** Loc****ation: Zoom/Real World**

### Program

Date | Time | Place | Title | Speaker |
---|---|---|---|---|

21.09.2021 | 14:15 CEST | MA A1 12 | A Coalgebra Model for p-adic Homotopy Types | Hadrian Heine, EPFL |

28.09.2021 | 14:15 CEST | MA A1 12/ Zoom | Fibrations and lax Limits of (∞,2)-Categories | Edoardo Lanari, Czech Academy of Sciences |

05.10.2021 | 14:15 CEST | MA A1 12 | Koszul Duality for Categories with a Fixed Object Set | Hadrien Espic, Stockholms Universitet |

19.10.2021 | 14:15 CEST | MA A1 12 | Infinity-Operads as Analytic Monads | Rune Haugseng, NTNU |

02.11.2021 | 14:15 CET | MA A1 12/ Zoom | Calculus of Fractions | Denis-Charles Cisinski, Universität Regensburg |

09.11.2021 | 14:15 CET | MA A1 12/ Zoom | Semisimple Field Theories and Stable Diffeomorphisms | David Reutter, MPIM |

16.11.2021 | 14:15 CET | MA A1 12 | Split Extensions, Actions and Crossed Modules in the Categories of Hopf Algebras | Florence Sterck, Université Catholique de Louvain |

23.11.2021 | 14:15 CET | MA A1 12 | Homotopy Fiber Sequences from a New Perspective | Alisa Govzmann, Universitéit Lëtzebuerg |

30.11.2021 | 14:15 CET | MA A1 12 | Synthetic Fibered (∞,1)-Category Theory | Jonathan Weinberger, University of Birmingham |

07.12.2021 | 14:15 CET | MA A1 12/ Zoom | Axioms for the Category of Hilbert Spaces | Chris Heunen, University of Edinburgh |

31.01.2022 | 14:00 CET | Zoom | Categorification of Link Homologies via Homotopy Types | Ilaria Rossinelli, EPFL |

### Abstracts

**Hadrian Heine,**

**A Coalgebra Model for p-adic Homotopy Types:**

By a theorem of Mandell connected p-complete nilpotent spaces of finite p-type are classified by a natural E_infty-algebra structure on cochains with coefficients in the algebraic closure of the field with p-elements.

In this talk I will discuss an extension of Mandell’s theorem classifying all connected p-complete nilpotent spaces not necessarily of finite p-type by a natural E_infty-coalgebra structure on chains with coefficients in the algebraic closure of the field with p-elements. This project is joint work in progress with Manfred Stelzer.

**Edoardo Lanari,**

**Fibrations and lax Limits of (∞,2)-Categories:**

We study four types of (co)cartesian fibrations of $\infty$-bicategories over a given base $B$, and prove that they encode the four variance flavors of $B$-indexed diagrams of $\infty$-categories. We then use this machinery to set up a general theory of 2-(co)limits for diagrams valued in an $\infty$-bicategory, capable of expressing lax, weighted and pseudo limits. When the $\infty$-bicategory at hand arises from a model category tensored over marked simplicial sets, we show that this notion of 2-(co)limit can be calculated as a suitable form of a weighted homotopy limit on the model categorical level, thus showing in particular the existence of these 2-(co)limits in a wide range of examples. Next, we extend this to $(\infty,2)$-category valued diagrams and the corresponding fibrations, and we provide motivating examples. We end by discussing a notion of cofinality appropriate to this setting and use it to deduce the unicity of 2-(co)limits, once they exist.

This is joint work with A.Gagna and Y.Harpaz.

**Hadrien Espic,**

**Koszul Duality for Categories with a Fixed Object Set:**

In this talk, we use the fact that categories with a fixed object set are the monoid objects in a category of “graphs” to develop a formalism for defining the Koszul dual in this context. We then explain its relationship with the theory of Koszul duality for operads in spaces and spectra developed by Michael Ching, through the functor associating to an operad P the corresponding prop, which is a category with the natural numbers as object set. We also mention potential applications and discuss how free algebras and trivial algebras are related in this version of Koszul duality.

