monadic functor



Category theory

Higher algebra

2-Category theory



A functor U:DCU:D\to C is monadic iff it has a left adjoint F:CDF:C\to D and the adjunction FUF\dashv U ‘comes from’ the induced monad on CC – that is, UU monadic iff FUF\dashv U is a monadic adjunction.

In this situation UU in some sense ‘looks like’ the forgetful functor from the Eilenberg-Moore category of the monad (UF,η,Uϵ F)(U\circ F,\eta,U\epsilon_F) on CC, and has ‘nice properties’ similar to these forgetful functors.

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


Given a pair of adjoint functors F:CD:UF: C \to D :U, FUF \dashv U, with unit η:Id CUF\eta: Id_C \to U \circ F and counit ϵ:FUId D\epsilon: F \circ U \to Id_D, one constructs a monad T=(T,μ,η)\mathbf{T}=(T,\mu,\eta) setting T=UF:CCT = U \circ F: C \to C, μ=UϵF:TT=UFUFUF=T\mu = U \epsilon F: T T = U F U F \to U F = T.

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

A functor U:DCU: D \to C is monadic (resp. strictly monadic) if it has a left adjoint F:CDF: C\to D and the comparison functor K T:DC TK^{\mathbf{T}}: D \to C^{\mathbf{T}} is an equivalence of categories (resp. an isomorphism of 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. A category DD is monadic over a category CC if there is a functor U:DCU: D \to C which is monadic.

Monadic functors are sometimes called functors of effective descent type. See the page on monadic descent for more details.


Various versions of Beck’s monadicity theorem (old-fashioned name of some schools: tripleability theorem) give sufficient, and sometimes necessary, conditions for a given functor to be monadic. There are also dual, comonadic versions.

A monadic functor is strictly monadic if and only if it is also an amnestic isofibration: clearly, a strictly monadic functor is an amnestic isofibration; and if a monadic functor UU is amnestic, then the comparison functor KK is also amnestic, and if UU is a monadic isofibration, so is KK; therefore in this case KK must be an isomorphism of categories.

Beware that monadic functors are not closed under composition. For a specific example: the category of reflexive quivers is monadic over SetSet via the functor RefGphSetRefGph \to Set sending a graph to its set of edges, and the category of categories is monadic over reflexive graphs via the forgetful functor CatRefGphCat \to RefGph, but CatCat is not monadic over SetSet (via any functor whatsoever, since such categories are regular categories but CatCat is not).

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

Last revised on November 10, 2021 at 03:59:47. See the history of this page for a list of all contributions to it.