nLab doubly indexed category

Idea

Informally a doubly indexed category is a unitary lax double functor $F : \mathbb{C}^\top \to \mathbf{\mathbb{C}at}$, where the latter is the proarrow equipment of categories, functors, profunctors and natural transformations.

Formally, it is a right $\mathbb{C}$-module, i.e. it’s a right action of the double category $\mathbb{C}$.

With this name, they were defined by David Jaz Myers in his book on categorical system theory.

Definition

Definition

Let $\mathbb{C} = \mathbf{C}_1 \overset{s,t}\rightrightarrows \mathbf{C}_0$ be a double category, presented as a pseudomonad in $\mathbf Span(Cat)$. A doubly indexed category over $\mathbb{C}$ is a right $\mathbb{C}$-module, meaning it consists of a functor $\pi : \mathbf F \to C_0$ together with a map $* : \mathbf{C}_1 \times_{C_0} \mathbf{F} \to \mathbf{F}$ over $\mathbf{C}_0$, satisfying the laws of unitality and functoriality of a module.

The reason one can see this data as given by a unitary lax double functor $F : \mathbb{C}^\top \to \mathbf{\mathbb{C}at}$ is the following. First, the data of the functor $\pi: \mathbf F \to C_0$ in itself is equivalent to that of a unitary lax 2-functor into $\mathbf{Prof}$, i.e. a displayed category (see there for an explanation). This means objects of $\mathbb{C}$ are sent to categories (the fibers of $\pi$) while horizontal maps (tight maps) in $\mathbb{C}$ act ‘profunctorially’ on them. This assignment is unitary and lax.

Then the action $*$ gives, for each vertical morphism (loose maps) in $\mathbb{C}$, a functor. Explicitly, if $f: x \to y$ is a vertical map in $\mathbb{C}$, and $a : F(x) = \pi^{-1}(x)$, then $F(f)(a) = f * a : F(y)$.

This part of the action is strict, or at least pseudofunctorial.

Examples

One of the central exampls of doubly indexed category is given by the ‘self double-indexing’ any finitely complete category $\mathbf{C}$ induces. Specifically, it’s the action $\mathbf{C}/-$ of $\mathbf{Span(C)}$.

It is defined as follows:

1. An object $x : \mathbf{C}$ is sent to the slice category $\mathbf{C}/x$
2. A morphism $f:x \to y$ is sent to the profunctor $\mathbf{C}/f: \mathbf{C}/x^{\mathrm{op}} \times \mathbf{C}/y \to \mathbf{C}$ that sends maps $p : a \to x$, $q:b \to y$ to the set of maps $h:a \to b$ that make the square commute:
   $$\mathbf{C}/f(p, q) := \left\{ \begin{matrix} \ a & \overset{h}\to & b\ \\ p \downarrow && \downarrow q\\ x & \overset{f}\to & y\end{matrix} \right\}$$

   The laxator is defined by sending squares on $f:x \to y$ and $g:y \to z$ agreeing on their $y$ boundary to the obvious composite square lying over $f;g : x \to z$ (thus forgetting the middle boundary). One can see this assignment is unitary, as $\mathbf{C}/1_x$ is exactly $\mathbf{C}/x(-,=)$ (the identity profunctor on $\mathbf{C}/x$).

3. A span $x \overset{l}\leftarrow s \overset{r}\to y$ is sent to the pull-push functor $r_* l^* : \mathbf{C}/x \to \mathbf{C}/y$ which, on a map $p : a \to x$, acts first by pullback along $l$ and then post-composition it with $r$:
   $$\mathbf{C}/(l, r)(p) = \sigma_r l^*p$$
4. On squares, it can be defined as…

Related concepts

• indexed category

• double category

• module over a monad

• categorical systems theory, systems theory?

References

• David Jaz Myers, Double Categories of Open Dynamical Systems, EPTCS 333 (2021) 154-167 [arXiv:2005.05956, doi:10.4204/EPTCS.333.11]

• David Jaz Myers, Categorical systems theory, book project [github, pdf]

Exposition:

• David Jaz Myers, Categorical systems theory, Topos Institute Blog (Nov 2021)

• David Jaz Myers, Double Categories of Dynamical Systems (2020) [pdf]

Actions of double categories are central in some approaches to dependent optics?:

• Matteo Capucci, Seeing double through dependent optics (2022), arXiv:2204.10708