**Rune Haugseng,**

**Infinity-Operads as Analytic Monads:**

Joyal proved that symmetric sequences in sets (or “species”) can be identified with certain endofunctors of Set, namely the “analytic” functors. Under this identification the composition product on symmetric sequences corresponds to composition of endofunctors, and this allows us to identify operads in Set with certain “analytic” monads. Moreover, the monad corresponding to an operad O is precisely the monad for free O-algebras in Set. In this talk I will explain how to obtain an analogous identification for infinity-operads: assigning to an infinity-operad O (in Lurie’s sense) the monad for free O-algebras in spaces identifies infinity-operads with analytic monads. This builds on previous work with Gepner and Kock where we developed the theory of analytic monads in the infinity-categorical setting.

**Denis-Charles Cisinski, **

**Calculus of Fractions:**

The purpose of abstract homotopy theory is to provide category theoretic constructions which are compatible with suitable notions of weak homotopy equivalences. One can revisit the concepts that have lead to Quillen’s notion of model category as devices to compute mapping spaces of localizations in terms of Kan extensions, which, in turns provide tools to compute (co)limits in localized infinity-categories as homotopy (co)limits. From there, one can produce a perfect dictionary between (co)complete infinity-categories and their models together with a good theory of derived functors as Kan extensions, revisiting the work of Szumiło, Kapulkin and Mazel-Gee. Reformulating homotopy theory properly as suggested above, using mainly the language of Kan extensions is not only a pleasant way to revisit classical constructions (although that would be good enough), but also a way to internalize homotopy theory in any higher topos (in fact in any directed type theory). This will have applications, for instance, to formulate and prove the universal property of Morel and Voevodsky’s motivic homotopy theory (possibly formulated within derived geometry, thus generalizing the contributions of Drew and Gallauer), as well as to study condensed/pyknotic mathematics (e.g. one can see the pro-étale topos of a scheme as a condensed/pyknotic presheaf topos on the the associated Galois category constructed by Barwick, Glasman and Haine).

**David Reutter,**

**Semisimple Field Theories and Stable Diffeomorphisms:**

A major open problem in quantum topology is the construction of a 4-dimensional topological field theory (TFT) in the sense of Atiyah-Segal which is sensitive to exotic smooth structure. More generally, how much manifold topology can a TFT see?

In this talk, I will outline an answer to this question for even-dimensional `semisimple’ TFT:

Such theories can at most see the stable diffeomorphism type of a manifold and conversely, if two sufficiently finite manifolds are not stably diffeomorphic, then they can be distinguished by a semisimple field theory. In this context, `semisimplicity’ is a certain algebraic condition satisfied by all currently known examples of linear algebraic TFT in more than two dimensions, and two 2n-manifolds are said to be stably diffeomorphic if they become diffeomorphic after connected sum with sufficiently many copies of S^n x S^n.

Along the way, I will introduce a number of semisimple TFTs built from homotopy types acted on by the orthogonal group O(n), and will discuss various implications, such as the fact that 4d oriented semisimple TFT cannot see smooth structure, while unoriented ones can.

This is based on arXiv:2001.02288 and joint work in progress with Christopher Schommer-Pries.

**Florence Sterck,**

** Split Extensions, Actions and Crossed Modules in the Categories of Hopf Algebras:**

In this talk, we will investigate some categorical properties of Hopf algebras. The first part of the talk will be devoted to the category of cocommutative Hopf algebras over a field. The fact that this category is semi-abelian will allow us to give a description of internal crossed modules (as defined by Janelidze). In the second part of the talk, we will study the notion of split extensions of general Hopf algebras. In the category of groups, split extensions have a lot of interesting properties. One of them is the fact that the category of split extensions is equivalent to the category of group actions. Unfortunately, this does not hold in any category, for example, the category of monoids does not have this property. Nevertheless, D. Bourn, A. Montoli, N. Martins-Ferreira and M. Sobral proved that there exists such an equivalence if the split extensions of monoids are « Schreier extensions ». We will answer the question: Which conditions on the split extensions of Hopf algebras do we need to have an equivalence with the category of actions of Hopf algebras?

