symmetric monoidal (∞,1)-category of spectra
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 $(T,\eta,\mu)$ be a monad in Cat, where $T \colon C\to C$ is an endofunctor with multiplication $\mu \colon T T\to T$ and unit $\eta \colon Id_C\to T$.
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
and
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
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
(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 $T$.
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 morphism $u^t \colon a^t \to a$ and a 2-cell $t u^t \Rightarrow u^t$ that mediate a natural isomorphism $K(x, a^t) \cong LMod(x,t)$ between morphisms $x \to a^t$ and $t$-modules $(m \colon x \to a, \lambda \colon t m \Rightarrow m)$. Not every 2-category admits Eilenberg–Moore objects.
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.
Every $T$-algebra $(A,\nu)$ is the coequalizer of the first stage of its bar resolution:
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
(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 $C_T$.
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 $C$.
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.
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 $C_T$.
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.
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 $\mathbf{G}$ and lax, colax or pseudo morphisms of such:
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
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 $[\Omega, A]$ on $[B,A]$ and an isomorphism of 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.
An accessible monad is a monad on an accessible category whose underlying functor is an accessible functor.
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.
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.
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).
General discussion is in
Ross Street, The formal theory of monads, Journal of Pure and Applied Algebra 2, 1972
Fred Linton, An outline of functorial semantics, in LNM 80, 1969
Fred Linton, Relative functorial semantics: adjointness results, Lecture Notes in Mathematics, vol. 99, 1969
Ross Street, Bob Walters, Yoneda structures, J. Algebra 50, 1978
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
Last revised on May 19, 2020 at 12:48:43. See the history of this page for a list of all contributions to it.