# Homotopy Theory Spring 2021

The organizers are Cindy Lester and Ettore Aldrovandi

Planned meeting times are on **Tuesday, 5pm Eastern.** Due to the speakers’ locations, some seminars might be held at the different times to accommodate for the time difference. These times are highlighted below.

The talks are held on Zoom.

- Paige North University of Pennsylvania
- Tuesday, April 27, 5pm EDT
*The Univalence Principle***Abstract:**Univalent foundations (a.k.a homotopy type theory) is a foundation of mathematics based on dependent type theory. It has interpretations in the classical model structure on simplicial sets and other model structures. Because of this, theorems and constructions made in univalent foundations can be ported to theorems and constructions in these models of homotopy theory. For example, notions of categories and univalent categories have been developed and studied in univalent foundations, and under the interpretation in simplicial sets, these become Segal spaces and complete Segal spaces, respectively. These univalent categories have the very convenient property that any two equivalent ones can be regarded as equal (note that the notion of equality in univalent foundations differs from that in set-based mathematics, and will be explained in the talk).

In this work, we have generalized this theory. We have developed a notion of univalence for many (higher) categorical structures (such as bicategories, dagger categories, etc) such that the appropriate notion of equivalence between two univalent structures is equivalent to equality.

This is joint work with Benedikt Ahrens, Mike Shulman, and Dimitris Tsementzis.

- Maximilien Péroux University of Pennsylvania
- Tuesday, April 13, 5pm EDT
*Coalgebras and comodules in stable homotopy theory***Abstract:**In higher algebra, we study algebraic objects endowed with a multiplication that is associative only up to (coherent) homotopy, or commutative up to (coherent) homotopy. In this Brave new algebra, we study algebras and modules that includes the classical theory of algebra. The ground ring is not the ring of integer anymore, it is the sphere spectrum. Rigidification results (or sometimes called rectification) state that some of these highly coherent algebras over some rings can have their multiplication rigidified into a strictly associative multiplication. This has been used in many instances using the tool of model categories. In fact, in the 90s, many symmetric monoidal model categories of spectra were introduced such that strictly associative associative algebras were representing \(A_\infty\)-algebras, and similarly \(E_\infty\)-algebras.

In this talk, we will explore the dual algebraic objects of coalgebras and comodules in higher algebra. Instead of a multiplication, we have a comultiplicationthat we now require to be co-associative up to higher homotopy. I will show that higher algebras are enriched over higher coalgebras and thus, coalgebras provide insight on the structure for algebras. However, we will see that these objects are much more mysterious than algebras. I will show that none of the current monoidal model categories of spectra represent well the higher coalgebras in spectra. This is will hint that the correct language to study higher coalgebra is infinity-categories. I will also show that it is challenging but possible to rigidify coaction of comodules when using connective spectra over a field. This result allows to define a derived cotensor product of comodules which has not been possible before.

- Alexander Campbell Centre of Australian Category Theory, Macquarie University
- Tuesday, April 6,
**6pm**EDT <– Note unusual time! *A model-independent construction of the Gray monoidal structure for (∞,2)-categories***Abstract:**In this talk I will describe joint work with Yuki Maehara in which we give a model-independent (i.e. a purely ∞-categorical) construction of the (non-symmetric) Gray monoidal structure on the ∞-category of (∞,2)-categories. Our construction is a generalisation to the ∞-categorical setting of a construction of the Gray monoidal structure for 2-categories due to Ross Street, which uses the techniques from Brian Day’s PhD thesis for extending a monoidal structure along a dense functor. The proof of our construction uses, among other things, the results from Yuki’s PhD thesis on the Gray tensor product for 2-quasi-categories. I will also mention a few of the open problems concerning the Gray monoidal structure for (∞,2)-categories, and explain how our results can be used to simplify (though not yet solve) one of these problems.

- Tuesday, April 6,
- Martina Rovelli, University of Massachusetts Amherst
- Tuedsday, March 23 5pm EDT
*\(n\)-complicial sets as a model for \((∞, n)\)-categories***Abstract:**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.

- Rhiannon Griffiths, Case Western Reserve University
- Tuesday, February 9, 5pm EST
*Enriched Model Categories for Functor Calculus***Abstract:**We present results necessary for developing model categories of enriched functors suitable for doing functor calculus. As a first example, we show how the discrete functor calculus of Bauer, Johnson and McCarthy may be placed into the context of simplicially enriched model categories.

- Christina Osborne, Cedarville University
- Tuesday, January 19, 5pm EST
*Decomposing the classifying diagram in terms of classifying spaces of groups***Abstract:**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.