Outer functors and a general operadic framework

Abstract

For O an operad in k-vector spaces, FO is defined to be the category of k-linear functors from the PROP associated to O to k-vector spaces. Given a binary operation μ∈O(2) that satisfies a right Leibniz condition, the full subcategory FOμ⊂FO is introduced here and its properties studied.This is motivated by the case of the Lie operad Lie, where μ is taken to be the generator. By previous results of the author, when k=Q, FLie is equivalent to the category of analytic functors on the opposite grop of the category gr of finitely-generated free groups. The main result shows that FLieμ identifies with the category of outer analytic functors on grop, as introduced in earlier work of the author with Vespa.Using this identification, this theory has applications to the study of the higher Hochschild homology functors related to work of Turchin and Willwacher.