nLab internal profunctor

Idea

For any category $\mathcal{E}$ with pullbacks, it is easy to define the notion of category in $\mathcal{E}$, 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 $\mathcal{E}$.

The solution lies in thinking of presheaves on an ordinary category $\mathcal{E}$, 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

Let $\mathbf{Span}(\mathcal{E})$ be the the bicategory of spans in a category $\mathcal{E}$ with pullbacks. The bicategory $\mathbf{Prof}(\mathcal{E})$ of internal categories and profunctors in $\mathcal{E}$ is defined to be the bicategory $\mathbf{Mod}(\mathbf{Span}(\mathcal{E}))$ of monads and modules in $\mathbf{Span}(\mathcal{E})$.

An internal profunctor $C \nrightarrow D$ between internal categories $C$ and $D$ is a module from $C$ to $D$. An internal presheaf on $C$, or an internal discrete fibration, is a module $C \nrightarrow 1$, where $1$ is the discrete category on the terminal object of $\mathcal{E}$. Dually, an internal discrete opfibration is a module $1 \nrightarrow D$.

An internal presheaf in $\mathcal{E}$ is the same thing as an internal diagram in $\mathcal{E}$.

Details

Let $C = (s, C_{1}, t) \colon C_{0} \nrightarrow C_{0}$ and $D = (s, D_{1}, t) \colon D_{0} \nrightarrow D_{0}$ be the underlying graphs of monads in the bicategory $\mathbf{Span}(\mathcal{E})$, that is, $C$ and $D$ are internal categories in $\mathcal{E}$. Consider a span $(f_{0}, M_{0}, g_{0}) \colon C_{0} \nrightarrow D_{0}$ as a $1$-cell in $\mathbf{Span}(\mathcal{E})$.

A left $C$-module consists of a morphism of spans $\lambda \colon (s, C_{1}, t) ; (f_{0}, M_{0}, g_{0}) \Rightarrow (f_{0}, M_{0}, g_{0})$, as depicted below, that is compatible with the unit and multiplication maps of the monad $C$.

Lemma

A left $C$-module determines an internal category $M_{C}$ and a span of internal functors $C \leftarrow M_{C} \rightarrow \mathsf{disc}(D_{0})$ whose left leg is an internal discrete fibration. The underlying morphism of internal graphs is depicted below.

A right $D$-module consists of a morphism of spans $\rho \colon (f_{0}, M_{0}, g_{0}) ; (s, D_{1}, t) \Rightarrow (f_{0}, M_{0}, g_{0})$, as depicted below, that is compatible with the unit and multiplication maps of the monad $D$.

Lemma

A right $D$-module determines an internal category $M_{D}$ and a span of internal functors $\mathsf{disc}(C_{0}) \leftarrow M_{D} \rightarrow D$ whose right leg is an internal discrete opfibration. The underlying morphism of internal graphs is depicted below.

An internal profunctor $(\lambda, \rho) \colon C \nrightarrow D$, or module in $\mathbf{Span}(\mathcal{E})$, between internal categories $C$ and $D$ consists of a span $(f_{0}, M_{0}, g_{0}) \colon C_{0} \nrightarrow D_{0}$ equipped with a left $C$-module $\lambda$ and a right $D$-module $\rho$ which are compatible, in the sense that the following diagram commutes.