Title: Representable tangent ∞-categories and functor calculi

Kristine Bauer, Matthew Burke and I have constructed a tangent structure T on a certain ∞-category of ∞-categories which is related to Goodwillie’s calculus of functors. This tangent structure is not representable, though it looks kind of like it almost is. I will try to describe a way in which we might view T as (co?)represented by a certain ∞-topos (the ∞-topos of parameterized spectra).

One of the reasons for taking this perspective is to try to fit other versions of functor calculus from homotopy theory into the tangent category framework. In particular, I will describe the “manifold calculus” (of Goodwillie and Michael Weiss) and “orthogonal calculus” (of Weiss), and ask to what extent they also can be viewed in terms of tangent categories or generalizations thereof.