nLab monadicity theorem



Category theory




In category theory and universal algebra, a monadicity theorem serves to characterize/recognize whether a given functor is a monadic functor.



Given a functor U:DCU \colon D \rightarrow C, then a parallel pair f,g:abf,g : a \rightarrow b in DD is called UU-split if the pair Uf,UgU f, U g has a split coequalizer in CC. Specifically, this means that there is a diagram in CC:

UaUfUgUbhcU a \;\underoverset{U f}{U g}{\rightrightarrows}\; U b \;\overset{h}{\rightarrow}\; c

such that hUf=hUgh \cdot U f = h \cdot U g, and hh and UfU f have respective sections ss and tt satisfying Ugt=shU g \cdot t = s \cdot h. This implies that the arrow hh is necessarily a coequalizer of UfU f and UgU g.

The functor UU is said to create coequalizers of UU-split pairs if for any such UU-split pair, there exists a coequalizer ee of f,gf,g in DD which is preserved by UU, and moreover any fork in DD whose image in CC is a split coequalizer must itself be a coequalizer (not necessarily split).


(Beck’s monadicity theorem, tripleability theorem)
A functor U:DCU \colon D \rightarrow C is monadic (or tripleable) if and only if

  1. UU has a left adjoint, and

  2. UU creates coequalizers of UU-split pairs, def. .

The proof is reproduced for instance in (MacLane, p. 147-150, Riehl 2017, 5/5/).

An equivalent, and sometimes easier, way to state these conditions is to say that


(Beck’s monadicity theorem – alternative formulation)
A functor U:DCU \colon D \to C is monadic precisely if

  1. UU has a left adjoint,

  2. UU reflects isomorphisms (i.e. it is conservative), and

  3. if for a parallel pair (f,g)(f,g) in DD the image (U(f),U(g))\big(U(f),\, U(g)\big) has a split coequalizer in CC then (f,g)(f,g) has a coequalizer in DD which is preserved by UU.

(e.g. Borceux, vol 2, Thm. 4.4.4)

These conditions are equivalent to those in Thm. , because a conservative functor reflects any limits or colimits which exist in its domain and which it preserves (by this Prop.), while monadic functors are always conservative.


The crude monadicity theorem

The crude monadicity theorem gives a sufficient, but not necessary, condition for a functor to be monadic. It states that a functor U:DCU : D \rightarrow C is monadic if

  1. UU has a left adjoint
  2. UU reflects isomorphisms
  3. DD has and UU preserves coequalizers of reflexive pairs.

(Recall that a parallel pair f,g:abf,g : a \rightarrow b is reflexive if ff and gg have a common section.) This sufficient, but not necessary, condition is sometimes easier to verify in practice. In contrast to the crude monadicity theorem, the necessary and sufficient condition above is sometimes called the precise monadicity theorem.

A further advantage of crude monadicity is this: while in general the composite of monadic functors need not be monadic, if U 1:DCU_1\colon D \to C satisfies the hypotheses of the crude monadicity theorem and U 2:CBU_2\colon C \to B is any monadic functor then U 2U 1U_2 \circ U_1 is monadic. Furthermore, if both U 1U_1 and U 2U_2 both obey the hypotheses of the crude monadicity theorem, so does U 2U 1U_2 \circ U_1. See (BarrWells, Proposition 3.5.1) for these and further results.

Duskin’s monadicity theorem

Duskin’s monadicity theorem gives a different sufficient condition, which is necessary under light conditions on the categories involved and which refers only to quotients of congruences.


(Duskin monadicity theorem)
A functor U:DCU \colon D \to C between finitely complete categories is monadic if

  1. UU has a left adjoint,

  2. UU creates coequalizers for congruences in DD whose images in CC have split coequalizers.

If CC admits all coequalizers of contractible pairs then these conditions are also necessary.

We can weaken the hypothesis a bit further to obtain the theorem:

As usual, we can also modify it by replacing reflection of limits by reflection of isomorphisms.

  • A conservative right adjoint U:DCU\colon D \to C between finitely complete categories is monadic if any congruence in DD which has a quotient in CC already has a quotient in DD, and that quotient that is preserved by UU.

If we view the objects of DD as underlying CC-objects with structure, this says that any congruence in DD induces a DD-structure on its quotient in CC. As with the crude monadicity theorem, this condition is sometimes easier to verify since quotients of congruences are often better-behaved than arbitrary coequalizers. This is the case in many “algebraic” situations.

