Category theory; homological algebra Monads and categorical algebra Theorem

Beck's monadicity theorem

Builds on Monadic functor Adjoint functors Coequalizer Eilenberg–Moore category

Statement

Let $U \colon \mathcal{D} \to \mathcal{C}$ be a functor. A pair of morphisms $f, g \colon d \to d'$ in $\mathcal{D}$ is $U$ -split if $U(f), U(g)$ have a split coequalizer in $\mathcal{C}$ . A split coequalizer of $p, q \colon x \to y$ is a morphism $e \colon y \to z$ with $s \colon z \to y$ and $t \colon y \to x$ such that $e p = e q$ , $e s = \id_z$ , $p t = \id_y$ and $q t = s e$ . Then $U$ is monadic if and only if these three conditions hold.

$U$ has a left adjoint.
$U$ reflects isomorphisms: if $U(h)$ is an isomorphism, then $h$ is an isomorphism.
Every $U$ -split pair in $\mathcal{D}$ has a coequalizer in $\mathcal{D}$ , and $U$ preserves these coequalizers.

Remarks

A split coequalizer is a coequalizer, and every functor preserves it, because the equations that define it are preserved. For the proof, let $T = U F$ and let $(A, a)$ be an algebra in the Eilenberg–Moore category. The pair $F(a), \varepsilon_{F(A)} \colon F U F (A) \to F(A)$ is $U$ -split, and its coequalizer defines a left adjoint of the comparison functor. Conditions 2 and 3 then make the unit and counit of this adjunction isomorphisms.

Example: the forgetful functor from groups to sets satisfies all three conditions, so it is monadic. The same holds for rings, modules over a ring, and every variety of universal algebras.

Without condition 2 the conclusion fails. The forgetful functor from topological spaces to sets has a left adjoint, but a continuous bijection need not be a homeomorphism, and this functor is not monadic.

Going deeper

For a student

Beck's theorem is the standard tool for proving monadicity, for example of compact Hausdorff spaces over sets or of modules over a ring. There is also a variant that uses reflexive coequalizers: a right adjoint that reflects isomorphisms and in whose domain reflexive coequalizers exist and are preserved is monadic. The dual statement characterizes comonadic functors, and in algebraic geometry it is the basis of descent theory. In the strict form, $U$ creates coequalizers of $U$ -split pairs, and the comparison functor is then an isomorphism.

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Tag	`FETZ0VZGMD`
URI	`m2://FETZ0VZGMD`
Kind	statement, theorem
Writing	W0 stub
MSC	18C15, 18C20
Status	This copy of the site has no build data. Run `m2 status FETZ0VZGMD`.
About	Monadic functor
Uses	Monadic functor, Adjoint functors, Coequalizer, Eilenberg–Moore category
Same as	Wikidata Q4878586 Wikipedia Beck's monadicity theorem
Changed	2 November 2026

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