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Warwick Algebraic Topology Seminar 21/22

A list of the seminar talks of the previous years can be found here.

Term 1

The talks will take place on Tuesday at 4pm in B3.03. Some of the talks will be in person, some will be given by a remote speaker on MS Teams and streamed in B3.03. You can join the teams talks with the code hud33su, or email me at emanuele.dotto@warwick.ac.uk.

Date Speaker Affiliation Title Notes
Oct 5 Nima Rasekh EPFL Shadows are bicategorical Traces Video
Oct 12 Andrew Macpherson University of Warwick Designer ∞-categories via homotopy theory of diagrams B3.03
Oct 19 Rajan Mehta Smith College Frobenius objects in categories of relations and spans B3.03/Teams
Oct 26 Anna Marie Bohmann Vanderbilt University   B3.03/Teams
Nov 2 Irakli Patchkoria University of Aberdeen   B3.03
Nov 9 Oscar Randal-Williams University of Cambridge   B3.03
Nov 16 Mingcong Zeng Utrecht University   B3.03/Teams
Nov 23 Angélica Osorno Reed college   B3.03/Teams
Nov 30 Tobias Barthel MPIM Bonn   B3.03?
Dec 7        
Abstracts
Nima Rasekh: Shadows are bicategorical Traces

Topological Hochschild homology (THH), first defined for ring spectra and then later dg-categories and spectrally enriched categories, is an important invariant with connections to algebraic K-theory and fixed point methods. The existence of THH in such diverse contexts motivated Ponto to introduce a notion that can encompass the various perspectives: a shadow of bicategories. On the other side, many versions of THH have been generalized to the homotopy coherent setting providing us with motivation to develop an analogous (∞,2)-categorical generalization of shadows.

The goal of this talk is to use an appropriate bicategorical notion of THH to prove that a shadow on a bicategory is equivalent to a functor out of THH of that bicategory. We then use this result to give an alternative conceptual understanding of shadows and, in particular, an alternative proof for the Morita invariance. Moreover, we use this alternative characterization to give an appropriate definition of a homotopy coherent shadow of (∞,2)-categories.

This is joint work with Kathryn Hess

Andrew Macpherson: Designer ∞-categories via homotopy theory of diagrams

In this talk, I will discuss two ways - Ehresmann's colimit sketches and Quillen's homotopy theory of categories - that *diagrams* are used in the construction of new categories with prescribed universal properties, and how in the world of higher categories, they merge and become essentially the same subject. Studying this as a general framework lets us give names to questions or phenomena - compositionality, filteredness, plus constructions - that pervade cocompletion constructions in category theory, and study the relations between them. Since all this is rather abstract, I'll try to couch it in terms of the examples that got me thinking about this.

Rajan Mehta: Frobenius objects in categories of relations and spans

Frobenius algebras can be given a category-theoretic definition in terms of the monoidal category of vector spaces, leading to a more general definition of Frobenius object in any monoidal category. In this talk, I will describe Frobenius objects in categories where the objects are sets and the morphisms are relations or spans. These categories can be viewed as toy models for the symplectic category. The main result is that, in both cases, it is possible to construct a simplicial set that encodes the data of the Frobenius structure. The simplicial sets that arise in this way satisfy conditions that are closely connected to the 2-Segal conditions of Dyckerhoff-Kapranov and Galvez-Carrillo-Kock-Tonks.

Commutative Frobenius objects in these categories give surface invariants that can be Boolean or natural number valued. I will give some explicit examples where the invariants can be computed. This work is a very small first step in a bigger program aimed at better understanding the relationship between Poisson/symplectic geometry and topological field theory. Part of the talk will be devoted to giving an overview of this program.

This is based on work with Ruoqi Zhang (arXiv:1907.00702), with Ivan Contreras and Molly Keller (arXiv:2106.14743), and work in progress with Ivan Contreras, Adele Long, and Sophia Marx