Duskin actually gave a slightly more precise version only assuming the categories CC and DD to have particular finite limits, rather than all of them.

Monadicity over Set

In the case when the base category CC is Set, one can further refine the requisite conditions. Linton proved that a functor U:DSetU\colon D\to Set is monadic if and only if:

  1. UU has a left adjoint,

  2. DD admits kernel pairs and coequalizers,

  3. A parallel pair RSR \rightrightarrows S in DD is a kernel pair if and only if its image under UU is so,

  4. A morphism ABA\to B in DD is a regular epimorphism if and only if its image under UU is so.

Strict monadicity

The version of the monadicity theorem given in MacLane 1971 uses a notion of “creation of limits” which fails to observe the principle of equivalence, concluding that the comparison functor is an isomorphism of categories, rather than merely an equivalence. But the versions mentioned above can be found in the exercises.

Note however that if U:DCU: D \to C is an amnestic isofibration, then UU is monadic iff it is strictly monadic. For an application of this observation, see for example the discussion of algebraically free monads at free monad.

Examples and Applications

Actions over sets


(base change of presheaves along essentially surjective functor)

F:𝒮𝒮 F \,\colon\, \mathcal{S} \overset{\phantom{---}}{\longrightarrow} \mathcal{S}'

be a functor between small categories which is essentially surjective. Then for 𝒞\mathcal{C} any bicomplete category (e.g. Set), the corresponding precomposition-functor on categories of 𝒞\mathcal{C}-valued presheaves

(1)PSh(𝒮;𝒞)F *PSh(𝒮;𝒞) PSh \big( \mathcal{S} ;\, \mathcal{C} \big) \overset{\;\; F^\ast \;\;}{\longleftarrow} PSh \big( \mathcal{S}' ;\, \mathcal{C} \big)

is monadic. To see this, we check the conditions in Thm. :

  1. By the assumption that 𝒞\mathcal{C} is bicomplete, the left and right Kan extensions of presheaves along FF exist and exhibit F *F^\ast as the middle part of a (“base change”) adjoint triple :

    F !F *F *:PSh(𝒮;𝒞)PSh(𝒮;𝒞). F_! \,\dashv\, F^\ast \,\dashv\, F_\ast \;\; \colon \;\; PSh \big( \mathcal{S} ;\, \mathcal{C} \big) \overset{\phantom{----}}{\leftrightarrow} PSh \big( \mathcal{S}' ;\, \mathcal{C} \big) \,.

    This shows that F *F^\ast has a left adjoint. In addition, it is a left adjoint and as such preserves all colimits, hence in particular all coequalizers.

  2. The assumption that FF is essentially surjective implies that F *F^\ast is conservative (because isomorphisms of presheaves are natural transformation of the underlying functors which are natural isomorphisms, which is the case iff their component morphisms for each object is an isomorphism.)


(group actions/G-sets)
Specializating Ex. to the case (where 𝒞=\mathcal{C} = Sets, just for definiteness and) where F:𝒮𝒮F \,\colon\,\mathcal{S} \longrightarrow \mathcal{S}' is the point-inclusion

pt BG:*BG. pt_{\mathbf{B}G} \,\colon\, \ast \overset{\phantom{---}}{\hookrightarrow} \mathbf{B}G \,.

into the delooping groupoid of a (discrete) group, and observing that

  1. PSh(*)PSh\big(\ast \big) \;\simeq\; Sets,

  2. PSh(BG)PSh\big(\mathbf{B}G \big) \;\simeq\; G G Sets

    is the category G-sets, i.e. of GG-group actions with equivariant functions between them,

gives that (1) is the forgetful functor

(pt BG) *:GSetsSets (pt_{\mathbf{B}G})^\ast \,\colon\, G Sets \overset{\phantom{--}}{\longrightarrow} Sets

which is hence monadic. This is of course the forgetful functor on the algebras of the monad

TG×():SetsSets T \,\coloneqq\, G \times (-) \;\colon\; Sets \longrightarrow Sets

which forms the Cartesian product with (the underlying set) of GG, and whose monad product is

TT=G×G×()G×()=T T \circ T \,=\, G \times G \times (-) \longrightarrow G \times (-) \,=\, T

is given by the group operation.


