Eilenberg-Moore category


2-Category theory

Higher algebra



The category of algebras over a monad (also: “modules over a monad”) is traditionally called its Eilenberg–Moore category (EM). Dually, the EM category of a comonad is its category of coalgebras (co-modules).

The subcategory of (co-)free (co-)algebras is traditionally called the Kleisli category of the (co-)monad.

The EM and Kleisli categories have universal properties which make sense for (co-)monads in any 2-category (not necessarily Cat).


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.


A (left) TT-module (or TT-algebra) in CC is a pair (A,ν)(A,\nu) of an object AA in CC and a morphism ν:T(A)A\nu\colon T(A)\to A which is a TT-action, in that

νT(ν)=νμ M:T(T(M))M \nu\circ T(\nu)=\nu\circ\mu_{M} \colon T(T(M))\to M


νη M=id M. \nu\circ\eta_M = id_M \,.

A homomorphism of TT-modules f:(A,ν A)(B,ν B)f\colon (A,\nu^A)\to (B,\nu^B) is a morphism f:ABf\colon A \to B in CC that commutes with the action, in that

fν M=ν NT(f):T(M)N. f\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. The resultiing category C TC^T of TT-modules/algebras is called the Eilenberg–Moore category of the monad TT, also be written Alg(T)Alg(T), or TAlgT\,Alg, etc.

By construction, there is a forgetful functor

U T:C TC U^T \colon C^T \to C

(which may be thought of as the universal TT-module) with a left adjoint free functor F TF^T such that the monad U TF TU^T F^T arising from the adjunction is isomorphic to TT.

More generally, for t:aat \colon a \to a is a monad in any 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.

As a colimit completion of the Kleisli category


Every TT-algebra (A,ν)(A,\nu) is the coequalizer of the first stage of its bar resolution:

(T 2A,μ TA)Tνμ A(TA,μ A)ν(A,ν). (T^2 A, \mu_{T A}) \stackrel{\overset{\mu_A}{\longrightarrow}}{\underset{T \nu}{\longrightarrow}} (T A, \mu_A) \stackrel{\nu}{\longrightarrow} (A,\nu) \,.

This is a reflexive coequalizer of TT-algebras. Moreover, the underlying fork in CC is a split coequalizer, hence in particular an absolute coequalizer (sometimes called the Beck coequalizer, due to its role in the Beck monadicity theorem). A splitting is given by

T 2Aη TATAη AA. T^2 A \stackrel{\eta_{T A}}{\longleftarrow} T A \stackrel{\eta_A}{\longleftarrow} A \,.

(e.g. MacLane, bottom of p. 148 and exercise 4 on p. 151) See also at split coequallizer – Beck coequalizer for algebras over a monad.

In particular this says that every TT-algebra is presented by free TT-algebras. The nature of TT-algebras as a kind of completion of free TT-algebras under colimits is made more explicit as follows.

Write C TC_T for the Kleisli category of TT, the category of free TT-algebras. Write F T:CC TF_T \colon C \to C_T the free functor. Observe that via the inclusion C TC TC_T \hookrightarrow C^T every TT-algebra represents a presheaf on C TC_T. Recall that the category of presheaves [C T op,Set][C_T^{op}, Set] is the free cocompletion of C TC_T.


The TT-algebras in CC are equivalently those presheaves on the category of free TT-algebras whose restriction along the free functor is representable in CC. In other words, the Eilenberg-Moore category C TC^T is the (1-category theoretic) pullback

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

of the category of presheaves on the Kleisli category along the Yoneda embedding YY of CC.

This statement appears as (Street 72, theorem 14). It seems to go back to (Linton 69), see (Melliès 10, p. 4). (Street-Walters 78) show that it holds in any 2-category equipped with a Yoneda structure?

By lax 2-limits

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.

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. 2, 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 representability condition of (Linton 69) 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 October 20, 2015 07:59:11 by Urs Schreiber (