nLab
internal profunctor

Idea

For any category $S$ with pullbacks, it is easy to define the notion of category in $S$, and the definition of an internal functor between such is similarly straightforward. But it is not so obvious how to define presheaves on internal categories, because they must land in the ambient category $S$.

The solution lies in thinking of presheaves on an ordinary category $C$, and more generally profunctors $C⇸D$, as giving sets equipped with an action of the arrows of $C,D$, i.e. as their categories of elements.

Definition

Here is the slick definition: let $S$ be a category with pullbacks. Then the bicategory $\mathrm{Prof}(S)$ of internal categories, profunctors and transformations in $S$ is defined to be $\mathrm{Mod}(\mathrm{Span}(S))$, the category of monads and bimodules in $\mathrm{Span}(S)$, the bicategory of spans in $S$.

So an internal profunctor $C⇸D$ between internal categories $C$ and $D$ is a bimodule from $C$ to $D$. An internal presheaf on $C$ is a right $C$-module, or equivalently a bimodule $1⇸C$, where $1$ is the discrete category on the terminal object of $S$ (as long as $S$ has one, of course).

An internal presheaf in $S$ is the same thing as an internal diagram in $S$.

Details

Recall that an internal category $C$ in $S$ is a monad in $\mathrm{Span}(S)$, and so has an underlying span $(s,t):C_1\rightrightarrows C_0$. Given $C,D\in \mathrm{Cat}(S)$, a bimodule $H:C⇸D$ must be given by a span $H:C_0\leftarrow H\to D_0$, together with a right action $H\circ C\to H$ of $C$ and a left action $D\circ H\to H$ of $D$ that are compatible in the sense described at module over a monad.

Composition of spans is given by pullback, so the action of $C$ on $H$ is given by a morphism of spans $H_r:C_1{\times }_{C_0}H\to H$, where the pullback is

$$\begin{array}{ccc}{C}_1{\times }_{C_0}H& \to & H\\ \downarrow & & \downarrow \\ C_1& \underset{t}{\to }& C_0\end{array}$$\array{ C_1 \times_{C_0} H & \to & H \\ \downarrow & & \downarrow \\ C_1 & \underset{t}{\to} & C_0 }

To say that $H_r$ is a morphism of spans is to require that

$$\begin{array}{ccc}{C}_1{\times }_{C_0}H& \to & H\\ \downarrow & & \downarrow \\ C_1& \underset{s}{\to }& C_0\end{array}$$\array{ C_1 \times_{C_0} H & \to & H \\ \downarrow & & \downarrow \\ C_1 & \underset{s}{\to} & C_0 }

commutes. This action must satisfy unit and associativity axioms expressing functoriality of the corresponding presheaf. Similarly, $H_{\ell }:H{\times }_{D_0}{D}_1\to H$, where the pullback is along $s$, so that this represents a copresheaf. Finally, compatibility of the two actions represents bifunctoriality of the profunctor.