the morphisms can be regarded as particular kinds of chain maps between the chain complexes of simplexes, or as cellular homotopy classes of particular kinds of maps between topological simplexes. I also hope to describe the internal structure of the category.

Tue, 19 July: 9:00  10:00
Centres
Street, Ross (Macquarie University, Sydney)\thanks{This is joint work with Brian Day and Elango Panchadcharam.}
%
The centre of a monoid is typical of constructions having higherdimensional
categorical analogues. The centre of a monoidal category has been studied
extensively with application to lowdimensional topology and quantum group
theory. The point usually was to create commutativity (a braiding) where none
existed. However, the centre of an already braided monoidal category can still
be of interest: even in the case where the monoidal structure is cartesian
product! Conditions under which the centre of a category of functors into a
monoidal category $\mathcal{V}$ is again a category of functors into
$\mathcal{V}$ will be discussed. Some centres will be calculated and the
advantage of having a monoid in a centre will be explained.

Tue, 19 July: 15:15  15:55
Developing the Theory of Weak Complicial Sets  A Roadmap
Verity, Dominic (Macquarie University, Sydney)Weak complicial sets were first described in Ross'\ 1987 Orientals paper, where they appear in concluding remarks regarding the nature of weak higher dimensional categorical structures. More recently, he dusted off this notion and pointed out that it deserved further study, if only as a way of studying the nerves of weak $\omega$categories.
In this talk we develop the theory of weak complicial sets, showing in particular that the category of such things may be (strictly) enriched over itself. To this structure we apply a generalisation of the homotopy coherent nerve construction in order to show that the universe of weak complicial sets bears a natural weak complicial structure. We also discuss a simple construction which allows us to derive Tamsamanilike globular $\omega$categories from weak complicial sets.
More tentatively, we foreshadow a coherence theorem, based on a Yoneda argument, by which it is possible to demonstrate that every weak complicial set is equivalent to the nerve of a (strict) category Grayenriched in weak complicial sets. 
Tue, 19 July: 10:30  11:10
Generalizations of braids: presentations, braided monoidal categories, homology
Vershinin, Vladimir (Université Montpellier II and Sobolev Institute of Mathematics, Novosibirsk)We study all sorts of generalizations of braids: surface braid groups, ArtinBrieskorn groups, virtual braids, braidpermutation groups, singular braids, etc., from various points of view. In particular, some of these generalizations form braided monoidal categories or even permutative categories like virtual braids and
braidpermutation groups. This gives the possibility to consider their homology as the homology of corresponding loop spaces.

Tue, 19 July: 14:30  15:10
Higher Hochschild homology and configuration spaces
Voronov, Alexander (University of Minnesota) 
Wed, 20 July: 17:00  17:40
Analytic cosmoi
Weber, Mark (Ottawa University)In the early 1970's Ross Street introduced the notion of ``cosmos", which is a 2categorical analogue of ``elementary topos". In this talk a 2categorical analogue of ``elementary topos with NNO (natural numbers object)", called ``analytic cosmos", will be presented.
The theory of operads and analytic functors works in general inside an analytic cosmos and applications to higher category theory will be discussed.