(monoid actions and monoid-oid actions)
In generalization of Ex. , if AA is a monoid and F:𝒮𝒮F \,\colon\, \mathcal{S} \longrightarrow \mathcal{S}' in Ex. is the point inclusion

pt:*BA pt \,\colon\,\ast \longrightarrow \mathbf{B}A

into the corresponding 1-object category (see there), then the monadic functor (1)

(pt BA) *:ASetsSets (pt_{\mathbf{B}A})^\ast \;\colon\; A Sets \longrightarrow Sets

is the forgetful functor on monoid actions, which are the algebras of the monad A×()A \times (-).

In this sense, the general situation of Ex. may be understood as monadicity of modules for a monoid-oid 𝒮\mathcal{S}' defined over a monoid-oid 𝒮\mathcal{S}.

Groups over sets

We will use Duskin’s monadicity theorem (Thm. ) to prove that the forgetful functor

U:GrpsSets U \,\colon\, Grps \overset{\phantom{---}}{\longrightarrow} Sets

from Groups to their underlying Sets is monadic.
Of course, this is also easy to show by explicit computation, but it serves as a useful example of how to use a monadicity theorem.

We first need it to have a left adjoint: this is easy to show by a direct construction of free groups, but we could also invoke the adjoint functor theorem.

It is also easy to show that it is conservative (a bijective group homomorphism is a group isomorphism).

So it remains to consider congruences:

Since limits in GrpGrp are created in SetSet, a congruence in GrpGrp on a group GG is an equivalence relation on GG which is also a subgroup of G×GG\times G. This latter condition means that if g 1g 2g_1\sim g_2 and h 1h 2h_1\sim h_2, then also g 1 1g 2 1g_1^{-1}\sim g_2^{-1} and g 1h 1g 2h 2g_1 h_1 \sim g_2 h_2. Since ggg\sim g for all gg, it follows that ghg\sim h if and only if 1hg 11\sim h g^{-1}, so \sim is determined by the subset HGH\subseteq G of those hGh\in G such that 1h1\sim h. This HH is clearly a subgroup of GG, and moreover a normal subgroup, since if hHh\in H and gGg\in G we have 1=g 1gg 1hg1 = g^{-1} g \sim g^{-1} h g, so g 1hgHg^{-1} h g\in H. Conversely, it is easy to construct a congruence from any normal subgroup, so the two notions are equivalent. It remains only to observe that the quotient of a group by a normal subgroup is, in fact, a quotient of its associated congruence in GrpGrp, which is preserved by UU.

Thus, by Duskin’s monadicity theorem (Thm. ), UU is monadic.

Categories over computads

The monadicity theorem becomes more important when the base category CC is more complicated and harder to work with explicitly, and when the objects of DD are not obviously defined as “objects of CC with extra structure.” For instance, the category of strict 2-categories is monadic over the category of 2-globular sets, essentially by definition, but it is much less trivial to show that it is also monadic over the category of 2-computads. This latter fact can, however, be proven using the monadicity theorem.

Monadic descent

The monadicity theorem also plays a central role in monadic descent.

In (,1)(\infty,1)-categories

There is a version of the monadicity theorem for (∞,1)-monads in section 3.4 of

There is also a 2-categorical approach using quasicategories in


The original reference for the (crude and precise) monadicity theorems is an untitled manuscript of Jon Beck that was distributed around 1966 – 1968. The following is a scan of a copy distributed at the Conference Held at the Seattle Research Center of the Battelle Memorial Institute in June – July 1968, provided by John Kennison:

see also:

  • Michael Barr, Coalgebras in a category of algebras, in: Category Theory, Homology Theory and their Applications I, Lecture Notes in Mathematics 86, Springer (1969) 1-12 [doi:10.1007/BFb0079381]

Textbook accounts:

Further references:

There is a version for Morita contexts instead of monads:

  • Tomasz Brzezinski, Adrian Vazquez Marquez, Joost Vercruysse, The Eilenberg-Moore category and a Beck-type theorem for a Morita context, Appl. Categ. Structures 19 (2011), no. 5, 821–858 MR2836546 doi

On Beck’s theorem for pseudomonads (see higher monadic descent for more details):

Discussion for (infinity,1)-monads:

and realized in the context of quasi-categories:

Last revised on June 22, 2024 at 06:37:59. See the history of this page for a list of all contributions to it.