EPFL Topology Seminar Spring 2021

 Location: Zoom

Program

Abstracts

Michael Mandell,
Large Scale Open Subsets of Configuration Spaces and the Foundations of Factorization Homology:
This project (joint with Andrew Blumberg) aims to adapt the foundations of factorization homology to be more amenable to equivariant generalizations for actions of positive dimensional compact Lie groups. This requires finding replacements for arguments that discretize configuration spaces (e.g., arguments in terms of quasi-categories) or that use a local-to-global approach (arguments in terms of a small neighborhood of a point in a configuration space).  In practice, each such theorem reduces (by a Quillen Theorem A argument) to showing that a comparison map from a certain homotopy coend (that depends on the specifics of  the statement) to a configuration space (or related space) is a weak equivalence; the replacement strategy is to construct a cover of the configuration space by “large scale” open subsets whose intersection combinatorics and homotopy types mirror those of the composition combinatorics and homotopy types in a bar construction for the homotopy coend.  (“Large scale” is descriptive rather than technical: the game to is to describe open subsets of configuration spaces C(n,M) that will be G-stable when M=G/H is an orbit space for a positive dimensional compact Lie group G.)  In other words, the project is to deduce the properties of factorization of homology directly from “large scale” open covers of configuration spaces.

Kristine Bauer,
Categorical Differentiation of Homotopy Functors:
The Goodwillie functor calculus tower is an approximation of a homotopy functor which resembles the Taylor series approximation of a function in ordinary calculus. In 2017, B., Johnson, Osborne, Tebbe and Riehl (BJORT, collectively) showed that the directional derivative for functors of an abelian category are an example of a categorical derivative in the sense of Blute-Cockett-Seely. The BJORT result relied on the fact that the target and source of the functors in question were both abelian categories. This leads one to the question of whether or not other sorts of homotopy functors have a similar structure.

To address this question, we instead use the notion tangent categories, due to Rosicky, Cockett-Cruttwell and Leung. The structure of a tangent category is highly reminiscent of the structure of a tangent bundle on a manifold. Iindeed, the category of smooth manifolds is a primary and motivating example of a tangent category. In recent work, B. Burke and Ching make precise the notion of a tangent infinity category, and show that the directional derivative for homotopy functors appears as the associated categorical derivative of a particular tangent infinity category. This ties together Lurie’s tangent bundle construction to the categorical literature on tangent categories.

In this talk, I aim to explain the categorical notions of differentiation and tangent categories, and explain their relationship to Goodwillie’s functor calculus. If time permits, I also hope to explain why this categorical framework is useful by explaining it is related to operad structures in functor calculus towers (work in progress with Johnson-Yeakel). The primary work discussed in this talk is joint work with Burke and Ching.

Daniel Fuentes-Keuthan,
Goodwillie Towers of Infinity Categories and Desuspension:
We reconceptualize the process of forming n-excisive approximations to ∞-categories, in the sense of Heuts, as inverting the suspension functor lifted to An-cogroup objects. We characterize n-excisive ∞-categories as those ∞-categories in which An-cogroup objects admit desuspensions. Applying this result to pointed spaces we reprove a theorem of Klein-Schwänzl-Vogt: every 2-connected cogroup-like A∞-space admits a desuspension.

Martina Rovelli,
n-Complicial Sets as a Model for (∞, n)-Categories:
With the rising significance of (∞, n)-categories, it is important to have easy-to-handle models for those and understand them as much as possible. In this introductory talk we will discuss how n-complicial sets provide a model for (∞, n)-categories, and how one can recover strict n-categories through a suitable nerve construction. We will focus on n = 2, for which more results are available, but keep an eye towards the general case. Time permitting, we will also discuss a few recent research directions and work in progress about n-complicial sets.

Mathieu Anel,
Enveloping infinity-Topoi:
Any 1-topos has an enveloping infinity-topos defined as sheaves of spaces over the 1-topos. But surprisingly, this construction does not send presheaf 1-topoi to presheaf infinity-topoi! I shall explain why and provide sufficient conditions for the envelope to stay a presheaf topoi. Applications includes the presheaves over the simplex category, or G-equivariant presheaves.

Akhil Mathew,
K-Theory and Polynomial Functors:
We show that the algebraic K-theory space of a stable ∞-category is functorial in polynomial functors. The argument is based on the universal property of algebraic K-theory and yields various non-additive operations in K-theory. Joint work with Clark Barwick, Saul Glasman, and Thomas Nikolaus.

