Outer functors and a general operadic framework

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.