On the Passi and the Mal'cev functors

TL;DR

This paper extends the equivalence between categories of polynomial functors on free groups and modules over Lie operad PROPs, introducing Mal'cev functors and applying the theory to bifunctors and topological categories.

Contribution

It generalizes the analytic functor equivalence to covariant functors, introduces Mal'cev functors, and connects to topological categories like Habiro-Massuyeau's category.

Findings

Category of covariant polynomial functors is equivalent to right modules over Lie PROP

Mal'cev functors serve as projective generators in the functor category

Application to the Casimir PROP and topological categories like bottom tangles

Abstract

The author has shown that the category of analytic contravariant functors on $\mathbf{gr}$, the category of finitely-generated free groups, is equivalent to the category of left modules over the PROP associated to the Lie operad, working over $\mathbb{Q}$. This exploited properties of the polynomial filtration of the category of contravariant functors on $\mathbf{gr}$. The first purpose of this paper is to strengthen the corresponding result for covariant functors on $\mathbf{gr}$. This involves introducing the appropriate analogue of the category of analytic contravariant functors, namely a certain category of towers of polynomial functors on $\mathbf{gr}$. This category is abelian and has a natural symmetric monoidal structure induced by the usual tensor product of functors. Moreover, the projective generators of this category are described in terms of the Mal'cev functors that are introduced here. It follows that this category is equivalent to the category of right modules over the PROP associated to the Lie operad. As a fundamental example, the Passi functors arising from the group ring functors are described explicitly. The theory is applied to consider bifunctors on $\mathbf{gr}$. This allows the $\mathbb{Q}$-linearization of the category of free groups to be described, up to polynomial filtration. As a stronger application of the theory, this is generalized to the Casimir PROP associated to the Lie operad, as studied by Hinich and Vaintrob. Up to polynomial filtration, this recovers the category $\mathbf{A}$ introduced by Habiro and Massuyeau in their study of bottom tangles in handlebodies.