Polynomial functors from Lawvere theories

For a category $𝒞$, a $𝒞$-module is a functor from $𝒞$ to the category of $k$-modules. Here, $k$ denotes a unital commutative ring unless otherwise specified. The (polynomial) degree of a $𝒞$-module serves as an invariant that allows for a systematic study of $𝒞$-modules when $𝒞$ is given as a monoidal category with zero unit. A polynomial $𝒞$-module is a $𝒞$-module with a finite polynomial degree. The origin of these notions goes back to Eilenberg and Mac Lane [1] where $𝒞$ is taken to be an additive category.

Several adjunctions between functor categories, respecting the degree of functors, have been studied.

1. 1.

   Let $R$ be a unital ring, and let ${𝐏}_R$ denote the category of finitely generated projective $R$-modules. There is a classical adjunction between the category of ${𝐏}_{\mathbb{Z}}$-modules and that of $𝔖$-modules, where $𝔖$ denotes the category of finite sets and bijections. When restricted to analytic ${𝐏}_{\mathbb{Z}}$-modules, and if $\mathrm{ch}(k)=0$, then this adjunction induces an equivalence compatible with degree of ${𝐏}_{\mathbb{Z}}$-modules.

2. 2.

   This example extends the previous one to general $R$. For $k=\mathbb{Z}$, Pirashvili [2] gave an equivalence between the category of ${𝐏}_R$-modules of polynomial degree at most $d$, modulo those of degree at most $d-1$, and the category of right $R\sim {𝔖}_d$-modules. Here, $R\sim {𝔖}_d$ denotes the wreath product.

3. 3.

   In [3], Powell constructs an adjunction between the category of ${𝐅}^{𝗈}$-modules and that of modules from the PROP associated with the Lie operad. Here, ${𝐅}^{𝗈}$ denotes the opposite category of finitely generated free groups. Powell also established that, when restricted to analytic ${𝐅}^{𝗈}$-modules, and if $\mathrm{ch}(k)=0$, then the adjunction induces an equivalence, compatible with degree of ${𝐅}^{𝗈}$-modules.

We develop a systematic method for constructing adjunctions that incorporate Powell’s adjunction and the adjunction implicit in Pirashvili’s equivalence. In addition to recovering these known examples, the present work also yields new adjunctions involving functor categories over Lawvere theories, compatible with polynomial degree. Recall that a Lawvere theory is a category with finite products, whose objects are $\mathbb{N}$, with the products on objects given by addition.

Main Result A.([5]) Given a Lawvere theory $𝒞$ with a zero object, we construct a natural $k$-linear PROP ${\tilde{\mathrm{\Phi }}}_{𝒞}$, and establish an (explicit) adjunction between ${\tilde{\mathrm{\Phi }}}_{𝒞}$-modules and $𝒞$-modules. Furthermore, if $𝒞$ satisfies some mild conditions, then (A) ${\tilde{\mathrm{\Phi }}}_{𝒞}(n,m)=0$ if , and (B) the adjunction corresponds polynomial $𝒞$-modules to truncated ${\tilde{\mathrm{\Phi }}}_{𝒞}$-modules, and conversely.

Main Result B.([5]) The result A recovers and refines known results by the following statements: (i) When $𝒞$ is the category of free $R$-modules of finite rank, ${\tilde{\mathrm{\Phi }}}_{𝒞}$ can be described as a category built from wreath products. (ii) When $𝒞$ is the opposite category of finitely generated free nilpotent groups of class $\le c$, ${\tilde{\mathrm{\Phi }}}_{𝒞}$ is isomorphic to the PROP associated with the operad for nilpotent Lie algebras of class $\le c$.

Remark. This work starts from a framework that the author presented at CT2024. The framework has also been applied to polynomial functor theory in a different context [4]. The long term goal of this line was to find an adjunction for functors on Habiro-Massuyeau category, which is currently under preparation.