nLab module over a monad

Idea

Just as the notion of a monad in a bicategory $K$ generalizes that of a monoid in a monoidal category, modules over monoids generalize easily to modules over monads.

Beware that modules over monads in Cat are often called *algebras* for the monad (see there for more), since they literally are algebras in the sense of universal algebra, see below. By extension, one might speak of modules over monads in any 2-category as “algebras for the monad”.

The formally dual concept is that of coalgebra over a comonad.

Definition

Modules

Let $K$ be a bicategory and $t \colon a \to a$ a monad in $K$ with structure 2-cells $\mu \colon t t \Rightarrow t$ and $\eta \colon 1_a \Rightarrow t$. Then a left $t$-module (or $t$-algebra) is given by a 1-cell $x \colon b \to a$ and a 2-cell $\lambda \colon t x \Rightarrow x$, where

$$\array{ t t x & \overset{\mu x}{\to} & t x \\ t\lambda\downarrow & & \downarrow \lambda \\ t x & \underset{\lambda}{\to} & x } \qquad \qquad \array{ x & \overset{\eta x}{\to} & t x \\ & 1\searrow & \downarrow \lambda \\ & & x }$$

commute. Similarly, a right $t$-module (or $t$-opalgebra) is given by a 1-cell $y \colon a \to c$ and a 2-cell $\rho \colon y t \Rightarrow y$, with commuting diagrams as above with $y$ on the left instead of $x$ on the right.

Clearly, a right $t$-module in $K$ is the same thing as a left $t$-module in $K^{\mathrm{op}}$. A left $t$-comodule or coalgebra is then a left $t$-module in $K^{\mathrm{co}}$, and a right $t$-comodule is a left $t$-module in $K^{\mathrm{coop}}$.

A $t$-module of any of these sorts is a fortiori an algebra over the underlying endomorphism $t$.

Bimodules

Given monads $s$ on $b$ and $t$ on $a$, an $s,t$-bimodule is given by a 1-cell $x\colon b \to a$, together with the structures of a right $s$-module $\rho \colon x s \Rightarrow x$ and a left $t$-module $\lambda \colon t x \Rightarrow x$ that are compatible in the sense that the diagram

$$\array{ t x s & \overset{t\rho}{\to} & t x \\ \lambda s \downarrow & & \downarrow \lambda \\ x s & \underset{\rho}{\to} & x }$$

commutes. Such a bimodule may be written as $x \colon s ⇸ t$.

A morphism of left $t$-modules $(x,\lambda) \to (x',\lambda')$ is given by a 2-cell $\alpha \colon x \Rightarrow x'$ such that $\lambda' \circ t\alpha = \alpha \circ \lambda$. Similarly, a morphism of right $t$-modules $(y,\rho) \to (y',\rho')$ is $\beta \colon y \Rightarrow y'$ such that $\rho' \circ \alpha s = \alpha \circ \rho$. A morphism of bimodules $(x,\lambda,\rho) \to (x',\lambda',\rho')$ is given by $\alpha \colon x \Rightarrow x'$ that is a morphism of both left and right modules.

More abstractly, the monads $s$ and $t$ in $K$ give rise to ordinary monads $s^*$ and $t_*$ on the hom-category $K(b,a)$, by pre- and post-composition. The associativity isomorphism of $K$ then gives rise to an invertible distributive law between these, so that the composite $s^* t_* \cong t_* s^* \colon x \mapsto t x s$ is again a monad. Then the category $Mod_K(s,t)$ of bimodules from $s$ to $t$ is the ordinary Eilenberg–Moore category $K(b,a)^{s^* t_*}$.

Algebras for monads in Cat

If $K =$ Cat and $(T,\eta,\mu)$ is a monad on a category $C$, then a left $T$-module $A \colon 1 \to C$, where $1$ is the terminal category, is usually called an algebra over $T$ or $T$-algebra (see there): it is given by an object $A \in C$ together with a morphism $\alpha \colon T A \to A$, such that

$$\array { T(T(A)) & \stackrel{\mu_A}\rightarrow & T(A) \\ T(\alpha) \downarrow & & \downarrow \alpha \\ T(A) & \stackrel{\alpha}\rightarrow & A }$$

and

$$\array { A & \stackrel{\eta_A}\rightarrow & T(A) \\ & id_A \searrow & \downarrow \alpha \\ & & A }$$

commute.

In particular, every algebra over a monad $(T,\eta,\mu)$ in $Cat$ has the structure of an algebra over the underlying endofunctor $T$.

