EPFL Topology Seminar Fall 2020

 Location: Zoom/Real World

Program

Abstracts

Nima Rasekh,
Shadows and THH of ∞-Categories:
Topological Hochschild Homology (THH) was originally conceived as a generalization of Hochschild homology to ring spectra. However, it has since been generalized to many other settings. Here are just some of the many examples:
> A generalization of THH to enriched categories and their profunctors that has been used, particularly by Ponto, in the study of fixed point theorems.
> An ∞-categorical approach to THH of ring spectra, which for example has been used successfully by Scholze and Nikolaus in their study of cyclotomic actions.
> Finally, there is now a notion of THH of enriched ∞-categories, due to Berman.

In this talk we want to discuss ongoing work, joint with Kathryn Hess, with the goal of reconciling the various notions of THH using an ∞-categorical approach. In particular, after giving some motivation, we will focus on comparing the axiomatic approach to THH introduced by Ponto, shadow functors, with the enriched ∞-categorical approach to THH due to Berman.

If time permits, we will illustrate how the ∞-categorical approach can help us recover classical results, such as Morita invariance of THH, using far more formal techniques.

Allen Yuan,
Integral Models for Spaces via the higher Frobenius:
We will describe a fully faithful integral model for spaces in terms of their E-infinity algebras of cochains which assembles Mandell’s p-adic homotopy theory with Sullivan’s rational homotopy theory.  The key input is the development of a homotopy coherent Frobenius action on a certain subcategory of p-complete E-infinity-rings for each prime p. Using this action, we will see that the data of a space X is the data of its E-infinity-algebra of spherical cochains together with a trivialization of the Frobenius action after completion at each prime.

​​We will then outline the construction of this Frobenius action. This involves constructions in equivariant homotopy theory, which produce an action of Quillen’s Q-construction (on the category of abelian groups) on certain E-infinity-rings with “genuine equivariant multiplication.”  The second main idea is a “pre-group-completed” variant of algebraic $K$-theory; we will state some basic results about this “partial K-theory” and briefly discuss the partial K-theory of F_p.

Bruno Vallette,
Higher Lie Theory:
The construction of the infinity-groupoid associated to homotopy Lie algebras due to Getzler, after Hinich’s works, can actually be presented in a more simple and more powerful way. While Getzler defines it intrinsically as the gauge (kernel) of the Dupont’s contraction, I will use the seminal approach of Kan to adjoint functors to/from simplicial sets. This will provide me with a new way to integrate homotopy Lie algebras which shows how Getzler’s function can be represented by a universal object, that it admits a left adjoint, that it is actually functorial with respect to infinity-morphisms. This approach will allow me to fully describe the sets of horn fillers: this will settle the Kan property in a canonical way which will make this object and algebraic infinity-groupoid, that is with given horn fillers. This change of paradigm will make us leave algebraic topology and enter algebra. In this way, we can perform explicit and algorithmic computations. For instance, the first horn filler gives the celebrated Baker—Campbell—Hausdorff formula. The higher horn fillers introduce for the first time higher BCH formulas. This also gives us homotopy Lie algebra models in rational homotopy theory.

Brandon Doherty,
Cubical Models of (∞,1)-Categories:
We describe a new model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical analogue of inner horns. We discuss the proof that this model structure is Quillen equivalent to the Joyal model structure on simplicial sets via the triangulation functor, and a new theory of cones in cubical sets which is used in this proof. We also introduce the homotopy category and mapping spaces of a fibrant cubical set, and characterize weak equivalences between fibrant cubical sets in terms of these concepts. This talk is based on joint work with Chris Kapulkin, Zachery Lindsey, and Christian Sattler, arXiv:2005.04853.

Luciana Bonatto,
An infinity Operad of Normalized Cacti:
Gluing surfaces along their boundaries allows us to define composition laws that have been used to define cobordism categories, as well as operads and props associated to surfaces. These have played an important role in recent years, for example in constructing topological field theory or computing the homology of the moduli space of Riemann surfaces.

Normalized cacti are a graphical model for the moduli space of genus 0 oriented surfaces. They are endowed with a composition that corresponds to glueing surfaces along their boundaries, but this composition is not associative. By introducing a new topological operad of bracketed trees, we show that this operation is associative up-to all higher homotopies and that normalized cacti form an $\infty$-operad. In particular, this provides one of the few examples in the literature of infinity operads that are not a nerve of an actual operad.

Duncan Clark,
An Intrinsic Operad Structure for the Derivatives of the Identity:
A long standing slogan in Goodwillie’s functor calculus is that the derivatives of the identity functor on a suitable model category should come equipped with a natural operad structure. A result of this type was first shown by Ching for the category of based topological spaces. It has long been expected that in the category of algebras over a reduced operad $\mathcal{O}$ of spectra that the derivatives of the identity should be equivalent to $\mathcal{O}$ as operads.

In this talk I will discuss my recent work which gives a positive answer to the above conjecture. My method is to induce a “highly homotopy coherent” operad structure on the derivatives of the identity via an pairing of underlying cosimplicial objects with respect to the box product. This cosimplicial object naturally arises by analyzing the derivatives of the Bousfield-Kan cosimplicial resolution of the identity via the stabilization adjunction for $\mathcal{O}$-algebras. Time permitting, I will describe some additional applications of these box product pairings. In particular, I will show how a similar box product pairing may be utilized to provide a new description of an operad structure on the derivatives of the identity in spaces.

