Monads - Overview

Summary

A monad is the “shadow” cast by an adjunction $F⊣U:\mathsf{C}\rightleftarrows \mathsf{D}$ on the category $\mathsf{C}$ serving as the codomain of its right adjoint: from $\mathsf{C}$ only the composite endofunctor $UF$, the unit, and a whiskered counit are visible. Abstractly, a monad packages an endofunctor $T$ with a unit $\eta$ and a multiplication $\mu$ satisfying associativity and unit laws — i.e. a monoid in the category of endofunctors. Monads encode “syntactic” algebraic structure on a category; their algebras (the Eilenberg–Moore category) recover the structured objects, and Beck’s Monadicity Theorem characterizes exactly when a category is the category of algebras for a monad.

Overview

Riehl’s Chapter 5 develops the theory in six movements, all organized around one slogan: a monad is the part of an adjunction that survives when you forget the “other” category. Consider an adjunction

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

Viewed from $\mathsf{C}$ (“home”), in ignorance of $\mathsf{D}$ (“abroad”), what remains visible is the endofunctor $UF:\mathsf{C}\to \mathsf{C}$, the unit $\eta$, and a whiskered version $UϵF$ of the counit. This triple of data is a monad on $\mathsf{C}$.

The remarkable converse — that the category $\mathsf{D}$ can often be reconstructed from the monad on $\mathsf{C}$ — is the heart of the chapter. When it can, objects of $\mathsf{D}$ are represented as algebras for the monad, and a monad is “a syntactic presentation of algebraic structure that is potentially borne by objects in the category on which it acts.”

The five notes in this folder cover:

1. Monads and the Monad Laws — the abstract definition: endofunctor $T$, unit $\eta :{\mathrm{id}}_{\mathsf{C}}\Rightarrow T$, multiplication $\mu :T^2\Rightarrow T$, and the associativity & unit coherence laws (Definition 5.1.1).
2. Adjunctions Induce Monads — every adjunction $F⊣U$ yields a monad $T=UF$ with $\mu =UϵF$ (Lemma 5.1.3).
3. Algebras for a Monad - Eilenberg-Moore and Kleisli — the Eilenberg–Moore category ${\mathsf{C}}^T$ of all algebras, the Kleisli category ${\mathsf{C}}_T$ of free algebras, and the comparison functor (§5.2).
4. Beck’s Monadicity Theorem — monadic adjunctions and the precise characterization via split coequalizers (Theorem 5.5.1), plus what monadicity buys you for limits and colimits (§5.6).

Main Content

The big picture

• Definition: a monad $(T,\eta ,\mu )$ is a monoid object in the strict monoidal category ${\mathsf{C}}^{\mathsf{C}}$ of endofunctors of $\mathsf{C}$, with composition as tensor and ${\mathrm{id}}_{\mathsf{C}}$ as unit. (Wadler’s joke: “a monad is a monoid in the category of endofunctors, what’s the problem?“)
• Source: every adjunction gives a monad (Adjunctions Induce Monads). The dual (“shadow” on the codomain of the left adjoint) is a comonad.
• Converse: every monad arises from an adjunction — in (at least) two universal ways. The Kleisli adjunction is initial; the Eilenberg–Moore adjunction is terminal, among all adjunctions inducing $T$.
• Recognition: Beck’s theorem says a right adjoint $U$ is monadic iff it creates coequalizers of $U$-split pairs — i.e. iff $\mathsf{D}$ is equivalent to the category of algebras.
• Payoff: monadic functors create all limits and reflect isomorphisms, so categories monadic over $\mathsf{Set}$ (groups, rings, modules, lattices, compact Hausdorff spaces, …) are automatically complete.

Examples

The categories that are monadic over $\mathsf{Set}$ form the chapter’s running motivation (Corollary 5.5.3):

• $\mathsf{Monoid}$, $\mathsf{Group}$, $\mathsf{Ab}$, $\mathsf{Ring}$, commutative rings, $R\text{-}\mathsf{Mod}$, ${\mathsf{Vect}}_{\mathbb{k}}$, affine spaces ${\mathsf{Aff}}_{\mathbb{k}}$;
• ${\mathsf{Set}}^{BG}$ (sets with a $G$-action), $\mathsf{Lattice}$, pointed sets ${\mathsf{Set}}_{\ast }$;
• compact Hausdorff spaces $\mathsf{cHaus}$ (Corollary 5.5.6, via the ultrafilter/Stone–Čech monad).

By contrast, fields do not form a category monadic over $\mathsf{Set}$ — which “explains why the category of fields shares few of the properties common to the categories just described.” Likewise $\mathsf{Poset}$ and $\mathsf{Top}$ are not monadic over $\mathsf{Set}$ (Lemma 5.6.1 / Corollary 5.6.4: there are bijective continuous maps that are not homeomorphisms).

Concrete monads to keep in mind (see Monads and the Monad Laws for details): the list/free-monoid monad $TA={\coprod }_{n\ge 0}{A}^n$, the maybe monad $(-{)}_{+}$, the (covariant) power-set monad $P$, and the double power-set monad $P^2$.

Connections

• Builds on adjunctions: requires Adjoint Functors and especially the counit (triangle identity) formulation — the triangle identities are exactly what make $UF$ a monad.
• Feeds into algebraic universal algebra: finitary monads ↔ Lawvere theories ↔ categories of models for an algebraic theory (§5.5).
• Limits & colimits: monadicity transfers completeness/cocompleteness; connects to Adjoint Functor Theorems (the construction of free algebras and of colimits of algebras uses the General Adjoint Functor Theorem).
• Cartesian closure: cartesian closed structure underlies the continuation monads $S^{S^{-}}$ and the self-adjointness of the contravariant power-set functor (used to prove ${\mathsf{Set}}^{op}$ is monadic, Theorem 5.5.9).

See Also

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