University of Chicago Algebraic Topology Seminar

On the unicity of the homotopy theory of higher categories
We will discuss joint work with Clark Barwick in which we propose four axioms that a quasicategory should satisfy to be considered a reasonable homotopy theory of (∞, n)-categories. This axiomatization requires that a homotopy theory of (∞, n)-categories, when equipped with a small amount of extra structure, satisfies a simple, yet surprising, universal property. We further prove that the space of such quasicategories is homotopy equivalent to B(Z/2)ⁿ. This generalizes a theorem of Toen when n = 1, and it verifies two conjectures of Simpson. In particular, any two such quasicategories are equivalent. We also provide a large class of examples of models satisfying our axioms, including those of Joyal, Kan, Lurie, Simpson, and Rezk.

Equivariant Localizations and Symmetric Monoidal Categories
In this talk, I'll discuss joint with with Hopkins on localizations of equivariant commutative ring spectra. I'll also sketch language for a candidate for G-symmetric monoidal categories, showing how this language makes the localization result intuitive and also provides a natural way to better understand the connection between Mackey functors, Tambara functors, and Eilenberg-MacLane spectra.

A homotopy characterization of p-completed classifying spaces of finite groups
The homotopy type of classifying spaces of finite groups is easy to characterize: they are the Eilenberg-MacLane spaces of type K(G,1). When working p-locally (as one often must) things are more complicated as the homotopy type of a p-completed classifying space does not depend on the group itself, but just on its p-local fusion system, which is the category of p-subgroups and homomorphisms induced by conjugation (this is the Martino--Priddy conjecture). Seeing fusion systems arise in different contexts, Puig axiomatized them and introduced abstract fusion systems. Thus, to characterize the homotopy type of p-completed classifying spaces, one should really study classifying spaces for fusion systems. To this end, Broto--Levi--Oliver introduced p-local finite groups as a rather complicated algebro-categorical model for the classifying space of a fusion system. Around the same time, Haynes Miller proposed a purely homotopy-theoretic characterization for a p-completed classifying spaces as a space that admits a transfer retract to the classifying space of a finite p-group, and tentatively suggested that they should be equivalent to the Broto--Levi--Oliver model. In this talk I will explain these two models and then go through an outline proof of Miller's conjecture. This is work in progress, joint with Matthew Gelvin.

Abelian properties of thee Anick spaces and the secondary EHP sequence.
I will indicate how the secondary EHP sequences are defined and discuss the issues involved with the construction of unstable compositions. This will be resolved into the question of Ableian H-space structures and I will give a general analysis of how to construct Abelian H-space structures. I will apply this method to the construction of an Abelian structure for the secondary EHP sequence.

Calculating homotopy n-types using strict higher homotopy groupoids of some spaces with structure: intuitions, results, limitations.
Even at level 2, homotopy groups are pale shadows of the 2-type, which is represented by a crossed module. To prove calculation of this under gluing (local-to-global) circumstances, one uses a strict homotopy double groupoid of a pair. Analogous results in higher dimensions are obtained using filtered spaces or n-cubes of spaces; the strict algebraic models do allow for some detailed calculations, and relations to classical homotopy theory, such as n-ads.

What is equivariant cohomology and what is it good for?
Pretalk: Let G be a topological group and let G act on a space X. What can one deduce about the G-fixed point space X^G and the orbit space X/G from information about X? I will explain classical cohomological results, P.A. Smith theory and the Connor conjecture, that give quite remarkable answers to these questions. I will use modern proofs of these results to describe equivariant cohomology theory, which means different things to different people. The proof of the Connor conjecture will lead us directly to the idea that cohomology should be graded on the real representation ring of G rather than just on the integers. In turn, this will lead us to a remarkable relationship between Mackey functors in algebra and equivariant stable homotopy groups. All concepts will be defined.

Talk: I'll give a slanted overview of equivariant cohomology, starting with the use of ordinary (Bredon) cohomology, of which Borel cohomology H^*(EG ×_G X) is a special case, to prove two classical results on the fixed point and orbit spaces of a G-space X, namely P.A. Smith theory and the Connor conjecture. For the second, we must extend our Z-graded cohomology theory to an RO(G)-graded theory, and that is also necessary for equivariant Poincaré duality. The equivariant analogue of an abelian group is a coefficient system, and an ordinary Z-graded theory extends to an RO(G)-graded theory if and only if its coefficient system extends to a "Mackey functor". That is a standard notion in representation theory when G is finite, but a topological reinterpretation in terms of equivariant stable homotopy groups extends the notion to compact Lie groups. I'll briefly describe the stable homotopy category of G-spectra, which is the natural home for RO(G)-graded cohomology. This is all ancient history, at least 30 years old, but I'll give a glimpse of the modern theory. As a matter of sheer good luck, the unreasonable effectiveness of equivariant ideas, the key calculational input to the recent solution of the Kervaire invariant problem by Hill, Hopkins, and Ravenel is an easy calculation of certain RO(G)-graded ordinary cohomology groups of a point.

Equivariant categories with multiplication and G-spectra
Recently, Peter May and I have provided a new model for the equivariant stable homotopy category, under a finite group of equivariance. This model involves equivariant infinite loop space theory and equivariant categories with multiplication. I will discuss this structure and provide examples.

NOTE THE TIME CHANGE

The homotopy theory of coalgebras over a comonad
Let K be a comonad on a model category M. We provide conditions under which the associated category M_K of K-coalgebras admits a model category structure such that the forgetful functor from M_K to M creates both cofibrations and weak equivalences. We will discuss an example in the differential graded context and mention on-going work in pointed sets and spectra. This is joint work with Kathryn Hess.

Directed Poincaré Duality
The max-flow min-cut theorem, traditionally applied to problems of maximizing the flow of commodities along a network (e.g. supply chains) and minimizing the costs of breaking apart networks (e.g. image segmentation), admits an algebro-topological interpretation and generalization as a 1-dimensional case of twisted Poincaré Duality for spaces equipped with some "direction". In this talk, I will describe the appropriate sheaf (co)homology theories, taking local coefficients and values in semigroups, on locally preordered spaces, and some possible applications in information theory and logistics. Conjecturally, such a Poincaré Duality should generalize Strong Duality in Linear Programming; I will also describe the intuition behind such a conjecture. No background in optimization or directed topology is assumed.

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