EPFL Topology Seminar Fall 2022

 Location: Real World (And sometimes Zoom)

Program

Abstracts

Victor Roca i Lucio,
The integration theory of curved absolute L-infinity algebras:

The integration procedure associates an infinity-groupoid to a (complete/nilpotent) homotopy Lie algebra. This essential tool in deformation theory was introduced by Hinich and later refined by Getzler. Recently, a new method was developed by Robert-Nicoud and Vallette: it relies on the representation of the Getzler functor by an universal object. The goal of this talk is to generalize their method to the integration of curved absolute homotopy Lie algebras. “Absolute algebras” are a new type of algebraic structures that come naturally equipped with infinite summations, without having a priori an underlying topology. We will explain why these new objects appear and how to integrate them, generalizing the above cases. We will then develop applications to rational homotopy theory and to deformation theory. 

Guille Carrion,
Higher limits of functors via Homotopical Algebra:

Classically, higher limits of functors $\operatorname{Fun}(\mathcal C,R\operatorname{-Mod})$ have been computed using the definition of the (right) derived functor of limit in homogical algebra. In this talk, we introduce a new approach to these computations. We consider a functor $F\colon \mathcal C\to R\operatorname{-Mod}$ as a functor taking values in cochain complexes of $R-$modules, $\operatorname{Ch}(R)$, where a module is just a complex concentrated in degree $0$. The category of functors $\operatorname{Fun(\mathcal{C},\operatorname{Ch}(R))$, $\mathcal{C}$ being a filtered poset and $R$ a commutative ring, admits a model category structure in which higher limits of a functor $F\colon \mathcal{C}\to R\operatorname{-Mod}$ are described as $\lim^iF=H^i(\lim \mathbf{R} F)$ where $\mathbf{R} F$ is a fibrant replacement of $F$. Using the Reedy structure on $\mathcal C$, we give a description of how to calculate a fibrant replacement and we obtain bounds for the vanishing of these higher limits.

The Martino-Priddy conjecture says that the p-fusion of G can be recovered (up to isomorphism) from the unstable homotopy type of BG^p. The same authors approached the stable analogue of that result, making strong use of the Segal conjecture (proved by Carlsson), which describes the homotopy classes of stable maps between BG^p and BH^p in terms of (G,H)-bisets. In this talk, I will introduce the notion of biset functors for fusion systems over finite p-groups and present some progress towards a generalization (possibly, also a correction) of the so-called stable Martino-Priddy conjecture.

In a famous paper, Sullivan showed that the rational homotopy theory of finite type nilpotent spaces can be encoded in a fully faithful manner by mapping it to the homotopy category of commutative differential graded algebras over the rational numbers. For integral homotopy theory, a result of Mandell shows that it is faithfully captured by the integral cochains equipped with their E-infinity structute. This functor is however not full. I will explain a way of fixing this problem inspired by work of Toën, using cosimplicial binomial rings instead of E-infty differential graded algebras.

Nicola Gambino,

The effective model structure

For a category E with finite limits and well-behaved countable coproducts, we construct a new Quillen model structure on the category of simplicial objects in E, which we call the effective model structure. The effective model structure generalises the Kan-Quillen model structure on simplicial sets; in particular, its fibrant objects can be viewed as Kan complexes in E. After introducing the main definitions and outlining the key steps of the proof of the existence of the effective model structure, I will describe some of its peculiar properties and what they mean in terms of its associated infinity-category. This is based on joint work with Simon Henry, Christian Sattler and Karol Szumiło (https://doi.org/10.1017/fms.2022.13).

Hadrian Heine,

Real K-theory for Waldhausen ∞-categories with genuine duality

We develop a new framework to study real algebraic K-theory in the context of ∞-categories. For this, we introduce Waldhausen ∞-categories with genuine duality, which will be the input for such K-theory. These are Waldhausen ∞-categories in the sense of Barwick equipped with a compatible duality and a refinement of their hermitian objects generalizing the concept of Poincaré ∞-categories of Lurie. In my talk I will explain the concept of Waldhausen ∞-category with genuine duality and describe how to construct real algebraic K-theory for these objects.

This is joint work with Markus Spitzweck and Paula Verdugo. 

Julie Rasmusen,

THR of Poincaré infinity-categories through enriched infinity-categories

In recent years work by Calmés-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle have moved the theory of Hermitian K-theory into the framework of stable infinity-categories. I will introduce the basic notions of this new theory and explain the ingredients that goes into constructing a functor on these, which describes the geometric fixed points of THR, generalising the formula for ring spectra with anti-involution of Dotto-Moi-Patchkoria-Reeh. This will be done by using a version of enriched modules in the setting of enriched infinity-categories as introduced by Gepner-Haugseng, expanding on the theory of enriched bimodules introduced by the latter.

Jérôme Scherer,

Floyd’s manifold is a conjugation space

This is joint work with Wolfgang Pitsch. Floyd showed in 1973 that there exist only two non-trivial cobordism classes that contain manifolds with three cells, and that they lie in dimensions 10 and 5. We prove that there is an action of the cyclic group C2 on the large Floyd manifold which turns it into a conjugation manifold in the sense of Hausmann, Holm, and Puppe. The submanifold of fixed points is the small one, whose cohomology is isomorphic to that of the large one, scaled down by dividing the cohomological degree by a factor two. I will construct for you this beautiful manifold, equip it with a C2-action, and provide background on conjugation spaces.

Thibault Décoppet,

Fusion 2-Categories and Fully Extended 4-Dimensional TQFTs

Through the Cobordism Hypotehsis of Baez-Dolan and Lurie, a fully extended framed topological quantum field theory can be constructed from any fully dualizable object of a symmetric monoidal higher category. By a result of Douglas-Schommer-Pries-Snyder, fusion 1-categories are fully dualizable objects in the symmetric monoidal 3-category of finite tensor 1-categories. Recently, fusion 2-categories were introduced by Douglas-Reutter as a categorification of the notion of fusion 1-category, and they conjectured that fusion 2-categories are fully dualizable objects of a symmetric monoidal 4-category. After reviewing the definition of a fusion 1-category, I will introduce the notion of fusion 2-category. Then, I will explain why fusion 2-categories are in fact fully dualizable objects of a symmetric monoidal 4-category.

David Gepner,

On stable infinity-categories in an infinity-topos

Globally equivariant spectra are a stabilization of globally equivariant spaces, but not the stabilization in the naive sense. This is because we want our stable objects to be locally (over the base infinity-topos) stable under loop and suspension by representation spheres. We formulate this precisely and describe the resulting stabilization functor. This is joint work with Thomas Nikolaus and Stefan Schwede.

Olivia Monjon,

Fiberwise localization and conditionally flat functors in the category of crossed modules

Categories of groups and crossed modules of groups have some similarities and are in particular examples of semi-abelian categories. We will define what is a localization functor in this context. Then we will see how these two categories behave differently with respect to a construction called fiberwise localization. We will use to study the notion of conditionally flat functors in the category of crossed modules.