nLab monadic functor

Idea

A functor $U \,\colon\, D\to C$ is monadic iff it has a left adjoint $F \,\colon\, C\to D$ such that – under the relation between adjunctions and monads – the adjunction $F\dashv U$ is that induced by the monad which it induces – in which case it is called a monadic adjunction.

In this situation $U$ is identified with the forgetful functor from the Eilenberg-Moore category $EM(U \circ F)$ of the monad $(U \circ F, \eta, U\epsilon_F)$ on $C$, and hence shares the properties of these forgetful functors.

The monadicity theorem characterizes monadic functors and makes these ‘nice properties’ precise.

Monadic functors are sometimes called functors of effective descent (type). See the page on monadic descent for more on this aspect.

Definition

Given a pair of adjoint functors $F \colon C \to D :U$, $F \dashv U$, with unit $\eta: Id_C \to U \circ F$ and counit $\epsilon: F \circ U \to Id_D$, one constructs a monad $\mathbf{T}=(T,\mu,\eta)$ setting $T = U \circ F: C \to C$, $\mu = U \epsilon F: T T = U F U F \to U F = T$.

Consider the Eilenberg-Moore category $C^{\mathbf{T}}$ of $T$-algebras ($T$-modules) in $C$. Clearly $U (\epsilon_M): T U M = U F U M \to U M$ is a $T$-action. In fact there is a canonical comparison functor $K^{\mathbf{T}} \colon D \to C^{\mathbf{T}}$ given on objects by $K(M) \coloneqq \big(U M, U (\epsilon_M) \big)$. We then say that we have a (resp. strictly) monadic adjunction iff $K$ is an equivalence (resp. isomorphism) of categories.

Definition

(monadic functor)
A functor $U \colon D \to C$ is monadic (resp. strictly monadic) if it has a left adjoint $F \colon C\to D$ and the comparison functor $K^{\mathbf{T}} \colon D \to C^{\mathbf{T}}$ is an equivalence of categories (resp. an isomorphism of strict categories).

In other words, up to equivalence, monadic functors are precisely the forgetful functors defined on Eilenberg-Moore categories for monads, and strictly monadic functors are the same as these forgetful functors up to isomorphism of strict categories.

• If the comparison functor is only fully faithful, a functor is sometimes called premonadic or said to be of descent type.

Definition

A category $D$ is called monadic over a category $C$ if there is any functor $U \colon D \to C$ which is monadic (Def. ).

Properties

Basic properties

Every monadic functor is

1. faithful (by the definition of Eilenberg-Moore category)

2. conservative (by Beck’s monadicity theorem).

Moreover:

Proposition

(monadic functors create limits) A monadic functor

1. creates all limits that exist in its codomain;

2. creates all colimits that exist in its codomain and are preserved by the corresponding monad (or, equivalently, by the monadic functor itself).

(e.g. MacLane 71, Exercise IV.2.2 (p. 138))

Remark

Beware that the class of monadic functors is not generally closed under composition.

For a specific counter-example: the category of reflexive graphs is monadic over Set via the functor $RefGph \to Set$ sending a graph to its set of edges, and the category of categories is monadic over reflexive graphs via the forgetful functor $Cat \to RefGph$, but $Cat$ is not monadic over $Set$ (via any functor whatsoever, since monadic categories over Set are regular categories which Cat is not).

This is an instance of a general phenomenon: Let $\mathcal{C}$ be a reflective subcategory of a presheaf category $\widehat{A}$ (e.g. any locally presentable category is of this form). Then the adjunction between $\mathcal{C}$ and $\widehat{A}$ is monadic, and the adjunction between $\widehat{A}$ and $\mathrm{Set}^{\mathrm{Ob} A}$ is also monadic. But the composite adjunction between $\mathcal{C}$ and $\mathrm{Set}^{\mathrm{Ob} A}$ is often not monadic. For instance, if it is monadic, then $\mathcal{C}$ must be a Barr-exact category.

Monadic functors have the following cancellation property:

Proposition

Consider a pair of adjunctions: If here $U' U$ is monadic, then $U$ is of descent type and the comparison functor has a left adjoint. If $U'$ is furthermore conservative (and in particular if it is monadic), then $U$ is monadic.

This is Propositions 4 and 5 of Bourn.

Proposition

A monadic functor is strictly monadic if and only if it is also an amnestic isofibration.

Proof

Clearly, a strictly monadic functor is an amnestic isofibration; and if a monadic functor $U$ is amnestic, then the comparison functor $K$ is also amnestic, and if $U$ is a monadic isofibration, so is $K$; therefore in this case $K$ must be an isomorphism of categories.

Monadicity theorem

Various versions of Beck’s monadicity theorem (also: “tripleability theorem” in older literature) give sufficient, and sometimes necessary, conditions for a given functor to be monadic. There are also dual, comonadic versions.

Monadic functors to Set

Monadic functors to the category Set have additional properties. For example:

• Every monadic functor $U \,\colon\, D \to \mathrm{Set}$ is a solid functor.

• A category is monadic over $\mathrm{Set}$ (i.e. it admits a monadic functor to $\mathrm{Set}$) if and only if is Barr exact, cocomplete, and has a projective generator.

[Vitale (1994)]

Examples

Example

Every reflective subcategory-inclusion is a monadic functor. See also there.

A proof is spelled out for instance in Borceux 1994, vol 2, cor. 4.2.4. A formal proof in cubical Agda is given in 1Lab. See also at idempotent monad – Properties – Algebras for an idempotent monad and localization.

Related concepts

• monadic adjunction, structure-semantics adjunction

• comonadic functor, monadicity theorem

• monadic decomposition

References

• Saunders Mac Lane, Section VI in: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

• Francis Borceux, Def. 4.4.1 in: Handbook of Categorical Algebra, Vol. 2: Categories and Structures, Encyclopedia of Mathematics and its Applications 50 Cambridge University Press (1994) [doi:10.1017/CBO9780511525865]

• Emily Riehl, §5.3 in: Category Theory in Context, Dover Publications (2017) [pdf]

• Dominique Bourn. Low dimensional geometry of the notion of choice. Category Theory 1991, CMS Conf. Proc. Vol. 13. 1992.

• Enrico Vitale, On the characterization of monadic categories over $SET$, Cahiers de topologie et géométrie différentielle catégoriques 35 4 (1994) 351-358. [numdam:CTGDC_1994__35_4_351_0, pdf]