Eilenberg-Moore category


2-Category theory

Higher algebra



The Eilenberg–Moore (EM) category of a monad is the category of its modules (aka algebras). Dually, the EM category of a comonad is its category of comodules. The subcategory of its free modules is one of the descriptions of the Kleisli category of the monad. The EM and Kleisli categories have universal properties which make sense in a general 2-category.


Let (T,η,μ)(T,\eta,\mu) be a monad in Cat, where T:CCT \colon C\to C is an endofunctor with multiplication μ:TTT\mu \colon T T\to T and unit η:Id CT\eta \colon Id_C\to T. Recall that a (left) TT-module (or TT-algebra) in CC is a pair (M,ν)(M,\nu) of an object MM in CC and a morphism ν:T(M)M\nu\colon T(M)\to M which is a TT-action, namely νT(ν)=νμ M:T(T(M))M\nu\circ T(\nu)=\nu\circ\mu_{M} \colon T(T(M))\to M and νη M=id M\nu\circ\eta_M = id_M, and that a morphism of TT-modules f:(M,ν M)(N,ν N)f\colon (M,\nu^M)\to (N,\nu^N) is a morphism f:MNf\colon M\to N in CC that commutes with the action: fν M=ν NT(f):T(M)Nf\circ\nu^M=\nu^N\circ T(f)\colon T(M)\to N. The composition of morphisms of TT-modules is the composition of underlying morphisms in CC.

TT-modules and their morphisms thus form a category C TC^T which is called the Eilenberg–Moore category of the monad TT. This may also be written Alg(T)Alg(T), TAlgT\,Alg, etc. It comes equipped with a forgetful functor U T:C TCU^T \colon C^T \to C which is the universal TT-module, and has a left adjoint F TF^T such that the monad U TF TU^T F^T arising from the adjunction is equal to TT.

In general, if t:aat \colon a \to a is a monad in a 2-category KK, then the Eilenberg–Moore object a ta^t of tt is, if it exists, the universal (left) tt-module. That is, there is a morphism u t:a tau^t \colon a^t \to a and a 2-cell tu tu tt u^t \Rightarrow u^t that mediate a natural isomorphism K(x,a t)LMod(x,t)K(x, a^t) \cong LMod(x,t) between morphisms xa tx \to a^t and tt-modules (m:xa,λ:tmm)(m \colon x \to a, \lambda \colon t m \Rightarrow m). Not every 2-category admits Eilenberg–Moore objects.


Universal properties

Apart from being the universal left TT-module, the EM category of a monad TT in CatCat has some other interesting properties.

There is a full subcategory RAdj(C)RAdj(C) of the slice category Cat/CCat/C on the functors XCX \to C that have left adjoints. For any monad TT on CC there is a full subcategory of this consisting of the adjoint pairs that compose to give TT. The functor U T:C TCU^T \colon C^T \to C is the terminal object of this category.

If C TC_T is the Kleisli category of TT and F T:CC TF_T \colon C \to C_T the canonical functor, then the EM category C TC^T can be constructed as the pullback

C T [C T op,Set] [F T op,Set] C Y [C op,Set] \array{ C^T & \to & [C_T^{op}, Set] \\ \downarrow & & \downarrow \mathrlap{[F_T^{op},Set]} \\ C & \underset{Y}{\to} & [C^{op}, Set] }

Thus a TT-algebra may be regarded as a presheaf on the Kleisli category of TT whose restriction to CC is representable. This observation seems to be due to Linton. Street–Walters show that it holds in any 2-category equipped with a Yoneda structure?.

Just as the Kleisli object of a monad tt in a 2-category KK can be defined as the lax colimit of the lax functor *K\ast \to K corresponding to tt, the EM object of tt is its lax limit.

By lax 2-limits

S. Lack has shown how Eilenberg-Moore objects C TC^T can be obtained as combinations of certain simpler lax limits, when the 2-category KK in question is the 2-category of 2-algebras over a 2-monad G\mathbf{G} and lax, colax or pseudo morphisms of such:

  • Steve Lack, Limits for lax morphisms , Applied Categorical Structures 13:3 (2005) , pp. 189–203(15)

This encompasses for example the theory of (op)monoidal monads and corresponding monoidal Eilenberg–Moore categories.

