nLab
Eilenberg-Moore category

Idea

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.

Definitions

Let $(T,\eta ,\mu )$ be a monad in Cat, where $T:C\to C$ is an endofunctor with multiplication $\mu :TT\to T$ and unit $\eta :{\mathrm{Id}}_C\to T$. Recall that a (left) $T$-module (or $T$-algebra) in $C$ is a pair $(M,\nu )$ of an object $M$ in $C$ and a morphism $\nu :T(M)\to M$ which is a $T$-action, namely $\nu \circ T(\nu )=\nu \circ {\mu }_M:T(T(M))\to M$ and $\nu \circ {\eta }_M={\mathrm{id}}_M$, and that a morphism of $T$-modules $f:(M,{\nu }^M)\to (N,{\nu }^N)$ is a morphism $f:M\to N$ in $C$ that commutes with the action: $f\circ {\nu }^M={\nu }^N\circ T(f):T(M)\to N$. The composition of morphisms of $T$-modules is the composition of underlying morphisms in $C$.

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

In general, if $t:a\to a$ is a monad in a 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 morphism $u^t:a^t\to a$ and a 2-cell $t{u}^t\Rightarrow u^t$ that mediate a natural isomorphism $K(x,a^t)\cong \mathrm{LMod}(x,t)$ between morphisms $x\to a^t$ and $t$-modules $(m:x\to a,\lambda :tm\Rightarrow m)$. Not every 2-category admits Eilenberg–Moore objects.

Properties

Universal properties

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

There is a full subcategory $\mathrm{RAdj}(C)$ of the slice category $\mathrm{Cat}/C$ 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 $U^T:C^T\to C$ is the terminal object of this category.

If $C_T$ is the Kleisli category of $T$ and $F_T:C\to C_T$ the canonical functor, then the EM category $C^T$ can be constructed as the pullback

$$\begin{array}{ccc}{C}^T& \to & [C_T^{\mathrm{op}},\mathrm{Set}]\\ \downarrow & & \downarrow [F_T^{\mathrm{op}},\mathrm{Set}]\\ C& \underset{Y}{\to }& [C^{\mathrm{op}},\mathrm{Set}]\end{array}$$\array{ C^T & \to & [C_T^{op}, Set] \\ \downarrow & & \downarrow \mathrlap{[F_T^{op},Set]} \\ C & \underset{Y}{\to} & [C^{op}, Set] }

Thus a $T$-algebra may be regarded as a presheaf on the Kleisli category of $T$ whose restriction to $C$ 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 $t$ in a 2-category $K$ can be defined as the lax colimit of the lax functor $*\to K$ corresponding to $t$, the EM object of $t$ is its lax limit.

By lax 2-limits

S. 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 $G$ 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 $[B,T](F)(b)=T(F(b))$, $b\in \mathrm{Ob}B$, $[B,T](F)(f)=T(F(f))$, $f\in \mathrm{Mor}B$, ${\mu }_F^{[B,T]}:\mathrm{TTF}\Rightarrow \mathrm{TF}$, $({\mu }_F^{[B,T]}{)}_b={\mu }_{\mathrm{Fb}}$ $({\eta }_F^{[B,T]}{)}_b={\eta }_{\mathrm{Fb}}:\mathrm{Fb}\to \mathrm{TFb}$. Then there is a canonical isomorphism of EM categories

$$[B,A^T]\cong [B,A{]}^{[B,T]}.$$[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 $G^{\rho }(b):{\mathrm{TG}}^A(b)\to G^A(b)$ is the $T$-action of $G^A(b)$ and the morphism part simply as $f\mapsto G(f)$. Then, $G^{\rho }:b\mapsto G^{\rho }(b):{\mathrm{TG}}^A\Rightarrow G^A$ is a natural transformation because for any morphism $f:b\to b\prime$, $G(f):(G^A(b),G^{\rho }(b))\to (G^A(b\prime ),G^{\rho }(b\prime ))$ 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 $b\in \mathrm{Ob}B$, $G^{\sigma }(b)$ 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 $[\Omega ,A]$ on $[B,A]$ and an isomorphism of categories

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

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.

EM-Categories on toposes

Let $C$ be a topos. Then

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

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

See topos of algebras over a monad for details.

References

• Ross Street, The formal theory of monads, JPAA 2, 1972
• Fred Linton, An outline of functorial semantics, in LNM 80, 1969
• Ross Street, Bob Walters, Yoneda structures, J. Algebra 50, 1978

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.

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