2

Definitions:
Given a category $\mathcal{C}$, and a double category $\mathbb{D}$, i.e. a category internal to $Cat$ with an “objects category” $\mathcal{D}_0$ and a “morphisms category” $\mathcal{D}_1$, call $\mathbb{D}$ a “double category over $\mathcal{C}$” if “$\mathcal{D}_0$ is $\mathcal{C}$”, i.e. if there are functors $\mathbf{F}: \mathcal{C} \to \mathcal{D}_0$, $\mathbf{F}^{-1}: \mathcal{D}_0 \to \mathcal{C}$, such that $\mathbf{F}^{-1} \circ \mathbf{F} = \mathbf{Id}_{\mathcal{C}}$ and $\mathbf{F} \circ \mathbf{F}^{-1} = \mathbf{Id}_{\mathcal{D}_0}$ (call this condition “strictly invertible”).

(The use of equality / "strict" invertibility is deliberate, a natural isomorphism is too weak here and incompatible with the motivation for the question, see below.)

The morphisms of $\mathcal{D}_0$ are called “vertical morphisms”, the objects of $\mathcal{D}_1$ are called “horizontal morphisms”. We can swap the roles of these two “morphisms” and still get valid categories, $\tilde{\mathcal{D}}_0$ whose objects are the objects of $\mathcal{D}_0$ and whose morphisms are the objects of $\mathcal{D}_1$, and $\tilde{\mathcal{D}}_1$ whose objects are the morphisms of $\mathcal{D}_0$ and whose morphisms are the morphisms of $\mathcal{D}_1$. Moreover, the combination of $\tilde{\mathcal{D}}_0$ and $\tilde{\mathcal{D}}_1$ is again a double category, $\tilde{\mathbb{D}}$, called the “transpose” of $\mathbb{D}$.

A double category $\mathbb{D}$ is called “edge-symmetric” if, given its transpose $\tilde{\mathbb{D}}$, one has strictly invertible functors between the objects category $\mathcal{D}_0$ of $\mathbb{D}$ and the objects category $\tilde{\mathcal{D}}_0$ of the transpose $\tilde{\mathbb{D}}$. (Presumably this definition is equivalent to the condition that there are strictly invertible double functors between $\mathbb{D}$ and its transpose $\tilde{\mathbb{D}}$, although I haven’t checked.)

Given a category $\mathcal{C}$, and a double category $\mathbb{D}$, unsurprisingly say that $\mathbb{D}$ is “an edge-symmetric double category over $\mathcal{C}$” when $\mathbb{D}$ is both “edge-symmetric” and a “double category over $\mathcal{C}$”.

For an arbitrary category $\mathcal{C}$, the class $\operatorname{Sq}(\mathcal{C})$ of squares of morphisms in $\mathcal{C}$ form the 2-cells (morphisms of the morphism category) of an edge-symmetric double category over $\mathcal{C}$. Likewise, the class $\operatorname{ComSq}(\mathcal{C})$ of commutative squares of morphisms in $\mathcal{C}$ also induces an edge-symmetric double category over $\mathcal{C}$.

Question: For an arbitrary category $\mathcal{C}$, is there any sense in which either $\operatorname{ComSq}(\mathcal{C})$ or $\operatorname{Sq}(\mathcal{C})$ is universal among edge-symmetric double categories over $\mathcal{C}$?

What about among edge-symmetric double categories over $\mathcal{C}$ with additional extra properties, e.g. that are both left-connected and right-connected?

Note: I’m unsure whether this question should be considered "research-level" and posted to MathOverflow instead. If nothing else, it seems like double categories remain a developing area of research, and that the only major monograph about them was written by a leading active researcher in the field to collect the most important current results.

  • Marco Grandis, Higher dimensional categories: from double to multiple categories, World Scientific, 2019, doi:10.1142/11406

Below I've added my motivation for asking the question as a community wiki "answer", but basically this question is a follow-up to / clarification of a previous question.

hasManyStupidQuestions
  • 1,384

1 Answers1

0

Motivation: The “minimal information guaranteed to be associated” to a 2-cell in a general double category is its source horizontal morphism and target horizontal morphism. But in an edge-symmetric double category, the minimal information guaranteed to be associated to a 2-cell is not only the source and target horizontal morphisms, but also the source and target vertical morphisms for the corresponding 2-cell in the transpose double category. This seems very similar to the minimal information guaranteed to be associated to a square of morphisms in $\mathcal{C}$, i.e. to a 2-cell in $\operatorname{Sq}(\mathcal{C})$ considered as a double category.

