Algebraic Topology and Category Theory

• Time and Place: Tuesdays and Thursdays 1:30-3:00 in Eckhart Room 203. Talks will be in the fields of
  topology, category theory, and their intersection.
• Audience: Graduate students, mostly in their second year or higher, with an interest in algebraic topology and/or category theory. Faculty interested in these topics will also attend. Talks will be contributed by everyone.
• Email List: http://zaphod.uchicago.edu:8080/mailman/listinfo/topology (also used for the topology/category theory seminar).

Schedule

• Tues, Jan 5: Organizational meeting
• Thurs, Jan 7: Locally presentable categories (Claire Tomesch)
• Abstract: Locally presentable categories appear in the interplay of universal algebra, model theory, and logic. They also are technically nice in certain model category applications. Basically, locally presentable categories are categories whose objects can be "nicely" described in terms of a _set_ of "nice" objects, where "niceness" has something to do with a kind of smallness and behavior of hom-functors. In this talk, we define locally presentable categories, describe their properties, and give relevant examples. We also discuss a representation theorem and the relation to Vopenka's principle.

• Tues, Jan 12: Trace methods in algebraic K-theory (Vigleik Angeltveit)
• Thurs, Jan 14: Locally presentable categories II (Claire Tomesch)
• Tues, Jan 19: Basic notions in equivariant stable homotopy theory (Anna Marie Bohmann)
• Abstract: This talk will be an introduction to several important constructions frequently used in equivariant stable homotopy theory. We will discuss equivariant spheres and spectra, fixed points and splitting spectra up based on isotropy type. The talk is meant to be accessible to people without much background in the field.

• Thurs, Jan 21: Orientals and cohomology (Daniel Schaeppi)
• Abstract: It is well-known that the 0th and the 1st ordinary cohomology of a space X with coefficients in A can be defined even if A is not commutative. These are useful for studying certain 'local to global' problems. For example, the cocycles in the general linear group are closely related to vector bundles over X. John Roberts suggested that, in order to define higher nonabelian cohomology, one should replace the coefficient object by an n-category; a group is, after all, a one object category, and an abelian group can be thought of as an n-category with exactly one k-cell for k<n.

  I will outline a definition of cohomology with coefficients in a strict omega-category due to Ross Street. This involves the notion of parity complexes and orientals; the latter are also used to define the nerve of a strict omega-category.

• Tues, Jan 26: The universal loop space operad and generalisations (Eugenia Cheng)
• Abstract: The problem of modelling homotopy types by ω-groupoids has been approached in many different ways in recent years. I will discuss the theory proposed by Batanin/Leinster using globular operads. There are various analogies with the use of operads to study loop spaces; I will recall the loop space operad E called "universal" by Salvatore, and the analogous globular operad G which can be used to recognise the fundamental ω-groupoid of a space. I will give a categorical proof of the universal property of E, enabling generalisation to prove a universal property of G. The helps us identify other, more tractable, operads for fundamental ω-groupoids.

• Thurs, Jan 28: Globular operads revisited (Mike Shulman)
• Tues, Feb 2: Kervaire invariant one (Rolf Hoyer -- topic presentation)
• Abstract: Until recently, the Kervaire invariant one problem was one of the more famous open problems in homotopy theory. My talk will summarize Hill-Hopkins-Ravenel's presentation of nonexistence in dimensions greater than 126. After an outline of the overall argument, I will provide a brief overview of each of the proof's major components (time permitting), emphasizing the parts of the proof utilizing equivariant stable homotopy theory.

• Thurs, Feb 4: What are parametrized spectra good for? (Peter May)
• Tues, Feb 9: Homotopical type theory (Mike Shulman)
• Abstract: Recently, a number of people have independently realized that there are surprisingly deep connections between homotopy theory and higher category theory on the one hand, and the intensional type theory of Martin-Löf on the other. The potential interaction flows both ways: homotopy theory provides a new class of models for type theory, and higher category theory provides a language in which to study identity types, while some believe type theory has the potential to provide a new syntax for abstract homotopy theory and perhaps even a "homotopical" foundation of mathematics. I'll give an introduction to some of these ideas from the perspective of a homotopy theorist, assuming no background in type theory.

