monadicity theorem




Given a functor U:DCU : 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

where hh has a section ss and UfU f has a section tt such that 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 : D \rightarrow C is monadic (tripleable) if and only if

  1. UU has a left adjoint, and
  2. UU creates coequalizers of UU-split pairs, def. 1.

A proof is reproduced at (Borceux, vol 2, theorem 4.4.4).

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


A functor U:DCU : D \to C is monadic precisely if

  1. UU has a left adjoint,
  2. UU reflects isomorphisms (i.e. it is conservative), and
  3. DD has, and UU preserves, coequalizers of UU-split pairs.

This is equivalent because a conservative functor reflects any limits or colimits which exist in its domain and which it preserves, 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.

Duskin’s monadicity theorem

Duskin’s monadicity theorem gives a different sufficient, but not necessary, condition which refers only to quotients of congruences. It says that a functor U:DCU \colon D \to C is monadic if

  1. UU has a left adjoint
  2. DD and CC are finitely complete
  3. UU creates coequalizers for congruences in DD whose images in CC have split coequalizers.

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

  • A right adjoint between finitely complete categories is monadic if it creates quotients for congruences.

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

There are other versions of this theorem, including generalizations to monadicity over presheaf categories, which can be viewed as analogues of Giraud's theorem.

Strict monadicity

The version of the monadicity theorem given in Categories Work uses an evil notion of “creation of limits” and concludes 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

Groups over sets

We will use Duskin’s variant to prove that the forgetful functor U:U\colonGrp\toSet 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, 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


Canonical textbook references include

Other references include:

There is a version for Morita contexts instead of monads:

  • Tomasz Brzeziński, 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

Discussion for (infinity,1)-monads is in

and realized in the context of quasi-categories in

Revised on September 9, 2015 06:21:16 by Urs Schreiber (