It seems like, given an arbitrary edge-symmetric double category $\mathbb{D}$ over $\mathcal{C}$, there should always be a double functor from $\mathbb{D}$ to $\operatorname{Sq}(\mathcal{C})$ defined in the manner suggested above. (Following the description of "right universal" properties from section 3.8 of Bergman's Cook's Tour of Other Universal Constructions.) Of course, commutative squares are better behaved than arbitrary squares, so it would be nice to restrict to $\operatorname{ComSq}(\mathcal{C})$, but I don’t see why that should always be possible.

Heuristically, given a double category $\mathbb{D}$ over $\mathcal{C}$, as long as it's edge-symmetric, we can identify the objects of the morphisms category $\mathcal{D}_1$ with the morphisms of the original category $\mathcal{C}$, and thus the 2-cells as “morphisms-of-morphisms of $\mathcal{C}$”.

Not only is the morphisms category of an edge-symmetric double category over $\mathcal{C}$ an example of an arbitrary “category whose objects are morphisms of $\mathcal{C}$ and whose morphisms are morphisms-of-morphisms of $\mathcal{C}$”, it is a “functorially well-behaved” example, i.e. one which “respects the composition/morphism structure of the original $\mathcal{C}$”, due to the source, target, identity, and composition functors in the definition of internal category.

So if $\operatorname{ComSq}(\mathcal{C})$, or at least $\operatorname{Sq}(\mathcal{C})$, was in any way a universal or “best” example of an edge-symmetric double category over $\mathcal{C}$, that would make it the universal/”best” example of a “category whose objects are morphisms of $\mathcal{C}$ and whose morphisms are morphisms-of-morphisms of $\mathcal{C}$ and that respects the structure of the original $\mathcal{C}$”.

Let $\operatorname{Arr}(\mathcal{C})$ denote any example of the (not well-defined) notion of a “category whose objects are morphisms of $\mathcal{C}$ and whose morphisms are morphisms-of-morphism of $\mathcal{C}$ (and that respects the structure of the original $\mathcal{C}$)”. Then (as long as we are agnostic about whether Cat is Cartesian-closed) one can argue (see my previous question on Math.SE) that functors $\mathcal{B} \to \operatorname{Arr}(\mathcal{C})$ make a reasonable definition of “morphism between functors $\mathcal{B} \to \mathcal{C}$” for any choice of $\operatorname{Arr}(\mathcal{C})$.

The only obstacle to this being that, given a morphism $b_1 \overset{h}{\to} b_2$ in $\mathcal{B}$, functors $\mathbf{F}, \mathbf{G}: \mathcal{B} \to \mathcal{C}$, and a functor $\mathbf{\eta}: \mathcal{B} \to \operatorname{Arr}(\mathcal{C})$, one would want the morphism-of-morphisms $\mathbf{\eta}(h)$ in $\mathcal{C}$, i.e. morphism in $\operatorname{Arr}(\mathcal{C})$, to not only (as guaranteed by functoriality) connect the morphisms $\mathbf{\eta}(b_1)$ and $\mathbf{\eta}(b_2)$, but also to connect $\mathbf{F}(h)$ and $\mathbf{G}(h)$.

But edge-symmetric double categories over $\mathcal{C}$ also happen to solve that problem, because (as discussed above) the 2-cell $\mathbf{\eta}(h)$ is associated not only to a pair of “horizontal” morphisms that we can take to be $\mathbf{\eta}(b_1)$ and $\mathbf{\eta}(b_2)$, but also to a pair of “vertical” morphisms that we can take to be $\mathbf{F}(h)$ and $\mathbf{G}(h)$.

In the case we choose the edge-symmetric double category over $\mathcal{C}$ to be $\operatorname{ComSq}(\mathcal{C})$, then we recover the usual definition of morphism between functors, namely natural transformations. (Cf. remark 3.2 of this nLab article.)

We can make other choices though, e.g. invertible commutative squares over $\mathcal{C}$ lead to natural isomorphisms, or pullback (pushout) squares over $\mathcal{C}$ lead to (co)cartesian natural transformations.

In other words, we’ve reduced the question of “why are natural transformations the best choice of morphism between functors” to the question of “why is $\operatorname{ComSq}(\mathcal{C})$ the best choice of edge-symmetric double category over $\mathcal{C}$”?

Moreover, the latter question only involves concepts (double categories) that can be defined exclusively in terms of categories and functors. This is unlike typical answers to the former question, which usually invoke concepts like Cartesian closedness or monoidal categories that themselves depend on the definition of natural transformation.

(That is analogous to answering “Why is the Euclidean norm the best norm on $\mathbb{R}^n$?” with the answer “It is the only norm that is invariant under the action of Euclidean isometries on $\mathbb{R}^n$ (including rotations)”. It’s technically accurate but largely useless / circular as motivation / justification unless one can define “Euclidean isometries” without reference to “Euclidean metric” or concepts equivalent to it.)

Related questions:

hasManyStupidQuestions
  • 1,384