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.

A special case of this (see below) is often called an algebra for a monad.

Definition

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

$$\begin{array}{ccc}ttx& \stackrel{\mu x}{\to }& tx\\ t\lambda \downarrow & & \downarrow \lambda \\ tx& \underset{\lambda }{\to }& x\end{array}\begin{array}{ccc}x\cong 1x& \stackrel{\eta x}{\to }& tx\\ & 1\searrow & \downarrow \lambda \\ & & x\end{array}$$\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 \cong 1 x & \overset{\eta x}{\to} & t x \\ & 1\searrow & \downarrow \lambda \\ & & x }

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

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

$$\begin{array}{ccc}txs& \stackrel{t\rho }{\to }& tx\\ \lambda s\downarrow & & \downarrow \lambda \\ xs& \underset{\rho }{\to }& x\end{array}$$\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 $x:s⇸t$.

A morphism of left $t$-modules $(x,\lambda )\to (x\prime ,\lambda \prime )$ is given by a 2-cell $\alpha :x\Rightarrow x\prime$ such that $\lambda \prime \circ t\alpha =\alpha \circ \lambda$. Similarly, a morphism of right $t$-modules $(y,\rho )\to (y\prime ,\rho \prime )$ is $\beta :y\Rightarrow y\prime$ such that $\rho \prime \circ \alpha s=\alpha \circ \rho$. A morphism of bimodules $(x,\lambda ,\rho )\to (x\prime ,\lambda \prime ,\rho \prime )$ is given by $\alpha :x\Rightarrow x\prime$ that is a morphism of both left and right modules.

Algebras for monads in Cat

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

$$\begin{array}{ccc}T(T(A))& \stackrel{{\mu }_A}{\to }& T(A)\\ T(\alpha )\downarrow & & \downarrow \alpha \\ T(A)& \stackrel{\alpha }{\to }& A\end{array}$$\array { T(T(A)) & \stackrel{\mu_A}\rightarrow & T(A) \\ T(\alpha) \downarrow & & \downarrow \alpha \\ T(A) & \stackrel{\alpha}\rightarrow & A }

and

$$\begin{array}{ccc}A& \stackrel{{\eta }_A}{\to }& T(A)\\ {\mathrm{id}}_A\downarrow & & \downarrow \alpha \\ A& \stackrel{{\mathrm{id}}_A}{\to }& A\end{array}$$\array { A & \stackrel{\eta_A}\rightarrow & T(A) \\ id_A \downarrow & & \downarrow \alpha \\ A & \stackrel{id_A}\rightarrow & A }

commute.

In particular, every algebra in this sense over a monad $(T,\eta ,\mu )$ 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 $\mathrm{End}(C)$. 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.

Tensor product

Given bimodules $x\prime :r⇸s$ and $x:s⇸t$, where $r,s,t$ are monads on $c,b,a$ respectively, we may be able to form the tensor product $x{\otimes }_{s}x\prime :r⇸t$ just as in the case of bimodules over rings. If the hom-categories of the bicategory $K$ have coequalizers that are preserved by composition on both sides, then the tensor product is given by the coequalizer in $K(c,a)$

$$\begin{array}{cccc}xsx\prime & \overrightarrow{\to }& xx\prime & \to x{\otimes }_{s}x\prime \end{array}$$\array{ x s x' & \overset{\to}{\to} & x x' & \to x \otimes_s x' }

where the parallel arrows are the two induced actions $\rho x\prime$ and $x\lambda$ on $s$. It is straightforward, though a little tedious, to show that this is an $r,t$-bimodule, the conditions following from those on $x,x\prime$ together with the fact that coequalizers are epimorphisms and that (because of the condition on $K$) whiskering one coequalizer diagram on either side results in another.

If $K$ satisfies the above conditions then there is a bicategory $\mathrm{Mod}(K)$ consisting of monads, bimodules and bimodule morphisms in $K$.

Example

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

More generally, $\mathrm{Mod}(\mathrm{Span}(C))$, 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.