nLab enriched monad




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

Kleisli presentation

The Kleisli presentation of a VV-enriched monad on a VV-category CC comprises

  • for every object XX, an object T(X)T(X);

  • for every object XX, a morphism η X:IC(X,T(X))\eta_X:I \to C(X,T(X)) in VV;

  • for objects X,YX,Y, a morphism *:C(X,T(Y))C(T(X),T(Y))\ast:C(X,T(Y))\to C(T(X),T(Y)) in VV;

such that

  • f *η=ff^\ast\circ \eta = f,

  • η *=id\eta^\ast = id, and

  • g *f *=(g *f) * g^\ast\circ f^\ast = (g^\ast f)^\ast.

In the setting of monad (in computer science), V=CV=C is typically a cartesian closed category and TT needs to exist in the internal language of VV, so TT is necessarily enriched. For example, if VV is the syntactic category of a programming language, then all the definable monads in the language are VV-enriched.

If CC is VV-enriched with copowers, e.g. if V=CV=C, then VV acts on CC. In this circumstance, a VV-enriched monad on CC is the same thing as a VV-strong monad on CC.



  • 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.