If (T,μ,η)(T,\mu,\eta) is a monad in a small category AA, and BB is another category, then consider the functor category [B,A][B,A]. There is a tautological monad [B,T][B,T] on [B,A][B,A] defined by [B,T](F)(b)=T(F(b))[B,T](F)(b) = T(F(b)), bObBb\in Ob B, [B,T](F)(f)=T(F(f))[B,T](F)(f) = T(F(f)), fMorBf\in Mor B, μ F [B,T]:TTFTF\mu^{[B,T]}_F : TTF\Rightarrow TF, (μ F [B,T]) b=μ Fb(\mu^{[B,T]}_F)_b = \mu_{Fb} (η F [B,T]) b=η Fb:FbTFb(\eta^{[B,T]}_F)_b = \eta_{Fb} : Fb\to TFb. Then there is a canonical isomorphism of EM categories

[B,A T][B,A] [B,T]. [B,A^T] \cong [B,A]^{[B,T]}.

Namely, write the object part of a functor G:BA TG : B\to A^T as (G A,G ρ)(G^A,G^\rho), where G A:BAG^A :B\to A and G ρ(b):TG A(b)G A(b)G^\rho(b) : TG^A(b)\to G^A(b) is the TT-action of G A(b)G^A(b) and the morphism part simply as fG(f)f\mapsto G(f). Then, G ρ:bG ρ(b):TG AG AG^\rho : b\mapsto G^\rho(b) : TG^A\Rightarrow G^A is a natural transformation because for any morphism f:bbf:b\to b', G(f):(G A(b),G ρ(b))(G A(b),G ρ(b))G(f) : (G^A(b),G^\rho(b))\to (G^A(b'),G^\rho(b')) is by the definition of GG, a morphism of TT-algebras. G ρG^\rho is, by the same argument, an action [B,T](G A)G A[B,T](G^A)\Rightarrow G^A. Conversely, for any [B,T][B,T]-module (G A,G σ)(G^A,G^\sigma) for any bObBb\in Ob B, G σ(b)G^\sigma(b) will evaluate to a TT-action on G A(b)G^A(b), hence b(G A(b),G σ(b))b\mapsto (G^A(b), G^\sigma(b)) is an object part of a functor in [B,A T][B,A^T] with morphism part again fG(g)f\mapsto G(g). The correspondence for the natural transformations, g:(G A,G σ)(H A,H τ)g: (G^A,G^\sigma)\Rightarrow (H^A,H^\tau) is similar.

Dually, for a comonad Ω\Omega in BB, there is a canonical comonad [Ω,A][\Omega, A] on [B,A][B,A] and an isomorphism of categories

[B Ω,A][B,A] [Ω,A] [B^\Omega, A] \cong [B,A]^{[\Omega,A]}

Limits and colimits in EM categories

  • The Eilenberg-Moore category of a monad TT on a category CC has all limits which exist in CC, and they are created by the forgetful functor.

  • In contrast, the subject of colimits in categories of algebras is less easy, but a good deal can be said.

Local presentability


An accessible monad is a monad on an accessible category whose underlying functor is an accessible functor.


The Eilenberg-Moore category of a κ\kappa-accessible monad, def. 1, is a κ\kappa-accessible category. If in addition the category on which the monad acts is a κ\kappa-locally presentable category then so is the EM-category.

(Adamek-Rosicky, 2.78)

Moreover, let CC be a topos. Then

  • if a monad T:CCT : C \to C has a right adjoint then TAlg(C)=C TT Alg(C)= C^T is itself a topos;

  • if a comonad T:CCT : C \to C is left exact, then TCoAlg(C)=C TT CoAlg(C) = C_T is itself a topos.

See at topos of algebras over a monad for details.



Given a reflective subcategory 𝒞iL𝒟\mathcal{C} \stackrel{\overset{L}{\leftarrow}}{\underset{\hookrightarrow}{i}} \mathcal{D} then the Eilenberg-Moore category of the induced idempotent monad iLi\circ L on 𝒟\mathcal{D} recovers the subcategory 𝒞\mathcal{C}.

For instance (Borceux, vol 2, cor. 4.2.4).


General discussion is in

Local presentability of EM-categories is discussed on p. 123, 124 of

The following paper of Melliès compares the Linton representability condition above with the Segal condition that distinguishes those simplicial sets that are the nerves of categories.

The example of idempotent monads is discussed also in

Discussion for (infinity,1)-monads realized in the context of quasi-categories is around def. 6.1.7 of

Revised on January 14, 2014 11:15:28 by Urs Schreiber (