nLab enriched monad

Idea

The concept monad in the context of enriched category theory, so a monad in the 2-category VCat of $V$-enriched categories.

Kleisli presentation

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 $C$.

Examples

• In the case in of enrichment by truth values, a monad is a closure operator on a poset.

References

• 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.

Related entries

• strong monad

• additive monad

• monad (in computer science)

• tensorial strength

• enriched adjunction

• enriched Lawvere theory