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.
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.
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:
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 $\mathbf{C}/x$).
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?:
Created on December 12, 2022 at 21:40:03. See the history of this page for a list of all contributions to it.