PROPs associated to Lawvere theories and their relation to polynomial functors
Minkyu Kim
TL;DR
The paper develops a unified framework linking Lawvere theories to linear PROPs via the primitivity ideal and the eigenmonad, yielding adjunctions between $\mathcal{C}$-modules and $\tilde{\Phi}_{\mathcal{C}}$-modules that respect polynomial degree. This construction recovers known adjunctions (e.g., Powell’s Lie/PROP and Pirashvili–Macdonald-type equivalences) and provides new ones across a broad class of theories, including module categories, free nilpotent groups, and free $\mathcal{R}$-semisimple groups. By explicit calculations for key examples, the work shows how polynomiality, analyticity, and primitivity interact to produce degree-sensitive correspondences, unifying disparate adjunctions under the eigenmonad paradigm. The framework paves the way for studying polynomial $\mathcal{C}$-modules in a principled way and has potential applications to Habiro–Massuyeau categories and related algebraic structures. Overall, the article offers a robust, transferable method for translating between Lawvere-theoretic and operadic/PROP-based perspectives in a degree-conscious setting.
Abstract
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 $\tildeΦ_{\mathcal{C}}$, which carries a canonical adjunction between functor categories over $\mathcal{C}$ and $\tildeΦ_{\mathcal{C}}$. The adjunction is compatible with polynomial degree, in the sense that it gives a correspondence between polynomial $\mathcal{C}$-modules and truncated $\tildeΦ_{\mathcal{C}}$-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 $\tildeΦ_{\mathcal{C}}$ 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.