Session S27 - Categories and Topology

Thursday, July 15, 15:00 ~ 15:30 UTC-3

Categorical differentiation of homotopy functors and applications

Kristine Bauer

University of Calgary, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

The Goodwillie functor calculus tower is an approximation of a homotopy functor which resembles the Taylor series approximation of a function in ordinary calculus. In 2017, Johnson, Osborne, Riehl, Tebbe and I (BJORT, collectively) showed that the directional derivative for functors of an abelian category are an example of a categorical derivative in the sense of Blute, Cockett and Seely. The BJORT result relied on the fact that the target and source of the functors in question were both abelian categories. This leads one to the question of whether or not other sorts of homotopy functors have a similar structure.

To address this question, Burke and Ching and I instead use the notion of tangent category, due to Rosicky, Cockett-Cruttwell and via an incarnation due to Leung. In recent work with Burke and Ching, we make precise the notion of a tangent infinity category, and show that the directional derivative for homotopy functors from Goodwillie's calculus of functors appears as the associated categorical derivative of a particular tangent infinity category.

In this talk I will give an overview of these structure, with an eye towards possible applications. These applications are being developed by Johnson, Yeakel and I.

Joint work with Matthew Burke (Lyryx Learning Inc.), Michael Ching (Amherst College), Brenda Johnson (Union College), Christina Osborne (Cedarville University), Emily Riehl (Johns Hopkins University), Amelia Tebbe (Indiana University Kokomo) and Sarah Yeakel (University of California, Riverside).

View abstract PDF