Beck’s Monadicity Theorem

Summary

A right adjoint $U:\mathsf{D}\to \mathsf{C}$ is monadic — meaning the comparison functor $K:\mathsf{D}\to {\mathsf{C}}^T$ to the Eilenberg–Moore category of its induced monad is an equivalence, so $\mathsf{D}$ “is” the category of algebras — if and only if $U$ creates coequalizers of $U$-split pairs (Beck, Theorem 5.5.1). The key technical tool is the split coequalizer, an absolute colimit (preserved by every functor), together with the fact that every algebra is canonically a coequalizer of free algebras (Proposition 5.4.2). Monadicity is powerful: a monadic functor reflects isomorphisms and creates all limits (and the colimits the monad preserves), so categories monadic over $\mathsf{Set}$ — groups, rings, modules, lattices, compact Hausdorff spaces — are automatically complete and (for finitary monads) cocomplete.

Overview

Having seen that every adjunction induces a monad (Adjunctions Induce Monads) and every monad has an Eilenberg–Moore adjunction (Algebras for a Monad - Eilenberg-Moore and Kleisli), the recognition problem is: when is a given adjunction (up to equivalence) the Eilenberg–Moore adjunction of its monad? Equivalently, when does $\mathsf{D}$ coincide with the category of algebras? Beck’s monadicity theorem answers this with a checkable condition about a special class of coequalizers.

Main Content

Monadic adjunction / monadic functor (Definition 5.3.1)

An adjunction $F⊣U:\mathsf{C}\rightleftarrows \mathsf{D}$ is monadic if the canonical comparison functor $K:\mathsf{D}\to {\mathsf{C}}^T$ (Proposition 5.2.13) to the Eilenberg–Moore category of the induced monad $T=UF$ is an equivalence of categories. A functor $U:\mathsf{D}\to \mathsf{C}$ is monadic if it admits a left adjoint defining a monadic adjunction. It is strictly monadic if $K$ is an isomorphism of categories.

Split coequalizer (Definition 5.4.4)

A split coequalizer consists of maps

$$x\underset{g}{\stackrel{f}{\rightrightarrows }}y\stackrel{h}{\to }z,\text{with sections }t:y\to x,s:z\to y,$$

satisfying $hf=hg,hs={\mathrm{id}}_z,gt={\mathrm{id}}_y,ft=sh.$ The condition $hf=hg$ makes $h$ a fork under the pair $f,g$.

Split coequalizers are absolute (Lemma 5.4.6)

The underlying fork of a split coequalizer is a coequalizer. Moreover it is an absolute colimit: it is preserved by every functor (the universal property is witnessed by equations, $ks=kft=kgt=k$, which any functor respects).

$U$-split coequalizer; creating coequalizers (Definition 5.4.8)

Given $U:\mathsf{D}\to \mathsf{C}$:

• A $U$-split coequalizer is a parallel pair $f,g:x\rightrightarrows y$ in $\mathsf{D}$ together with an extension of $Uf,Ug$ to a split coequalizer in $\mathsf{C}$.
• $U$ creates coequalizers of $U$-split pairs if any such pair admits a coequalizer in $\mathsf{D}$ whose image under $U$ is (isomorphic to) the given split coequalizer, and any such fork in $\mathsf{D}$ is a coequalizer.
• $U$ strictly creates them if there is a unique lift to a coequalizer in $\mathsf{D}$.

Every algebra is a coequalizer of free algebras (Proposition 5.4.2)

For a monad $(T,\eta ,\mu )$ and a $T$-algebra $(A,a:TA\to A)$, the diagram

$$(T^{2}A,{\mu }_{TA})\underset{{\mu }_A}{\stackrel{Ta}{\rightrightarrows }}(TA,{\mu }_A)\stackrel{a}{\to }(A,a)$$

is a coequalizer in ${\mathsf{C}}^T$ — the “canonical presentation” of an algebra as a quotient of free algebras (generalizing presentation by generators and relations). Its underlying fork in $\mathsf{C}$ is the split coequalizer $T^{2}A\rightrightarrows TA\to A$ of Example 5.4.7 (with splittings ${\eta }_{TA},{\eta }_A$). The monadic forgetful functor $U^T:{\mathsf{C}}^T\to \mathsf{C}$ strictly creates coequalizers of $U^T$-split pairs (Proposition 5.4.9).

Beck's Monadicity Theorem (Theorem 5.5.1)

A right adjoint functor $U:\mathsf{D}\to \mathsf{C}$ is monadic if and only if it creates coequalizers of $U$-split pairs.

More precisely, for the canonical comparison $K:\mathsf{D}\to {\mathsf{C}}^T$, the following are equivalent:

1. $K$ is an equivalence (respectively, isomorphism) of categories;
2. $U$ creates (respectively, strictly creates) coequalizers of $U$-split pairs.

Due to Beck [Bec67]; sometimes called the PTT (“precise tripleability theorem” — “triple” is an old synonym for “monad”; “precise” because the condition is necessary and sufficient).