• Thurs, Feb 11: Computads (Daniel Schaeppi)
• Abstract: Over the past few weeks we have seen several objects on which we can define free ω-categories. Given a globular set X, there is a free ω-category TX, and for a parity complex C, there is a free ω-category OC. For a globular set, the domain and codomain of any n-cell consists of a single (n-1)-cell, while the domain and codomain of an n-cell in a parity complex are given by sequences of (n-1)-cells. A parity complex is required to be loop-free, but a globular set can have lots of loops (e.g., the terminal globular set). Computads are a common generalization of these two situations. Batanin later realised that for any monad A on globular sets, we can define a notion of A-computads. I will give a survey of some of the important results about computads and A-computads.

• Tues, Feb 16: Generalized characters for Morava K-theories (Anna Marie Bohmann)
• Abstract: Over the next two Tuesdays, we will discuss work by Hopkins, Kuhn and Ravenel about Morava K-theories and generalized characters. We begin with a reminder about the Atiyah-Segal theorem for complex K-theory, which relates the represention ring of a group to the K-theory of its classifying space. We will then discuss the Morava K-theories and Morava K-theory Euler characteristic. Next week, John Lind will discuss generalized characters as a way of calculating Morava K-theories.
• Thurs, Feb 18: Higher Group Cohomology (Evan Jenkins)
• Abstract: Ordinary abelian group cohomology has a simple and elegant topological description in terms of fibrations of Eilenberg-Mac Lane spaces, which can be thought of as higher categorical analogues of group extensions. Much of the computational power of abelian group cohomology stems from the fact that abelian groups are infinite loop spaces. The countably many deloopings of an abelian group give rise to countably many cohomology groups, and these groups fit together nicely into long exact sequences. Ordinary groups are less fortunate; they are only one-fold loop spaces. Still, some interesting information can be gotten out of their ordinary and "nonabelian" cohomology groups.

  We can pass from the world of groups to the world of n-groupoids (or even infinity-groupoids) and still make sense of extensions as fibrations. In this world, however, "abelianness" is no longer just a property but rather a variety of possible structures, and hence an extension will be more than a mere fibration. In my talk, I will extend the notion of group extension to k-tuply groupal n-groupoids, give an algebraic description of the corresponding cohomology n-groupoids, and show how short exact sequences give rise to long exact sequences in cohomology. I will focus in particular on the simplest case that does not fit into the existing paradigm, namely, extensions of groups by braided 2-groups. This particular problem closely parallels work by Etingof, Nikshych, and Ostrik on G-extensions of braided fusion categories.

• Tues, Feb 23: Generalized Character Theory (John Lind)
• Abstract: Last week Anna Marie told us about the work of Hopkins, Kuhn and Ravenel on counting the Euler characteristic of K(n)^*(BG) in terms of generalized conjugacy classes in the group G. I will talk about how these generalized conjugacy classes are the natural domain for generalized characters. This will allow a description of E^*(BG) for appropriate theories E related to K(n). Following recent work of Ganter and Kapranov, I will try to convince you that a representation theory expressed using higher categories is the natural source of the HKR character theory.

• Thurs, Feb 25: Giraud-type theorems (Mike Shulman)
• Abstract: The classical Giraud theorem gives a concise list of categorical properties which, if satisfied by some category C, imply that C is equivalent to the category of sheaves for some Grothendieck topology—i.e. that C is a Grothendieck topos. I'll start by describing these properties, which can be seen as a "nonabelian version of exactness," and sketching a proof of Giraud's theorem. Then (as far as time allows) I'll talk about some generalizations to Giraud theorems for n-topoi for some values of n>1, where there are a few surprises in store.
• Tues, Mar 2: No seminar
• Thurs, Mar 4: Bousfield localization (Peter May)
• Abstract: I will give an overview of some central parts of model category theory, focusing on dual resolution and localization model structures. In particular, I'll show how to localize spaces at homology theories, following Bousfield.
• Tues, Mar 9: The Adams conjecture (Rolf Hoyer)
• Thurs, Mar 11: Pretalk for "Higher dimensional operads and categories" (Dennis Borisov)