Elena Dimitriadis Bermejo,
The Homotopy Theory of DG-Categories:
Differentially-graded categories are essential to Derived Algebraic Geometry; but they don’t behave as well as we would like them to. In 2005, Bertrand Toën proved that all theories of ∞-categories are Quillen equivalent to the category of complete Segal spaces, basically meaning that we can see any ∞-category as a presheaf from the simplex category to the simplicial sets, satisfying certain conditions. In this talk, we will try and apply a similar logic to dg-categories: we will define a certain “linearized version of the simplex category” and sketch the proof that every dg-category is a simplicial presheaf from that linear simplex category to the category of simplicial sets. We will finish with a few possible applications of the result.

Kursat Sozer,
Two-Dimensional Extended HQFTs with arbitrary Targets:
Inspired by theoretical physics, topological quantum field theories (TQFTs) produce manifold invariants behaving well under gluing. Homotopy quantum field theories (HQFTs), introduced by Turaev, generalize TQFTs to manifolds equipped with continuous maps to fixed target space. A different generalization of TQFTs is given by extended TQFTs which includes lower-dimensional manifolds utilizing higher categories. In this talk, we define and classify 2-dimensional extended HQFTs with arbitrary targets generalizing the earlier work on K(G,1)-targets using the methods introduced for TQFTs by Chris Schommer-Pries in 2009.

Anna Cepek,
The Combinatorics of Configuration Spaces of R^n:
We approach manifold topology by examining configurations of finite subsets of manifolds within the homotopy-theoretic context of ∞-categories by way of stratified spaces. Through these higher categorical means, we identify the homotopy types of such configuration spaces in the case of Euclidean space in terms of the category Θ_n.

Lorenzo Guerra,
Homotopy E_∞ cooperads:
I will discuss an ongoing joint research project with Benoit Fresse. We study a notion of homotopy cooperads in chain complexes defined via bar duality and we lift this bar duality approach to Segal cooperads in the category of E_∞ algebras. I will present some homotopical properties of these objects.

Pedro Tamaroff,
Differential Forms for Smooth Affine Algebras over Operads:
The Hochschild–Kostant–Rosenberg theorem is a classical algebro-geometric result: Hochschild homology and cohomology of a smooth commutative algebra can be computed as the space of differential forms and of poly-vector fields on it, respectively. It says, in other words, how to compute the cohomology of a commutative algebra if we pull it back through the canonical projection of operads p : Ass –> Com and consider it merely to be associative. This result was then exploited to study the deformation theory of smooth commutative algebras, and obtain the celebrated Kontsevich formality theorem (cf. D. Tamarkin’s proof). In this talk, I will explain how to answer the following natural generalization of this question: given a map of operads f : P –> Q and a smooth Q-algebra A, how can one construct a space Ω*(A) of ‘differential forms for A’ and when can one produce an isomorphism from the (Hochschild) homology of the pullback P-algebra A to Ω*(A)? In particular, I’ll explain how to recover the usual Hochschild–Kostant–Rosenberg theorem using operadic homological algebra. This is joint work with Ricardo Campos (IMAG, Universite Montpellier, CNRS), arXiv:2010.08815.

Sarah Klanderman,
Topological CoHochschild Homology Calculations:
In recent work, Hess and Shipley defined an invariant of coalgebra spectra called topological coHochschild homology (coTHH). In 2018, Bohmann-Gerhardt-Høgenhaven-Shipley-Ziegenhagen developed a coBökstedt spectral sequence to compute the homology of coTHH for coalgebras over the sphere spectrum. However, examples of coalgebras over the sphere spectrum are limited, and we would like to have computational tools to study coalgebras over other ring spectra. In this talk I’ll describe a relative coBökstedt spectral sequence that I developed in order to study the topological coHochschild homology of more general coalgebra spectra. We’ll look at a few examples of coalgebra spectra and use the additional algebraic structure of the spectral sequence to complete coTHH computations.

Christina Osborne,
Decomposing the Classifying Diagram in Terms of Classifying Spaces of Groups:
The classifying diagram was defined by Rezk and is a generalization of the nerve of a category; in contrast to the nerve, the classifying diagram of two categories is equivalent if and only if the categories are equivalent. In this talk, we will show that the classifying diagram of any category is characterized in terms of classifying spaces of stabilizers of groups. We will also prove explicit decompositions of the classifying diagrams for the categories of finite ordered sets, finite dimensional vector spaces, and finite sets in terms of classifying spaces of groups.

Clover May,
Decomposing C2-Equivariant Spectra:
Computations in RO(G)-graded Bredon cohomology can be challenging and are not well understood, even for G=C_2, the cyclic group of order two. A recent structure theorem for RO(C_2)-graded cohomology with Z/2 coefficients substantially simplifies computations. The structure theorem says the cohomology of any finite C2-CW complex decomposes as a direct sum of two basic pieces: cohomologies of representation spheres and cohomologies of spheres with the antipodal action. This decomposition lifts to a splitting at the spectrum level. In joint work with Dan Dugger and Christy Hazel we extend this result to a classification of compact modules over the genuine equivariant Eilenberg-MacLane spectrum HZ/2.