Enriched algebraic theories and monads for a system of arities

Abstract

Under a minimum of assumptions, we develop in generality the basic theory of universal algebra in a symmetric monoidal closed category $\mathcal{V}$ with respect to a specified system of arities $j:\mathcal{J} \hookrightarrow \mathcal{V}$. Lawvere's notion of algebraic theory generalizes to this context, resulting in the notion of single-sorted $\mathcal{V}$-enriched $\mathcal{J}$-cotensor theory, or $\mathcal{J}$-theory for short. For suitable choices of $\mathcal{V}$ and $\mathcal{J}$, such $\mathcal{J}$-theories include the enriched algebraic theories of Borceux and Day, the enriched Lawvere theories of Power, the equational theories of Linton's 1965 work, and the $\mathcal{V}$-theories of Dubuc, which are recovered by taking $\mathcal{J} = \mathcal{V}$ and correspond to arbitrary $\mathcal{V}$-monads on $\mathcal{V}$. We identify a modest condition on $j$ that entails that the $\mathcal{V}$-category of $\mathcal{T}$-algebras exists and is monadic over $\mathcal{V}$ for every $\mathcal{J}$-theory $\mathcal{T}$, even when $\mathcal{T}$ is not small and $\mathcal{V}$ is neither complete nor cocomplete. We show that $j$ satisfies this condition if and only if $j$ presents $\mathcal{V}$ as a free cocompletion of $\mathcal{J}$ with respect to the weights for left Kan extensions along $j$, and so we call such systems of arities eleutheric. We show that $\mathcal{J}$-theories for an eleutheric system may be equivalently described as (i) monads in a certain one-object bicategory of profunctors on $\mathcal{J}$, and (ii) $\mathcal{V}$-monads on $\mathcal{V}$ satisfying a certain condition. We prove a characterization theorem for the categories of algebras of $\mathcal{J}$-theories, considered as $\mathcal{V}$-categories $\mathcal{A}$ equipped with a specified $\mathcal{V}$-functor $\mathcal{A} \rightarrow \mathcal{V}$.