$T$-algebras can also be defined as left modules over $T$ qua monoid in $End(C)$. There the object $A$ is represented by the constant endofunctor at $A$.

The Eilenberg-Moore category of $T$ is the category of these algebras. It has a universal property that allows the notion of Eilenberg-Moore object to be defined in any bicategory.

In a virtual double category

The notion of (bi)module makes sense in virtual double categories, generalizing the previous definition.

A monad in a virtual double category is a loose endomorphism $t : a \nrightarrow a$ together with a binary map (i.e. a square in the virtual double category) $\mu:t,t \to t$ giving the multiplication and a nullary map $\eta:() \to t$ giving the unit:

This data satisfies strict unitality and associativity equations.

A left module $m$ over a monad $t:a \nrightarrow a$ is another loose cell $m:a \nrightarrow b$ together with a binary map $\rhd : t,m \to m$:

This again satisfies standard equations stating compatibility with the monad structure on $t$.

A right module is equipped with a binary map $\lhd : m, t \to m$ instead, and a bimodule has a ternary map $\rhd - \lhd : t,m,t \to m$.

Properties

Colimits

see at colimits in categories of algebras

Tensor product

Given bimodules $x' \colon r ⇸ s$ and $x \colon s ⇸ t$, where $r,s,t$ are monads on $c,b,a$ respectively, we may be able to form the tensor product $x \otimes_s x' \colon r ⇸ t$ just as in the case of bimodules over rings. If the hom-categories of the bicategory $K$ have reflexive coequalizers that are preserved by composition on both sides, then the tensor product is given by the reflexive coequalizer in $K(c,a)$

$$\array{ x s x' & \overset{\to}{\to} & x x' & \to x \otimes_s x' }$$

where the parallel arrows are the two induced actions $\rho x'$ and $x \lambda$ on $s$. Indeed, under the hypothesis on $K$ the forgetful functor $Mod_K(r,t) = K(c,a)^{r^* t_*} \to K(c,a)$ reflects reflexive coequalizers, because the monad $r^* t_*$ preserves them, and so $x \otimes_s x'$ is an $r,t$-bimodule.

If $K$ satisfies the above conditions then there is a bicategory $Mod(K)$ consisting of monads, bimodules and bimodule morphisms in $K$. The identity module on a monad $t$ is $t$ itself, and the unit and associativity conditions follow from the universal property of the above coequalizer. There is a lax forgetful functor $Mod(K) \to K$, with comparison morphisms $1_a \to t$ the unit of $t$, and $x x' \to x \otimes_s x'$ the coequalizer map.

Examples

If $K = Span(Set)$, the bicategory of spans of sets, then a monad in $K$ is precisely a small category. Then $Mod(K) = Prof$, the category of small categories, profunctors and natural transformations.

More generally, $Mod(Span(C))$, for $C$ any category with coequalizers and pullbacks that preserve them, consists of internal categories in $C$, together with internal profunctors between them and transformations between those.

Related concepts

• algebra over a monad, algebra over an endofunctor, coalgebra over an endofunctor, algebra over a profunctor

  ∞-algebra over an (∞,1)-monad

  • model structure on algebras over a monad

• algebra over an algebraic theory

  ∞-algebra over an (∞,1)-algebraic theory

  • homotopy T-algebra / model structure on simplicial T-algebras

• algebra over an operad

  ∞-algebra over an (∞,1)-operad

  • model structure on algebras over an operad

• Eilenberg-Moore category, Kleisli category, Eilenberg-Moore object, Kleisli object

References

• John Isbell, Generic algebras Transactions of the AMS, vol 275, number 2 (pdf)

• H. Lindner, Commutative monads in Deuxiéme colloque sur l’algébre des catégories. Amiens-1975. Résumés des conférences, pages 283-288. Cahiers de topologie et géométrie différentielle catégoriques, tome 16, nr. 3, 1975.

• R. Guitart, Tenseurs et machines, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 21(1):5-62, 1980.

• A. Kock. Closed categories generated by commutative monads, Journal of the Australian Mathematical Society, 12(04):405-424, 1971.

• G. J. Seal. Tensors, monads and actions, Theory Appl. Categ., 28:No. 15, 403-433, 2013.

Discussion of model category structures on categories of coalgebras over comonads is in

• Kathryn Hess, Brooke Shipley, The homotopy theory of coalgebras over a comonad (arXiv:1205.3979)