nLab 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 A:T(T(A))A \nu\circ T(\nu)=\nu\circ\mu_{A} \colon T(T(A))\to A


νη A=id A. \nu\circ\eta_A = id_A \,.

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 A=ν BT(f):T(A)B. f\circ\nu^A=\nu^B\circ T(f)\colon T(A)\to B \,.

The composition of morphisms of TT-modules is the composition of underlying morphisms in CC. The resulting 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.

Eilenberg–Moore object

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 “forgetful” 1-cell u t:a tau^t \colon a^t \to a and a 2-cell β:tu tu t\beta \colon t 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 h:xa th \colon x \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.


Let REL be the (locally posetal) 2-category of sets and relations. A monad on a set XX is just an endorelation RR satisfying id XRid_X\subseteq R (reflexivity) and RRRR\circ R \subseteq R (transitivity) i.e. a preorder on XX. When RR arises from an adjunction it is necessarily of the form R=FF opR=F\circ F^{op} implying R=R opR=R^{op} (symmetry) since adjunctions in RELREL consist of functional relations as the left adjoint with their opposite relation as right adjoint. In other words, only equivalence relations admit Eilenberg–Moore objects which then consist of the set X RX^R of equivalence classes with “free algebra functor” F RF^R relating xXx\in X to its equivalence class [x]X R[x]\in X^R.


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 coequalizer – 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

of the category of presheaves on the Kleisli category along the Yoneda embedding YY of CC. (The top arrow is given by a functor isomorphic to the nerve of the inclusion of the Kleisli category into the Eilenberg–Moore category.)

This statement appears as (Linton 69, Observation 1.1) (cf. (Street 72, Theorem 14)). (Street-Walters 78) establish a generalisation for a 2-category equipped with a Yoneda structure. Arkor–McDermott ‘24 establish a generalisation to a virtual equipment, which also captures relative monads with dense roots.

Sketch of proof

It is easy to see that the square commutes. To see that it is a pullback, assume that P:C T opSetP:C_T^{op}\to Set is a presheaf on the Kleisli category and AA is an object of CC such that YA=PF T opYA=P\circ F_T^{op}. Then a TT-algebra structure α:TAA\alpha:TA\to A on AA is given by α=P(1 TA)(1 A)\alpha=P(1_{TA})(1_A), where 1 TA1_{TA} is viewed as a Kleisli morphism from TATA to AA in C TC_T.

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.

Steve 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 [A,Ω][A, \Omega] on [A,B][A, B] and an isomorphism of categories

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

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. , 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).


Original reference:

General discussion:

On local presentability of EM-categories:

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 of the universal property as the final adjoint decomposition of the monad:

  • Anthony Voutas, The basic theory of monads and their connection to universal algebra, (2012) [pdf, pdf]

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

An analogue of the pullback theorem for lax algebras (there confusingly called “pseudoagebras”) of a 2-monad is given in:

  • Albert Burroni. Structures pseudo-algébriques (1ère partie). Cahiers de topologie et géométrie différentielle catégoriques 16.4 (1975): 343-393. (pdf)

Last revised on April 6, 2024 at 11:03:54. See the history of this page for a list of all contributions to it.