Outer functors and a general operadic framework

TL;DR

This paper develops a general operadic framework for outer functors, connecting them to analytic functors and applying this to higher Hochschild homology, thus advancing the understanding of functor categories in algebraic topology.

Contribution

It introduces a new subcategory of functors associated with operads satisfying a Leibniz condition and relates it to outer analytic functors, extending previous work on Lie operads.

Findings

Identification of the subcategory _\u03a9^ with outer analytic functors

Application to higher Hochschild homology functors

Extension of functor category theory in operadic contexts

Abstract

For $\mathcal{O}$ an operad in $k$-vector spaces, the category $\mathcal{F}_\mathcal{O}$ is defined to be the category of $k$-linear functors from the PROP associated to $\mathcal{O}$ to $k$-vector spaces. Given $\mu \in \mathcal{O} (2)$ that satisfies a right Leibniz condition, the full subcategory $\mathcal{F}_\mathcal{O}^\mu \subset \mathcal{F}_\mathcal{O}$ is introduced here and its properties studied. This is motivated by the case of the Lie operad, where $\mu$ is taken to be the generator. By previous results of the author, when $k = \mathbb{Q}$, $\mathcal{F}_{Lie}$ is equivalent to the category of analytic functors on the opposite of the category $\mathbf{gr}$ of finitely-generated free groups. The main result shows that $\mathcal{F}_{Lie}^\mu$ identifies with the category of outer analytic functors, 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.