nLab monadic functor

Idea

A functor $U \,\colon\, D\to C$ is monadic iff it has a left adjoint $F \,\colon\, C\to D$ and the adjunction $F\dashv U$ “comes from: the induced monad on $C$” – that is, $U$ monadic iff $F\dashv U$ is a monadic adjunction.

In this situation $U$ “looks like” the forgetful functor from the Eilenberg-Moore category of the monad $(U\circ F, \eta, U\epsilon_F)$ on $C$, and has ‘nice properties’ similar to 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.

• 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

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.

Remark

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

For a specific counter-example: the category of reflexive quivers 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 such categories are regular categories but $Cat$ 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. every 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.

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.

Related concepts

• monadic adjunction, structure-semantics adjunction

• comonadic functor, monadicity theorem

• monadic decomposition

References

• 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]