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enriched monad in nLab

Contents

Context

Enriched category theory

enriched category theory

Background

Basic concepts

Universal constructions

weighted limit
end, coend

Homotopical enrichment

Relation to strong and monoidal monads

Idea

The notion of enriched monads is that of monads in the context of enriched category theory: For VV a base of enrichment, a VV-enriched monad is a monad internal to the 2-category VCat of VV-enriched categories.

Generally (except in the base case V=V = Set) the structure of a VV-enriched monad on a VV-enriched category C\mathbf{C} is stronger than that of the underlying monad on the underlying Set-category CC, 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 VV 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→YX \to Y are to be interpreted not as external hom-sets but as internal homs in the ambient closed monoidal category VV of data types — is really a VV-enriched monad (hence typically a strong monad).

Definition

Let

(V,⊗,I)(V, \otimes, I) be a symmetric monoidal category which serves as the base of enrichment,

with II denoting its unit object.
C\mathbf{C} be a VV-enriched category

with underlying Set-category denoted CC, and

with hom-objects between any pais of objects X,YX, Y denoted C(X,Y)∈V\mathbf{C}(X,Y) \,\in\, V.

Definition

A VV-enriched monad on C\mathbf{C} is, in Kleisli triple-presentation (eg. McDermott & Uustalu 2022, Def. 5.8):

for every object X∈CX \,\in\, \mathbf{C}, an object T(X)∈CT(X) \,\in\,\mathbf{C};
for every object X∈CX \,\in\,\mathbf{C}, a morphism in VV of the form

ret X:I→C(X,T(X)) ret_X \;\colon\; I \to \mathbf{C}\big(X,T(X)\big)

(the monad unit)
for all pairs of objects X,YX,Y of mathbfCmathbf{C}, a morphism in VV of the form

bind X,Y:C(X,T(Y))→C(T(X),T(Y)) bind_{X,Y} \;\colon\; \mathbf{C}\big(X,T(Y)\big) \to \mathbf{C}\big(T(X),T(Y)\big)

(the Kleisli extension or bind-operation)

such that the structural equations on a Kleisli triple

bind(f)∘ret X=fbind(f) \circ ret_X = f,
bind(ret X)=idbind(ret_X\big) = id
bind(g)∘bind(f)=bind(bind(g)∘f)bind(g) \circ bind(f) = bind\big(bind(g) \circ f\big)

hold for generalized elements ff, gg of the hom-objects C(X,T(Y))\mathbf{C}\big(X,\,T(Y)\big), C(Y,T(Z))\mathbf{C}\big(Y,\,T(Z)\big), which means that the following diagrams commute in VV for all objects X,Y,ZX, Y, Z of C\mathbf{C}:

Properties

Relation to strong and monoidal monads

(Ratkovic 2012, §3.2; McDermott & Uustalu 2022, Prop. 5.4, 5.8)

Moreover:

(Kock 1972, Thm. 3.2, review in GLLN08, §7.3, §A.4, Ratkovic 2012, Prop. 3.3.9)

Hence from combining Prop. with Prop. we get:

Examples

References

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]