nLab monadicity theorem

Idea

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

Statement

Definition

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

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

such that $h \cdot U f = h \cdot U g$, and $h$ and $U f$ have respective sections $s$ and $t$ satisfying $U g \cdot t = s \cdot h$. This implies that the arrow $h$ is necessarily a coequalizer of $U f$ and $U g$.

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

Theorem

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

1. $U$ has a left adjoint, and

2. $U$ creates coequalizers of $U$-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

Theorem

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

1. $U$ has a left adjoint,

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

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

(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.

Variants

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 $U : D \rightarrow C$ is monadic if

1. $U$ has a left adjoint
2. $U$ reflects isomorphisms
3. $D$ has and $U$ preserves coequalizers of reflexive pairs.

(Recall that a parallel pair $f,g : a \rightarrow b$ is reflexive if $f$ and $g$ 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 $U_1\colon D \to C$ satisfies the hypotheses of the crude monadicity theorem and $U_2\colon C \to B$ is any monadic functor then $U_2 \circ U_1$ is monadic. Furthermore, if both $U_1$ and $U_2$ both obey the hypotheses of the crude monadicity theorem, so does $U_2 \circ U_1$. 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.

Theorem

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

1. $U$ has a left adjoint,

2. $U$ creates coequalizers for congruences in $D$ whose images in $C$ have split coequalizers.

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

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 $U\colon D \to C$ between finitely complete categories is monadic if any congruence in $D$ which has a quotient in $C$ already has a quotient in $D$, and that quotient that is preserved by $U$.

If we view the objects of $D$ as underlying $C$-objects with structure, this says that any congruence in $D$ induces a $D$-structure on its quotient in $C$. 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 $C$ and $D$ to have particular finite limits, rather than all of them.

Monadicity over Set

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

1. $U$ has a left adjoint,

2. $D$ admits kernel pairs and coequalizers,

3. A parallel pair $R \rightrightarrows S$ in $D$ is a kernel pair if and only if its image under $U$ is so,

4. A morphism $A\to B$ in $D$ is a regular epimorphism if and only if its image under $U$ 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 $U: D \to C$ is an amnestic isofibration, then $U$ 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

Example

(base change of presheaves along essentially surjective functor)
Let

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

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

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

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

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

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

   This shows that $F^\ast$ 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 $F$ is essentially surjective implies that $F^\ast$ 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.)

Example

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

$$pt_{\mathbf{B}G} \,\colon\, \ast \overset{\phantom{---}}{\hookrightarrow} \mathbf{B}G \,.$$

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

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

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

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

gives that (1) is the forgetful functor

$$(pt_{\mathbf{B}G})^\ast \,\colon\, G Sets \overset{\phantom{--}}{\longrightarrow} Sets$$

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

$$T \,\coloneqq\, G \times (-) \;\colon\; Sets \longrightarrow Sets$$

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

$$T \circ T \,=\, G \times G \times (-) \longrightarrow G \times (-) \,=\, T$$

is given by the group operation.

Example

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

$$pt \,\colon\,\ast \longrightarrow \mathbf{B}A$$

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

$$(pt_{\mathbf{B}A})^\ast \;\colon\; A Sets \longrightarrow Sets$$

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

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

Groups over sets

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

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

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 $Grp$ are created in $Set$, a congruence in $Grp$ on a group $G$ is an equivalence relation on $G$ which is also a subgroup of $G\times G$. This latter condition means that if $g_1\sim g_2$ and $h_1\sim h_2$, then also $g_1^{-1}\sim g_2^{-1}$ and $g_1 h_1 \sim g_2 h_2$. Since $g\sim g$ for all $g$, it follows that $g\sim h$ if and only if $1\sim h g^{-1}$, so $\sim$ is determined by the subset $H\subseteq G$ of those $h\in G$ such that $1\sim h$. This $H$ is clearly a subgroup of $G$, and moreover a normal subgroup, since if $h\in H$ and $g\in G$ we have $1 = g^{-1} g \sim g^{-1} h g$, so $g^{-1} h g\in H$. 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 $Grp$, which is preserved by $U$.

