The concept monad in the context of enriched category theory, so a monad in the 2-category VCat of $V$-enriched categories.
The Kleisli presentation of a $V$-enriched monad on a $V$-category $C$ comprises
for every object $X$, an object $T(X)$;
for every object $X$, a morphism $\eta_X:I \to C(X,T(X))$ in $V$;
for objects $X,Y$, a morphism $\ast:C(X,T(Y))\to C(T(X),T(Y))$ in $V$;
such that
$f^\ast\circ \eta = f$,
$\eta^\ast = id$, and
$g^\ast\circ f^\ast = (g^\ast f)^\ast$.
In the setting of monad (in computer science), $V=C$ is typically a cartesian closed category and $T$ needs to exist in the internal language of $V$, so $T$ is necessarily enriched. For example, if $V$ is the syntactic category of a programming language, then all the definable monads in the language are $V$-enriched.
If $C$ is $V$-enriched with copowers, e.g. if $V=C$, then $V$ acts on $C$. In this circumstance, a $V$-enriched monad on $C$ is the same thing as a $V$-strong monad on $C$.
Max Kelly and John Power, Adjunctions whose counits are coequalizers and presentations of finitary enriched monads, Journal of Pure and Applied Algebra vol 89, 1993. (pdf).
John Power, Enriched Lawvere theories, (tac)
Eduardo Dubuc, Kan Extensions in Enriched Category Theory, Springer, 1970.
Last revised on June 23, 2022 at 08:15:23. See the history of this page for a list of all contributions to it.