**Alisa Govzmann,**

**Homotopy Fiber Sequences from a New Perspective:**

Quillen introduced fibration sequences in the homotopy category of a pointed model category. Up to a (non-canonical) Ho(M)-isomorphism a fibration sequence is then a Ho(M)–sequence K → F → G that is implemented by the kernel K of a fibration F → G between fibrant objects F and G. Using the fact that the loop space functor Ω^QF of a fibrant object F is a group object in the homotopy category Ho(M), Quillen shows that there is an action of Ω^QG on K which induces a connecting Ho(M)–morphism Ω^QG → K and the sequence Ω^QG → K → F is again a fibration sequence. I want to present an alternative approach to construct homotopy fiber sequences without using the concept of an action. We define a loop space functor Ω and for every morphism f : F → G we define its homotopy fiber K_f such that K_f → F → G is a homotopy fiber sequence. We get a universal connecting morphism ΩF → K_f such that ΩF → K_f → G is also a homotopy fiber sequence. It turns out that the loop space functor we define as well as the connecting homomorphism coincide with the one proposed by Quillen. At the beginning of this talk I will explain how to construct an equivalence of categories between the homotopy category of morphisms (arrows) in M, denoted by Ho(M^→) and the homotopy category of long homotopy fiber sequences, denoted by Ho(l(M)). This will lead to a canonical choice of an isomorphism in Ho(M) between two different homotopy fibers of a morphism in Ho(M).

**Jonathan Weinberger,**

**Synthetic Fibered (∞,1)-Category Theory:**

As an alternative to set-theoretic foundations, homotopy type theory is a logical system which allows for reasoning about homotopical structures in an invariant and more intrinsic way.

Specifically, for the case of higher categories there exists an extended framework, due to Riehl-Shulman, to develop (∞,1)-category theory synthetically. The idea is to work internally to simplicial spaces, where one can define predicates witnessing that a type is (complete) Segal. This had also independently been suggested by Joyal.

Generalizing Riehl-Shulman’s previous work on synthetic discrete fibrations, we discuss the case of synthetic cartesian fibrations in this setting, leading up to a 2-Yoneda Lemma. In developing this theory, we are led by Riehl–Verity’s model-independent higher category theory, therefore adapting results from ∞-cosmos theory to the type-theoretic setting. Time permits, we’ll briefly point out generalizations to the two-sided case.

In fact, by Shulman’s recent work on strict universes, the theory at hand has semantics in Reedy fibrant diagrams in an arbitrary (∞,1)-topos, so all type-theoretically formulated results semantically translate to statements about internal (∞,1)-categories.

This is based on joint work with Ulrik Buchholtz

(https://arxiv.org/abs/2105.01724) and the speaker’s recent PhD thesis.

**Chris Heunen,**

**Axioms for the Category of Hilbert Spaces:**

We provide axioms that guarantee a category is equivalent to that of continuous linear functions between Hilbert spaces. The axioms are purely categorical and do not presuppose any analytical structure such as probabilities, convexity, complex numbers, continuity, or dimension. We’ll discuss the axioms, sketch the proof of the theorem, and survey open questions, further directions, and context. (Based on joint work with Andre Kornell arxiv:2109.07418.)

**Ilaria Rossinelli,**

**Categorification of Link Homologies via Homotopy Types:**

Links appear in many areas of mathematics, and the study of their invariants is a well-established problem. In particular, it is an interesting question if and how invariants can be recovered from higher invariant structures that enhance them and possibly provide more information.

In this expository talk, we will present one example of link invariant (the triply-graded link homology by Khovanov and Rozansky) and one possible categorification via equivariant homotopy types. This is based on the work by Kitchloo (https://arxiv.org/abs/1910.07443 and https://arxiv.org/abs/1910.07444).

If time permits, we will also briefly discuss how this construction can be obtained from the categorical setting provided by traces and bicategories.