The notion of enriched monads is that of monads in the context of enriched category theory: For $V$ a base of enrichment, a $V$-enriched monad is a monad internal to the 2-category VCat of $V$-enriched categories.
Generally (except in the base case $V =$ Set) the structure of a $V$-enriched monad on a $V$-enriched category $\mathbf{C}$ is stronger than that of the underlying monad on the underlying Set-category $C$, whence one also speaks of strong monads (a priori a different notion, which however coincides with that of enriched monads under mild conditions, such as when $V$ is closed, see there).
The concept of enriched monads is key for the application of monads in computer science, since a monad coded verbatim in a functional programming language — where function types $X \to Y$ are to be interpreted not as external hom-sets but as internal homs in the ambient closed monoidal category $V$ of data types — is really a $V$-enriched monad (hence typicall a strong monad).
Let
$(V, \otimes, I)$ be a symmetric monoidal category which serves as the base of enrichment,
with $I$ denoting its unit object.
$\mathbf{C}$ be a $V$-enriched category
with underlying Set-category denoted $C$, and
with hom-objects between any pais of objects $X, Y$ denoted $\mathbf{C}(X,Y) \,\in\, V$.
A $V$-enriched monad on $\mathbf{C}$ is, in Kleisli triple-presentation (eg. McDermott & Uustalu 2022, Def. 5.8):
for every object $X \,\in\, \mathbf{C}$, an object $T(X) \,\in\,\mathbf{C}$;
for every object $X \,\in\,\mathbf{C}$, a morphism in $V$ of the form
(the monad unit)
for all pairs of objects $X,Y$ of $mathbf{C}$, a morphism in $V$ of the form
(the Kleisli extension or bind-operation)
such that the structural equations on a Kleisli triple
$bind(f) \circ ret_X = f$,
$bind(ret_X\big) = id$
$bind(g) \circ bind(f) = bind\big(bind(g) \circ f\big)$
hold for generalized elements $f$, $g$ of the hom-objects $\mathbf{C}\big(X,\,T(Y)\big)$, $\mathbf{C}\big(Y,\,T(Z)\big)$, which means that the following diagrams commute in $V$ for all objects $X, Y, Z$ of $\mathbf{C}$:
(equivalence between $V$-strength and $V$-enrichment) For
$V$ a closed monoidal category
$\mathbf{C}$, $\mathbf{D}$ a pair of $V$-enriched categories
then there is a bijection between the structures of
on the underlying functors,
and for any pair $F$, $G$ of such a bijection between
It follows in particular that there is a bijection between
on such $\mathbf{C}$.
For $V$ a closed monoidal category the further assumptions in Prop. apply to $\mathbf{C} = \mathbf{D} \coloneqq \mathbf{V}$ being the canonical self-enrichment of $V$.
Moreover:
For $V$ a symmetric monoidal closed category, there is a bijection between the structures of
on underlying monads on $V$.
Hence from combining Prop. with Prop. we get:
For $V$ a symmetric closed monoidal category with $\mathbf{V}$ denoting its self-enriched category, every symmetric monoidal monad on $V$ gives the structure of a $V$-enriched monad on $\mathbf{V}$.
In the case in of enrichment by truth values, a monad is a closure operator on a poset.
In the application of monads in computer science, $\mathbf{C} = \mathbf{V}$ is typically the base of enrichment itself, canonically enriched over itself. For classical computing this is typically a cartesian closed category.
Since declaring a monad in a functional programming language means to define it in the internal language of $V$, such monads in computer science are actually enriched, see the discussion there.
For example, if $V$ is the syntactic category of a programming language, then all the definable monads in the language are $V$-enriched.
Kruna S. Ratkovic, Strength and Enrichment, Section 3.2 of: Morita theory in enriched context (2012) [arXiv:1302.2774, hal:tel-00785301]
Dylan McDermott, Tarmo Uustalu, Def. 5.6 in: What Makes a Strong Monad?, EPTCS 360 (2022) 113-133 [arXiv:2207.00851, doi:10.4204/EPTCS.360.6]
See also:
Max Kelly, John Power, Adjunctions whose counits are coequalizers and presentations of finitary enriched monads, Journal of Pure and Applied Algebra 89 (1993) [doi:10.1016/0022-4049(93)90092-8]
John Power, Enriched Lawvere theories, Theory and Applications of Categories 6 7 (1999) 83-93 [tac:6-07, pdf]
Eduardo Dubuc, Kan Extensions in Enriched Category Theory, Lecture Notes in Mathematics 145, Springer (1970) [doi:10.1007/BFb0060485]
Last revised on August 23, 2023 at 11:09:32. See the history of this page for a list of all contributions to it.