Kristine Bauer

Title: Operad structures in abelian functor calculus
Abstract: Abelian categories are a cartesian differential category, and the derivative corresponds to the same derivative which is used in functor calculus (a branch of homotopy theory). In 2011, Cockett and Seely showed that any Cartesian differential category has a higher-order chain rule for the derivative.
B. Johnson, S. Yeakel and I have identified this higher order chain rule in the abelian functor calculus example. Furthermore, we have shown that a consequence of the higher-order chain rule is that higher order derivatives of a functor of R-modules form an operad (a monoid in the category of symmetric sequences). The existence of this operad was predicted by a similar result for functors of topological spaces (discovered by G. Arone and M. Ching). In the case of abelian calculus, we have identified this operad as a consequence of the existence of a (lax) functor from abelian categories to the category Faa(AbCat), as defined by Cockett and Seely. We see our result as a kind of translation between the homotopy theoretic and category theoretic results.
In this talk, I will define the abelian functor calculus derivative, explain the higher order chain rule and produce the resulting operad.