Title: Homotopy invariance and derivatives

This talk involves joint work with Matthew Burke, Michael Ching, Brenda Johnson and Sarah Yeakel

Abstract: Goodwillie’s functor calculus provides a tower of polynomial approximations to functors of homotopical categories which resemble the Taylor series in ordinary calculus of functions. These towers are often useful because they contain a lot of data which is interesting to homotopy theorists, or because the tower make computations simpler.

Practitioners of differential category theory may naturally ask whether the resemblance to the Taylor series is indicative that Goodwillie calculus is related to some kind of categorical differentiation. For functors of abelian categories, this was answered in the affirmative in BJORT 2017, which showed that the category of abelian categories is itself a differential category (up to chain homotopy equivalence). For homotopy functors, this is answered by work of myself, Matthew Burke and Michael Ching proving that there is a tangent structure on the infinity category of infinity categories.

The purpose of this talk is to review the role of homotopy in both of these contexts and provide an introduction to these ideas for newcomers. In the case of abelian functor calculus, BJORT 2017 sought to establish a type of higher order chain rule up to homotopy. From this chain rule we were able to extract two kinds of operads. However, higher coherence data makes the extraction process difficult – a process which is reflected in my joint work with Johnson and Yeakel. In the case of homotopy functors, higher coherence data is already encoded in infinity categories. What is required in that case is ensuring that tangent structures can be defined to be homotopy invariant. To do this, one needs to examine a related model structure in which the category Weil – which is used to define tangent structures – is itself cofibrant. In this talk I’ll attempt to compare and contrast the abelian and homotopy calculus stories.