PROPs associated to Lawvere theories and their relation to polynomial functors

Several adjunctions between functor categories have been studied and applied previously. These include Powell’s adjunction between functor categories on free groups and on the linear PROP associated with the Lie operad, as well as those implicit in the equivalence of Pirashvili between functors on projective modules and modules over wreath products. In this paper, for a Lawvere theory $\mathcal{C}$ with a zero object, we construct a natural linear PROP ${\tilde{\mathrm{\Phi }}}_{\mathcal{C}}$, which carries a canonical adjunction between functor categories over $\mathcal{C}$ and ${\tilde{\mathrm{\Phi }}}_{\mathcal{C}}$. The adjunction is compatible with polynomial degree, in the sense that it gives a correspondence between polynomial $\mathcal{C}$-modules and truncated ${\tilde{\mathrm{\Phi }}}_{\mathcal{C}}$-modules. We suggest that this framework provides a useful step toward studying polynomial $\mathcal{C}$-modules. To support this perspective, a large part of this paper is devoted to explicit calculations of ${\tilde{\mathrm{\Phi }}}_{\mathcal{C}}$ for a specific Lawvere theory $\mathcal{C}$. This construction unifies the previously known examples, and also yields new ones, including adjunctions for functor categories on modules over a ring, as well as on free nilpotent groups and, more generally, on free $\mathcal{R}$-semisimple groups, where $\mathcal{R}$ is a radical functor for groups.