Correo interno
Categorical differentiation of homotopy functors
Ponente: Kristine Bauer
Institución: University of Calgary
Tipo de Evento: Investigación
Institución: University of Calgary
Tipo de Evento: Investigación
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, B., Johnson, Osborne, Tebbe and Riehl (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 tangent categories, due to Rosicky, Cockett-Cruttwell and via an incarnation due to Leung. The structure of a tangent category is highly reminiscent of the structure of a tangent bundle on a manifold. Indeed, the category of smooth manifolds is a primary and motivating example of a tangent category. 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 appears as the associated categorical derivative of a particular tangent infinity category. This ties together Lurie’s tangent bundle construction to the categorical literature on tangent categories.
In this talk, I aim to explain the categorical notions of differentiation and tangent categories, and explain their relationship to Goodwillie’s functor calculus.