monadic functor


Category theory

Higher algebra

2-Category theory



A functor is monadic if it is equivalent to the forgetful functor from a category of algebras over a monad. In this case it is part of a monadic adjunction.

The monadicity theorem characterizes monadic functors.


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 monadic adjunction.

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.


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).

Revised on June 26, 2017 12:28:48 by Mike Shulman (