David Jaz Myers,
Modal Fibrations in Homotopy Type Theory:
Spaces are not the same thing as homotopy types; after all, there is more than one function from R to R. Moreover, there are many different sorts of spaces — topological spaces, condensed sets, smooth manifolds, schemes — each with their own peculiar theory. We may extract a homotopy type from a space, usually by localizing at some family of “contractible” spaces, and use this homotopy type to study the space with algebraic methods.

Homotopy type theory is a logical system for working directly with sheaves of homotopy types as fundamental objects. Sheaves of homotopy types include spaces as representable sheaves, as well as higher spaces like orbifolds, Lie groupoids, and Deligne-Mumford stacks. The operation of taking the homotopy type forms a modality on the oo-topos of such sheaves.

In this talk, we will introduce a notion of modal fibration suitable for doing algebraic topology in modal homotopy type theory. In one sense, a modal fibration closely resembles the classical definition of a quasi-fibration — its fibers are weakly equivalent to its homotopy fibers. This is a simple definition, but it is classically ill behaved. What we want from a fibration is the monodromy action of the homotopy type of the base on the homotopy types of the fibers, which is to say that the homotopy types of the fibers should form a local system over the base. With the magic of modal homotopy type theory, we will show that these two conditions are equivalent when both are interpreted in homotopy type theory. We will then see a trick for proving that a map is a modal fibration which may be summarized by saying that “if it can be written as F –> E –> B, then it is a fibration”. This definition and trick work as well for higher stacks (including orbifolds) as for spaces.

José Cantarero,
The Twisted K-Theory of p-Completed Classifying Spaces:
In this talk we will give a description of the twisted K-theory of the p-completion of the classifying space of a finite group, and more generally, for classifying spaces of p-local finite groups. We will begin by introducing representations of p-groups which are invariant under fusion, some examples and a completion theorem relating them to the K-theory of a p-completed classifying space. After this, we consider rings of twisted representations and an alternative interpretation of twisted K-theory. Both objects will be interpreted in terms of central extensions of p-local finite groups, allowing us to give a completion theorem for twisted K-theory. This is joint work with Noe Barcenas.
 
We will end with some questions regarding the behaviour of fusion-invariant representations.

Özgür Bayındır,
Algebraic K-Theory of THH(Fp):
In this work, we study THH(Fp) from various perspectives. We start with a new identification of THH(Fp) as an E_2-algebra. Following this, we compute the K-theory of THH(Fp).

The first part of my talk is going to consist of an introduction to the Nikolaus Scholze approach to trace methods. In the second part, I will introduce our results and the tools we develop to study the topological Hochschild homology of graded ring spectra.

This is a joint work with Tasos Moulinos.

Anna Marie Bohmann,
Algebraic K-Theory for Lawvere Theories: Assembly and Morita Invariance:
Much like operads and monads, Lawvere theories are a way of encoding algebraic structures, such as those of modules over a ring or sets with a group action.  In this talk, we discuss the algebraic K-theory of Lawvere theories, which contains information about automorphism groups of these structures.  We’ll discuss both particular examples and general constructions in the K-theory of Lawvere theories, including examples showing the limits of Morita invariance and the construction of assembly-style maps.  This is joint work with Markus Szymik.

Jordan Williamson,
Torsion Models for Tensor-Triangulated Categories:
In this talk I will describe how to construct a model for suitably nice tensor-triangulated categories from the data of local and torsion objects. The idea is to use a variation on the Hasse-Tate square, and mimic constructions such as local cohomology and localization from commutative algebra in this more general context. I will then discuss some examples, the main example being the construction of an algebraic torsion model for rational SO(2)-equivariant spectra. This is joint work with S. Balchin, J.P.C. Greenlees and L. Pol.

Matt Feller,
Conjecturing a Characterization of Joyal Fibrations:
The Joyal model structure on simplicial sets has been an important setting for developing the theory of higher category theory. The fibrant objects are the quasi-categories, a model for (∞,1)-categories, which have a straightforward characterization: they are the simplicial sets with lifts of inner horn inclusions. However, so far in the literature there has been no such characterization of the fibrations in terms of a countable set of concrete maps. We give a conjecture of such a characterization, describing a class of “special horn” extensions. We use Cisinski’s theory to build a model structure whose fibrant objects are simplicial sets with special horn extensions.

Cindy Lester,
The Homotopical Canonical Grothendieck Topology:
Every category can be equipped with the canonical Grothendieck topology, which can be explicitly described using colimits. However, colimits are not homotopy invariant. By instead using homotopy colimits we can define a homotopy invariant canonical Grothendieck topology on any simplicial model category. I will explain the homotopical version of the canonical Grothdendieck topology, showcase some examples and discuss some open questions.