Under a minimum of assumptions, we develop in generality the basic theory of universal algebra in a symmetric monoidal closed category $V$ with respect to a specified system of arities $j:J \hookrightarrow V$. Lawvere's notion of algebraic theory generalizes to this context, resulting in the notion of single-sorted $V$-enriched $J$-cotensor theory, or $J$-theory for short. For suitable choices of $V$ and $J$, such $J$-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 $V$-theories of Dubuc, which are recovered by taking $J = V$ and correspond to arbitrary $V$-monads on $V$. We identify a modest condition on $j$ that entails that the $V$-category of $T$-algebras exists and is monadic over $V$ for every $J$-theory $T$, even when $T$ is not small and $V$ is neither complete nor cocomplete. We show that $j$ satisfies this condition if and only if $j$ presents $V$ as a free cocompletion of $J$ with respect to the weights for left Kan extensions along $j$, and so we call such systems of arities eleutheric. We show that $J$-theories for an eleutheric system may be equivalently described as (i) monads in a certain one-object bicategory of profunctors on $J$, and (ii) $V$-monads on $V$ satisfying a certain condition. We prove a characterization theorem for the categories of algebras of $J$-theories, considered as $V$-categories $A$ equipped with a specified $V$-functor $A \rightarrow V$.
Keywords: algebraic theory; Lawvere theory; universal algebra; monad; enriched category theory; free cocompletion
2010 MSC: 18C10, 18C15, 18C20, 18C05, 18D20, 18D15, 08A99, 08B99, 08B20, 08C05, 08C99, 03C05, 18D35, 18D10, 18D25, 18A35
Theory and Applications of Categories, Vol. 31, 2016, No. 5, pp 101-137.
Published 2016-01-31.
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