University of Chicago
Department of Mathematics

In the late 80's, Graeme Segal mathematically described conformal field theory as a symmetric monoidal functor from a cobordism category to Hilbert spaces. However, this definition was not entirely rigorous. This talk deals with the 2-category theory necessary for a complete definition of conformal field theory. I will discuss pseudo algebras over theories and 2-theories, pseudo limits, and biadjoints, always motivated by the algebraic structure on the geometric category of worldsheets.

A category is a monoid with many objects. A bicategory is a monoidal category with many objects. The most interesting monoidal categories are ``closed'', meaning that they have internal homs, and symmetric. There is a standard and well-developed duality theory in closed symmetric monoidal categories that is used implicitly or explicitly throughout mathematics. This talk will discuss closed symmetric bicategories and their duality theory, which can be viewed as a 2-categorification of classical duality theory. This theory should have been discovered long ago, since it is very easy and is descriptive of very commonly occurring structures. In fact, the theory was discovered by staring at the analogue of Spanier-Whitehead duality for parametrized homology and cohomology theories, but I will say nothing about that. It is joint work with Johann Sigurdsson. The ideas are most easily understood in terms of the bicategory of rings, bimodules, and maps of bimodules. This specialization works just as well in the DG-world, where it seems to give exactly the right framework for Morita duality. In general, this kind of structure appears whenever one has any kind of base change functors around, so should be highly relevant to sheaf theory and algebraic geometry. The talk will be entirely elementary, and everyone is welcome.

In this talk we will describe various ways of combining two monads on the same category. Many mathematical structures can be described as algebras for a monad, for example groups, rings, monoids, R-modules, operads, compact Hausdorff spaces, complete semilattices, categories, strict n-categories. We may then ask how to combine two sorts of mathematical structure at the same time. In some examples the two structures are required to interact well via a distributive law, but we will be interested in combining structures freely, which amounts to taking a coproduct in the category of monads. Our motivating example is the construction of the monad for $n$-categories according to Leinster, Batanin or Penon. The existence of each of these monads can be deduced from general abstract arguments, but in order to be able to work with the definitions we give an actual construction of the monad by "interleaving" the two component monads. Other examples of such "mixed" monads appear to include Steenrod's description of the lean realisation of the nerve of a topological group, Berger's description of the W-construction, and a logic of "mixed choice" which was presented by Beaulieu in Ottawa this weekend, combining non-deterministic and probabilistic operators.

Quasicategories are one of many notions of "weak category" or "category up to homotopy". The idea is simple: a quasicategory is a simplicial set which is like the nerve of a category, but only "up to homotopy". It turns out that much of basic category theory can be done for quasicategories as well. I will try to give an idea of how this works for concepts such as limits and fibrations ("pre-stacks"), drawing from Joyal's as-yet-unpublished draft, but leaving out the technical details that take up most of the space.

Orbifolds arise naturally in many branches of mathematics, such as geometry, Gromov-Witten theory, and equivariant topology, to name a few. Recent progress in understanding these singular spaces hinges upon using categorical structures to organize the data. In this expository talk, I'll discuss two of these modern frameworks for defining the category of orbifolds: the first uses certain internal groupoids in the category of smooth manifolds, the second certain stacks over the category of smooth manifolds. I will also discuss the equivalence between the two approaches, which involves defining principal groupoid bundles. Friendly examples and definitions will be given, many details will be withheld. A good reference is D. Metzler's preprint "Topological and Smooth Stacks," available at arXiv:math.DG/0306176.

This will be an introductory talk on stacks.

In any symmetric monoidal category we can define duality for objects and trace for endomorphisms. For vector spaces this trace is the usual (linear algebra) trace and for (suspension spectra of) compact manifolds this is the Lefschetz number or index. There are some traces that are similar to those coming from symmetric monoidal categories, but do not actually fit into that definition. One example is trace for endomorphisms of left modules over a non commutative ring. I will give a definition of trace for bicategories that does describe some of these examples.

Operads, cyclic operads and modular operads may be considered in any symmetric monoidal category C, and the category of -/cyclic/modular operads is monadic if C is cocomplete. In this talk, building on a construction of Kevin Costello, I introduce a new structure (which I call a "design"), a small symmetric monoidal category T with some additional data, such that the category of symmetric monoidal functors from T to C has the same properties as in the above three examples. For example, the symmetric monoidal category underlying the design associated to cyclic operads has as its morphisms all forests. The theory of designs admits a straightforward extension to the enriched setting, which we explain. We give some naturally occurring examples of simplicial designs, which are closely related to topological field theories of different flavors.

(Joint work with Nick Gurski.) We discuss an approach to constructing a weak $n$-category of cobordisms. First we present a generalisation of Trimble's definition of $n$-category which seems most appropriate for this construction; in this definition composition is parametrised by a contractible operad. Then we discuss the problem of defining an $n$-category &ob;\bfseries nCob&cb;, whose $k$-cells are $k$-cobordisms, possibly with corners. We show how to make some preliminary constructions as ``stepping stones" towards the desired goal. We follow Baez and Langford in using ``manifolds embedded in cubes" rather than general manifolds.

I will give an introductory talk on the topic of tricategories, a very hands-on approach to weak 3-categories. I will introduce the basic ideas present in the definition, give some trivial examples, discuss in some detail the example Bicat, and present the basic coherence result. Along the way we will discuss the Gray tensor product, an interesting tensor product on the category of 2-categories that is strongly related to the braid groups.

To prevent a scheduling conflict, the original talk has been moved.

In 1946, Whitehead introduced the notion of crossed module in his study of adding relations to homotopy groups. A few years later, Mac Lane and Whitehead proved that crossed modules model homotopy 2-types. Since crossed modules are equivalent to certain double groupoids, these classical results allude to the significance of double categories. In this talk I will recall Ehresmann's 1963 notion of double category, examine some examples, and describe thin structures, connection pairs, and folding structures on double categories. These double structures arise naturally when one categorifies the notion of category to the notion of pseudo algebra over the 2-theory of categories as in the context of conformal field theory. After a long gestation period, double categories (and their weakened version) are finding more and more applications, such as 2-dimensional Van Kampen theorems and extended field theory. This talk will require no background and will be useful to anyone who is curious about double categories, their classical examples such as crossed modules, and modern applications.

Anchored bicategories are a new type of two-level categorical structure. An anchored bicategory has an underlying bicategory, but it also includes more data designed to capture the structure seen in numerous important examples. These examples, which include rings and modules, spaces and parametrized spectra, manifolds and cobordisms, and categories and presheaves, are often represented as bicategories, but this is only possible by disregarding important structure which is present. This extra structure, which is included in an anchored bicategory, allows the construction of base change objects and of the "cancellation" operations needed to define nonabelian traces, which Kate discussed last quarter. It also allows a study of the correct notion of "equivalence" for such structures, which cannot even be stated in the language of bicategories. This talk will begin with an explanation of the defects of bicategories. Then I will give the definition of anchored bicategories, as it presently exists, and discuss how it solves some of the problems associated with bicategories. I will end with some remarks about progress towards a coherence theorem for anchored bicategories. Only a basic acquaintance with bicategories will be assumed.

I will discuss different statements of coherence for bicategories and tricategories. The first part of the talk will review three of the common ways to state coherence for bicategories. I will also discuss how to move between these statements. The second part of the talk will focus on adapting these different statements to the context of tricategories. We will see how making the definition of tricategory fully algebraic is crucial to this adaptation.

Thesis defence.