University of Chicago
Department of Mathematics

Duality in bicategories, such as the bicategory of bimodules and the bicategory of parametrized spectra, has recently proven to be a fruitful extension of duality in monoidal categories. There are problems with using bicategories for this purpose, however, and most are due to the fact that in these bicategories, the `maps between 0-cells' (such as ring homomorphisms and continuous maps) are missing from the structure, since the 1-cells are used for something different. Framed bicategories are a special sort of double category in which the maps between 0-cells, and their action via `base change' on the 1-cells, is incorporated, using the notion of a `categorical fibration' which we introduced 2 weeks ago. On Tuesday, we'll define framed bicategories, prove some basic results, and try to get some intuition and familiarity with the structure. We'll show how framed bicategories solve many of the problems with ordinary bicategories, giving simpler and better notions of equivalence, adjunction, monad, monoidal structure, and so on. We'll also discuss how framed bicategories provide a natural framework for enriched category theory, in the same way that 2-categories provide a natural framework for ordinary category theory. On Thursday, we'll talk about a couple important ways to construct framed bicategories. One starts with a `monoidal fibration', a structure which encodes a number of monoidal categories parametrized by another category; this is how the bicategory of parametrized spectra, among others, is constructed. The other starts with one framed bicategory and constructs another framed bicategory of `monoids and modules' in the original one. Finally, we'll discuss how combining the two constructions unifies the notions of internal category and enriched category, and points the way to a new hybrid theory of `categories which are both internal and enriched'. The case of categories which are both internal to spaces and enriched over spectra has already appeared in the work of Kate Ponto.

This talk will introduce two contructions on double categories which were motivated by a problem in 2-categories. The problem was to describe and understand the $\Pi_2$-construction for 2-categories. This is the free construction to add right adjoints to a class of arrows in a category. In order to understand the equivalence relation on the 2-cells in this construction, one needs to view this construction as the composite of the Path and Span constructions on double categories. We will describe these constructions and discuss their universal properties.

Let G and H be groups and let H--->G be a homomorphism. I will describe how one could come up with the notion of crossed modules, by examining the "quotient" of G by the action of H . From this point of view, it becomes clear that a crossed module is the "same thing" as a 2-group (i.e., a strict monoidal category in which all objects are invertible). I will then explain a recent result of B. Noohi, which describes all monoidal functors between 2-groups, in terms of their corresponding crossed modules. Much of the motivation for these results is based on the fact that a crossed module is the same thing as a connected homotopy 2-type. However, the details of this relationship will not be discussed. This talk will be very elementary. There are no prerequisites.

In an $n$-category, $k$-cells have $k$ different kinds of composition - along bounding cells of each lower dimension - and these different kinds of composition must interact coherently. The axioms for this interaction generalise the middle four interchange law in a 2-category governing the interaction between horizontal and vertical composition of 2-cells. We will examine the case when these axioms are satisfied strictly, and show how to express this via distributive laws. Ordinary distributive laws between two monads on a category give us a way of combining two different algebraic structures on the same category, in a coherent manner. We will generalise this to combine more than two structures on the same category. We will then use $n$ monads on the category of $n$-dimensional globular sets to describe $n$-categories in which interchange is strict; each monad will give composition along a different dimension of bounding cell. Such "semi-strict" $n$-categories are expected to be of interest for proving coherence for $n$-categories and also for studying homotopy types.