Commutants for enriched algebraic theories and monads

Abstract

We define and study a notion of $\textit{commutant}$ for $\mathcal{V}$-enriched $\mathcal{J}$-algebraic theories for a system of arities $\mathcal{J}$, recovering the usual notion of commutant or centralizer of a subring as a special case alongside Wraith's notion of commutant for Lawvere theories as well as a notion of commutant for $\mathcal{V}$-monads on a symmetric monoidal closed category $\mathcal{V}$. This entails a thorough study of commutation and Kronecker products of operations in $\mathcal{J}$-theories. In view of the equivalence between $\mathcal{J}$-theories and $\mathcal{J}$-ary monads we reconcile this notion of commutation with Kock's notion of commutation of cospans of monads and, in particular, the notion of commutative monad. We obtain notions of $\mathcal{J}$-$\textit{ary commutant}$ and $\textit{absolute commutant}$ for $\mathcal{J}$-ary monads, and we show that for finitary monads on Set the resulting notions of finitary commutant and absolute commutant coincide. We examine the relation of the notion of commutant to both the notion of $\textit{codensity monad}$ and the notion of $\textit{algebraic structure}$ in the sense of Lawvere.