nLab Eilenberg-Moore category

Idea

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).

Definition

Let $(T,\eta,\mu)$ be a monad in Cat, where $T \colon C\to C$ is an endofunctor with multiplication $\mu \colon T T\to T$ and unit $\eta \colon Id_C\to T$.

Definition

A (left) $T$-module (or $T$-algebra) in $C$ is a pair $(A,\nu)$ of an object $A$ in $C$ and a morphism $\nu\colon T(A)\to A$ which is a $T$-action, in that

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

and

$$\nu\circ\eta_A = id_A \,.$$

A homomorphism of $T$-modules $f\colon (A,\nu^A)\to (B,\nu^B)$ is a morphism $f\colon A \to B$ in $C$ that commutes with the action, in that

$$f\circ\nu^A=\nu^B\circ T(f)\colon T(A)\to B \,.$$

The composition of morphisms of $T$-modules is the composition of underlying morphisms in $C$. The resulting category $C^T$ of $T$-modules/algebras is called the Eilenberg–Moore category of the monad $T$, also be written $Alg(T)$, or $T\,Alg$, etc.

By construction, there is a forgetful functor

$$U^T \colon C^T \to C$$

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

Eilenberg–Moore object

More generally, for $t \colon a \to a$ is a monad in any 2-category $K$, then the Eilenberg–Moore object $a^t$ of $t$ is, if it exists, the universal (left) $t$-module. That is, there is a “forgetful” 1-cell $u^t \colon a^t \to a$ and a 2-cell $\beta \colon t u^t \Rightarrow u^t$ that mediate a natural isomorphism $K(x, a^t) \cong LMod(x,t)$ between morphisms $h \colon x \to a^t$ and $t$-modules $(m \colon x \to a, \lambda \colon t m \Rightarrow m)$. Not every 2-category admits Eilenberg–Moore objects.

(Non)example

Let REL be the (locally posetal) 2-category of sets and relations. A monad on a set $X$ is just an endorelation $R$ satisfying $id_X\subseteq R$ (reflexivity) and $R\circ R \subseteq R$ (transitivity) i.e. a preorder on $X$. When $R$ arises from an adjunction it is necessarily of the form $R=F\circ F^{op}$ implying $R=R^{op}$ (symmetry) since adjunctions in $REL$ 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 $X^R$ of equivalence classes with “free algebra functor” $F^R$ relating $x\in X$ to its equivalence class $[x]\in X^R$.

Properties

Universal properties

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

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

As a colimit completion of the Kleisli category

Proposition

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

$$(T^2 A, \mu_{T A}) \stackrel{\overset{\mu_A}{\longrightarrow}}{\underset{T \nu}{\longrightarrow}} (T A, \mu_A) \stackrel{\nu}{\longrightarrow} (A,\nu) \,.$$

This is a reflexive coequalizer of $T$-algebras. Moreover, the underlying fork in $C$ 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

$$T^2 A \stackrel{\eta_{T A}}{\longleftarrow} T A \stackrel{\eta_A}{\longleftarrow} A \,.$$

(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 $T$-algebra is presented by free $T$-algebras. The nature of $T$-algebras as a kind of completion of free $T$-algebras under colimits is made more explicit as follows.

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

Proposition

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

of the category of presheaves on the Kleisli category along the Yoneda embedding $Y$ of $C$. (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 $P:C_T^{op}\to Set$ is a presheaf on the Kleisli category and $A$ is an object of $C$ such that $YA=P\circ F_T^{op}$. Then a $T$-algebra structure $\alpha:TA\to A$ on $A$ is given by $\alpha=P(1_{TA})(1_A)$, where $1_{TA}$ is viewed as a Kleisli morphism from $TA$ to $A$ in $C_T$.

By lax 2-limits

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

Steve Lack has shown how Eilenberg–Moore objects $C^T$ can be obtained as combinations of certain simpler lax limits, when the 2-category $K$ in question is the 2-category of 2-algebras over a 2-monad $\mathbf{G}$ 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 $(T,\mu,\eta)$ is a monad in a small category $A$, and $B$ is another category, then consider the functor category $[B,A]$. There is a tautological monad $[B,T]$ on $[B,A]$ defined by $[B,T](F)(b) = T(F(b))$, $b\in Ob B$, $[B,T](F)(f) = T(F(f))$, $f\in Mor B$, $\mu^{[B,T]}_F : TTF\Rightarrow TF$, $(\mu^{[B,T]}_F)_b = \mu_{Fb}$ $(\eta^{[B,T]}_F)_b = \eta_{Fb} : Fb\to TFb$. Then there is a canonical isomorphism of EM categories

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

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

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

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

Limits and colimits in EM categories

• The Eilenberg–Moore category of a monad $T$ on a category $C$ has all limits which exist in $C$, 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

Definition

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

Proposition

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

(Adamek-Rosicky, 2.78)

Moreover, let $C$ be a topos. Then

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

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

See at topos of algebras over a monad for details.

Examples

Examples

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

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

Related concepts

• (∞,1)-category of algebras over an (∞,1)-monad
• Eilenberg-Moore category, Kleisli category
• Eilenberg-Moore object, Kleisli object
• module over a monad

References

Original reference:

• Samuel Eilenberg, John Moore, Adjoint functors and triples, Illinois J. Math. 9 3 (1965) 381-398 [doi:10.1215/ijm/1256068141]

General discussion:

• Fred Linton, §7 in: An outline of functorial semantics, in Seminar on Triples and Categorical Homology Theory, Lecture Notes in Mathematics 80, Springer (1969) 7-52 [doi:10.1007/BFb0083080]

• Fred Linton, Relative functorial semantics: adjointness results, Lecture Notes in Mathematics, 99 (1969)

• Dieter Pumplün, Eine Bemerkung über Monaden und adjungierte Funktoren, Mathematische Annalen 185 (1970) 329-337 [eudml:161964, pdf]

• Saunders MacLane, §VI.2 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971) [doi:10.1007/978-1-4757-4721-8]

• Ross Street, The formal theory of monads, Journal of Pure and Applied Algebra 2 2 (1972) 149-168 [doi:10.1016/0022-4049(72)90019-9]

• Michael Barr, Charles Wells, §3.2 of: Toposes, Triples, and Theories, Grundlehren der math. Wissenschaften 278, Springer (1983) [TAC:12]

• Ross Street, Bob Walters, Yoneda structures, J. Algebra 50, 1978

• Nathanael Arkor, Dylan McDermott, The pullback theorem for relative monads (2024) [arXiv:2404.01281]

• Stephen Lack, Ross Street, The formal theory of monads II, Journal of Pure and Applied Algebra

  175 1–3 (2002) 243-265 [doi:10.1016/S0022-4049(02)00137-8]

On local presentability of EM-categories:

• Jiří Adámek, Jiří Rosický, p. 123, 124 of: Locally presentable and accessible categories, Cambridge University Press, (1994)

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.

• Paul-André Melliès, Segal condition meets computational effects, LICS 2010 (pdf)

The example of idempotent monads is discussed also in

• Francis Borceux, Handbook of Categorical Algebra, vol.2, p. 196.

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

• Emily Riehl, Dominic Verity, Homotopy coherent adjunctions and the formal theory of monads (arXiv:1310.8279)

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)