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

Categories over computads

The monadicity theorem becomes more important when the base category $C$ is more complicated and harder to work with explicitly, and when the objects of $D$ are not obviously defined as “objects of $C$ 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 $(\infty,1)$-categories

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

• Jacob Lurie, Noncommutative algebra (arXiv)

There is also a 2-categorical approach using quasicategories in

• Emily Riehl, Dominic Verity The 2-category theory of quasi-categories (arXiv), Homotopy coherent adjunctions and the formal theory of monads (arXiv)

Related concepts

• monadic functor, comonadic functor

• functor of descent type

• monadic decomposition

• Bénabou-Roubaud's theorem

• monadic descent

References

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:

• Jon Beck, untitled manuscript, 1968 (pdf)

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:

• Saunders MacLane, §VI.7 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971) [doi:10.1007/978-1-4757-4721-8]

• Francis Borceux, Section 4 in volume 2 of Handbook of Categorical Algebra, in 3 vols.

• Chapter 3 of Michael Barr, Charles Wells, Toposes, Triples, and Theories Grundlehren der math. Wissenschaften 278, Springer-Verlag 1983. Available as TAC Reprint 12.

Further references:

• FGA explained, SGA 1

• Fred Linton, Some aspects of equational categories, Proceedings of the Conference on Categorical Algebra, Springer (1966) [doi:10.1007/978-3-642-99902-4_3]

• Jack Duskin, Variations on Beck’s tripleability criterion, in: Reports of the Midwest Category Seminar III, Lecture Notes in Mathematics 106, Springer (1969) [doi:10.1007/BFb0059143]

• Fred Linton, Applied functorial semantics, I, Annali di Matematica Pura ed Applicata 86 (1970) 1-13 [doi:10.1007/BF02415703]

• Fred Linton, Applied functorial semantics, II, in: Seminar on Triples and Categorical Homology Theory: ETH 1966/67, Lecture Notes in Mathematics 80, Springer (1969) 53-74 [doi:10.1007/BFb0083081]

• Francis Borceux, Brian Day, A monadicity theorem, Bulletin of the Australian Mathematical Society 17 2 (1977) 291-296 [doi:10.1017/S0004972700010480]

• Yasuo Kawahara, A Relation Theoretic Proof of a Tripleability Theorem Over Exact Categories, Bulletin of the Kyushu Institute of Technology. Mathematics, Natural Science 25 (1978) 31-40 [pdf]

• Francis Borceux, Brian Day, Universal algebra in a closed category, Journal of Pure and Applied Algebra 16 2 (1980) 133-147 [doi:10.1016/0022-4049(80)90012-2]

• Jean Bénabou, Jacques Roubaud, Monades et descente, C. R. Acad. Sc. Paris, Ser. A 270 (1970) 96-98 [gallica:12148/bpt6k480298g/f100, pdf, English translation (by Peter Heinig): MO:q/279152]

• Duško Pavlović, Categorical interpolation: descent and the Beck-Chevalley condition without direct images, Category theory Como 1990, pp. 306–325, Lecture Notes in Mathematics 1488, Springer 1991

• Pierre Deligne, Catégories Tannakiennes , Grothendieck Festschrift, vol. II, Birkhäuser Progress in Math. 87 (1990) pp. 111-195.

• Wikipedia: monadicity theorem

• John Bourke, Two dimensional monadicity, arxiv/1212.5123

• Emily Riehl, §5.5 in: Category Theory in Context, Dover Publications (2017) [pdf, book website]

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):

• I. J. Le Creurer, Francisco Marmolejo, Enrico M. Vitale, Beck’s theorem for pseudo-monads, J. Pure Appl. Algebra 173 3 (2002) 293-313 [doi:10.1016/S0022-4049(02)00038-5]

• Claudio Hermida, Descent on 2-fibrations and strongly 2-regular 2-categories, Appl. Categ. Structures 12 5-6 (2004) 427-459 [doi:10.1023/B:APCS.0000049311.17100.da]

Discussion for (infinity,1)-monads:

• Jacob Lurie, section 4.7 of Higher Algebra

and realized in the context of quasi-categories:

• Emily Riehl, Dominic Verity, section 7.2 of Homotopy coherent adjunctions and the formal theory of monads (arXiv:1310.8279)