Adjunctions Induce Monads

Summary

Every adjunction induces a monad. Given $F⊣U:\mathsf{C}\rightleftarrows \mathsf{D}$ with unit $\eta :{\mathrm{id}}_{\mathsf{C}}\Rightarrow UF$ and counit $ϵ:FU\Rightarrow {\mathrm{id}}_{\mathsf{D}}$, the data $(T=UF,\eta ,\mu =UϵF)$ is a monad on $\mathsf{C}$ — the codomain of the right adjoint. The endofunctor is the composite $UF$, the unit is the adjunction’s unit, and the multiplication is the whiskered counit $UϵF:UFUF\Rightarrow UF$. The monad laws are precisely the triangle identities (for the unit law) plus naturality of $ϵ$ (for associativity). (Writing the right adjoint as $G$, this is the familiar construction $T=GF$, $\mu =GϵF$.)

Overview

This is the source of “almost all” monads in practice. Riehl frames it as the shadow an adjunction casts on the “home” category $\mathsf{C}$: from $\mathsf{C}$, in ignorance of $\mathsf{D}$, only the composite endofunctor $UF$, the unit $\eta$, and a whiskered version of the counit are visible — and exactly that data is a monad. The dual statement (Definition 5.1.6 / comonad) is that an adjunction casts a comonad on $\mathsf{D}$, the codomain of the left adjoint.

The converse question — does every monad arise this way? — is answered “yes” in Algebras for a Monad - Eilenberg-Moore and Kleisli (two canonical such adjunctions: Kleisli and Eilenberg–Moore).

Main Content

Every adjunction gives a monad (Lemma 5.1.3)

Any adjunction

$$\mathsf{C}\underset{U}{\stackrel{F}{\rightleftarrows }}\mathsf{D},\eta :{\mathrm{id}}_{\mathsf{C}}\Rightarrow UF,ϵ:FU\Rightarrow {\mathrm{id}}_{\mathsf{D}}$$

gives rise to a monad on the category $\mathsf{C}$ serving as the codomain of the right adjoint $U$, with

• the endofunctor $T:=UF$,
• the unit $\eta :{\mathrm{id}}_{\mathsf{C}}\Rightarrow UF$ of the adjunction serving as the monad unit $\eta :{\mathrm{id}}_{\mathsf{C}}\Rightarrow T$, and
• the whiskered counit $UϵF:UFUF\Rightarrow UF$ serving as the multiplication $\mu :T^2\Rightarrow T$.

Equivalently, with the right adjoint named $G$: $T=GF$ and $\mu =GϵF$.

Proof idea

The two unit-law triangles for the monad,

$$UF\stackrel{\eta UF}{\to }UFUF\stackrel{UF\eta }{\leftarrow }UF,UϵF\text{ down the middle, }{\mathrm{id}}_{UF}\text{ on the diagonals,}$$

commute by the triangle identities of the adjunction ($ϵF\cdot F\eta ={\mathrm{id}}_F$ and $Uϵ\cdot \eta U={\mathrm{id}}_U$, whiskered appropriately). The associativity square,

$$UFUFUF\stackrel{UFUϵF}{\to }UFUF,UϵFUF\downarrow \downarrow UϵF,UFUF\stackrel{UϵF}{\to }UF,$$

commutes by naturality of the vertical natural transformation $Uϵ:UFU\Rightarrow U$. The two composites $UϵF\cdot UFUϵF$ and $UϵF\cdot UϵFUF$ together define the horizontal composite of $ϵ$ with itself. ∎

Examples

$\mathsf{Set}\rightleftarrows \mathsf{Ab}$ → free abelian group monad

Take the free $⊣$ forgetful adjunction $F⊣U:\mathsf{Set}\rightleftarrows \mathsf{Ab}$, then “forget abelian groups entirely.” What remains on $\mathsf{Set}$: the endofunctor $UF$ sends a set to the set of finite formal integer sums of its elements; the unit ${\eta }_S:S\to UFS$ sends an element to its singleton sum; and the multiplication $UϵF$ has components that are instances of the evaluation map (regarded as a function, not a group homomorphism) on free groups. The counit ${ϵ}_A:FUA\to A$ itself (evaluate a formal sum to its actual sum in $A$) is not visible from $\mathsf{Set}$ — only its whiskering is.

Free $⊣$ forgetful adjunctions

Each canonical monad on $\mathsf{Set}$ (see Examples) arises this way:

• pointed sets ⊣ → maybe monad $(-{)}_{+}$;
• monoids ⊣ → list / free-monoid monad ${\coprod }_{n\ge 0}{A}^n$;
• $R$-modules ⊣ → free $R$-module monad $R[-]$; groups ⊣ → free group monad;
• $\mathsf{Set}\to \mathsf{Top}\to \mathsf{cHaus}$ (composite adjunction) → Stone–Čech / ultrafilter monad $\beta$;
• quivers ⊣ $\mathsf{Cat}$ → the free-category monad on quivers.

Adjunctions need not be of “free ⊣ forgetful” type: the self-adjoint contravariant power-set functor $P:{\mathsf{Set}}^{op}\rightleftarrows \mathsf{Set}$ induces the double power-set monad $P^2$ (Example 5.1.4(vii)).

Reflective subcategories → idempotent monads

When $F⊣U$ exhibits $\mathsf{D}$ as a reflective subcategory of $\mathsf{C}$, the induced monad is idempotent: its multiplication $\mu :T^2\Rightarrow T$ is a natural isomorphism (Exercise 5.1.iii). Idempotent monads have a very simple algebra theory — see Algebras for a Monad - Eilenberg-Moore and Kleisli and Proposition 5.3.3.

Connections

• Upstream: this is the concrete realization of the abstract Monads and the Monad Laws; the verification is the triangle identities of Adjoint Functors.
• Downstream — the converse: every monad arises from some adjunction. §5.2 constructs the two canonical ones (Kleisli $F_T⊣U_T$ and Eilenberg–Moore $F^T⊣U^T$); both have underlying monad exactly $(T,\eta ,\mu )$.
• Recognition: an adjunction that recovers $\mathsf{D}$ from its monad is called monadic — see Beck’s Monadicity Theorem.
• Comonad dual: the same construction on ${\mathsf{C}}^{op}$ casts a comonad on $\mathsf{D}$ (the codomain of the left adjoint).

See Also

• Monads and the Monad Laws
• Algebras for a Monad - Eilenberg-Moore and Kleisli
• Beck’s Monadicity Theorem
• Adjoint Functors · Units and Counits · Monads - Overview
• Source: Riehl - Category Theory in Context.pdf