Enriched Lawvere theories

John Power

We define the notion of enriched Lawvere theory, for enrichment over a monoidal biclosed category $V$ that is locally finitely presentable as a closed category. We prove that the category of enriched Lawvere theories is equivalent to the category of finitary monads on $V$. Moreover, the $V$-category of models of a Lawvere $V$-theory is equivalent to the $V$-category of algebras for the corresponding $V$-monad. This all extends routinely to local presentability with respect to any regular cardinal. We finally consider the special case where $V$ is $Cat$, and explain how the correspondence extends to pseudo maps of algebras.

Keywords: Lawvere theory, monad.

1991 MSC: 18C10, 18C15, 18D05.

Theory and Applications of Categories, Vol. 6, 1999, No. 7, pp 83-93.


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