**Title:**Abelian functor calculus, derivatives, and operads

**Abstract:
**The first chain rules in functor calculus were established by Arone-Ching and Klein-Rognes, who considered functors of spaces or spectra. The Arone-Ching chain rule stemmed from earlier work of M. Ching, in which he established that the derivatives of the identity functor form an operad, whose homology is the classical Lie operad. The chain rule, in conjunction with Ching’s earlier work, lead to a classification theorem for all functors of spaces or spectra. That is, it explained how you could produce a functor from its derivatives. However, Arone-Ching’s work was quite complicated and difficult. One of the motivations for establishing a chain rule for abelian functor calculus was to try to look for a simplification of their work. Indeed, in BJORT, we established the chain rule by first establishing that the category of abelian categories is a cartesian differential category. This, together with the associated tangent structure, lead to a much simpler proof of the chain rule. Following the program laid out by Arone and Ching, we are now looking for the expected operad structure and classification theorems. In this talk, I will explain the derivative (as opposed to the directional derivative) for abelian functor calculus, and the candidate for our operad.