Proof skeleton

$(1)\Rightarrow (2)$: a monadic adjunction is, up to the equivalence $K$, the Eilenberg–Moore adjunction, and $U^T$ strictly creates such coequalizers (Proposition 5.4.9 / Corollary 5.4.10). $(2)\Rightarrow (1)$: assuming $U$ creates coequalizers of $U$-split pairs, build an inverse equivalence $L:{\mathsf{C}}^T\to \mathsf{D}$ by setting $L(TA,{\mu }_A):=FA$ on free algebras and, for a general algebra $(A,a)$, letting $L(A,a)$ be the coequalizer in $\mathsf{D}$ of $FUFA\rightrightarrows FA$ (which exists because the parallel pair $Fa,{ϵ}_{FA}$ is $U$-split by Example 5.4.7). One checks $LK\cong {\mathrm{id}}_{\mathsf{D}}$ and $KL\cong {\mathrm{id}}_{{\mathsf{C}}^T}$. ∎

Variants: crude/vulgar/reflexive tripleability

Inverse-equivalence constructions adapt to other hypotheses; the literature attaches three-letter acronyms (PTT, plus the crude (CTT) and vulgar (VTT) tripleability theorems, [ML98a §VI.7]). One practical variant:

• Reflexive (crude) tripleability — Proposition 5.5.8. If $U:\mathsf{D}\to \mathsf{C}$ has a left adjoint and (i) $\mathsf{D}$ has coequalizers of reflexive pairs, (ii) $U$ preserves coequalizers of reflexive pairs, and (iii) $U$ reflects isomorphisms, then $U$ is monadic. (A parallel pair $f,g:A\rightrightarrows B$ is reflexive if $f,g$ admit a common section $s:B\to A$.)
• Alternate form (Exercise 5.6.i, via Exercise 3.4.iii): $U$ is monadic iff $U$ has a left adjoint, reflects isomorphisms, and $\mathsf{D}$ has and $U$ preserves coequalizers of $U$-split pairs.

Examples

Monadic over $\mathsf{Set}$ (Corollary 5.5.3, 5.5.6, 5.5.7)

The free $⊣$ forgetful adjunctions make the following monadic over $\mathsf{Set}$: $\mathsf{Monoid}$, $\mathsf{Group}$, $\mathsf{Ab}$, $\mathsf{Ring}$, $\mathsf{CRing}$ (and non-unital variants); ${\mathsf{Set}}^{BG}$, $R\text{-}\mathsf{Mod}$, ${\mathsf{Vect}}_{\mathbb{k}}$, affine spaces ${\mathsf{Aff}}_{\mathbb{k}}$; $\mathsf{Lattice}$ (and meet/join semilattices); pointed sets ${\mathsf{Set}}_{\ast }$; and compact Hausdorff spaces $\mathsf{cHaus}$ (Corollary 5.5.6, proof via Kuratowski closure operators — uses that split coequalizers are absolute). The proof for $\mathsf{Monoid}$ verifies Beck’s condition directly: $U:\mathsf{Monoid}\to \mathsf{Set}$ strictly creates the coequalizer of any pair of homomorphisms whose underlying functions extend to a split coequalizer.

${\mathsf{Set}}^{op}$ is monadic (Paré, Theorem 5.5.9)

The contravariant power-set functor $P:{\mathsf{Set}}^{op}\to \mathsf{Set}$ is monadic. The proof applies the reflexive tripleability theorem (Prop 5.5.8), using that $P$ is self-adjoint (from the cartesian closed structure $\mathsf{Set}(A,PB)\cong \mathsf{Set}(B,PA)$), that ${\mathsf{Set}}^{op}$ has the needed coequalizers, and that $P$ reflects isomorphisms (via the subobject classifier $\Omega$). The argument generalizes to any elementary topos.

Non-examples

$U:\mathsf{Field}\to \mathsf{Set}$ is not monadic (it has no left adjoint). $U:\mathsf{Top}\to \mathsf{Set}$ and $U:\mathsf{Poset}\to \mathsf{Set}$ are not monadic: a monadic functor reflects isomorphisms (Lemma 5.6.1), but there are bijective continuous maps that are not homeomorphisms and bijective monotone maps that are not order-isomorphisms.

What monadicity buys you (§5.6)

Limits and colimits in categories of algebras

Let $U:\mathsf{A}\to \mathsf{C}$ be monadic.

• Lemma 5.6.1: $U$ reflects isomorphisms (is conservative).
• Theorem 5.6.5: $U$ creates (i) any limits $\mathsf{C}$ has, and (ii) any colimits $\mathsf{C}$ has that the monad $T$ and $T^2$ preserve.
• Corollary 5.6.7: any category monadic over $\mathsf{Set}$ is complete, with limits created by the forgetful functor (and $\mathsf{Set}$ is cocomplete, Corollary 5.6.9).
• Corollary 5.6.6: a reflective subcategory of a complete category is complete.
• Proposition 5.6.11 / Theorem 5.6.12: if $\mathsf{C}$ is cocomplete and $U$ monadic, then $\mathsf{A}$ is cocomplete iff it has coequalizers; a finitary monad on a complete, cocomplete, locally small $\mathsf{C}$ yields a complete and cocomplete ${\mathsf{C}}^T$. (A functor is finitary if it preserves filtered colimits; categories of models for an algebraic theory are finitary-monadic over $\mathsf{Set}$, Def 5.5.4–5.5.5, and are cocomplete, Corollary 5.6.14.)

Connections

• Directly continues Algebras for a Monad - Eilenberg-Moore and Kleisli: “monadic” means the comparison functor $K:\mathsf{D}\to {\mathsf{C}}^T$ there is an equivalence.
• Uses Adjunctions Induce Monads: the monad $T=UF$ being recognized is the one induced by the adjunction.
• Idempotent case: the inclusion of a reflective subcategory is monadic (Proposition 5.3.3), giving the cleanest examples; the induced monad is idempotent.
• Connects to Adjoint Functor Theorems: constructing colimits of algebras (Theorem 5.6.12) invokes the General Adjoint Functor Theorem and the solution-set condition; finitary monads correspond to locally finitely presentable categories / Lawvere theories.
• Relies on Cartesian Closed Categories for Paré’s theorem (self-adjoint power-set functor).

See Also

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