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Stream: theory: category theory

Topic: The original definition of double category

Bryce Clarke (Feb 29 2024 at 14:20):

The standard definition of a (strict) double category is as an internal category in $\mathbf{Cat}$. Therefore a double category $\mathbb{D}$ has a category $D_{0}$ of objects and tight morphisms, and a category $D_{1}$ of loose morphisms and squares together with domain, codomain, unit, and composition functors which satisfy suitable axioms. This definition was given by Charles Ehresmann in Catégories structurées, Annales scientifiques de l'École Normale Supérieure 80.4 (1963): 349-426.

However, Ehresmann also gave another "naive" definition of double category which I learnt yesterday while reading Andrée Bastiani and Charles Ehresmann, Multiple functors. I. Limits relative to double categories, Cahiers de topologie et géométrie différentielle, Volume 15 (1974) no. 3, pp. 215-292. On page 25 of that paper, a double category is defined as a pair of categories $(\Sigma^{\circ}, \Sigma^{\bullet})$ with the same set of morphisms satisfying two axioms (the idea being that the shared set of morphisms is the set of squares of the double category). This definition is also attributed to the 1963 paper of Ehresmann.

It seems like this "naive" definition has been largely forgotten by the community in favour of the standard definition as an internal category. I wonder if there are constructions in double category theory which are more naturally stated in the non-standard definition? Or if there are interesting generalisations using this other definition of a double category. It is also not really clear to me which of the two definitions "came first".

Bryce Clarke (Feb 29 2024 at 14:24):

Definition of double category

Bryce Clarke (Feb 29 2024 at 14:40):

Actually, I just remembered remarks by Andrée Ehresmann in the video here which seem to confirm that the naive definition of double category above was the first definition given in 1961.

Nathanael Arkor (Feb 29 2024 at 16:21):

It seems like this "naive" definition has been largely forgotten by the community in favour of the standard definition as an internal category.

I suspect that many people are aware of the presentation, but it tends not to be a useful one in practice, and so it is rarely explicitly mentioned. For instance, it came up in this Zulip thread, and plays a role in this characterisation of one-object double categories. Arguably, it is also the perspective that is taken with [[proarrow equipments]] and [[F-categories]], which are presented as certain identity-on-objects (pseudo) functors, but may be viewed equivalently as (pseudo) double categories.

Nathanael Arkor (Feb 29 2024 at 16:23):

But it does seem plausibe that, due to its relative obscurity, there are some constructions on double categories that are more naturally presented in this "coincident categories" style, but are not typically presented that way.

Brendan Murphy (Feb 29 2024 at 16:30):

Huh, I actually had no idea about this when I made that other thread!

Bryce Clarke (Feb 29 2024 at 16:53):

Nathanael Arkor said:

It seems like this "naive" definition has been largely forgotten by the community in favour of the standard definition as an internal category.

I suspect that many people are aware of the presentation, but it tends not to be a useful one in practice, and so it is rarely explicitly mentioned. For instance, it came up in this Zulip thread, and plays a role in this characterisation of one-object double categories. Arguably, it is also the perspective that is taken with [[proarrow equipments]] and [[F-categories]], which are presented as certain identity-on-objects (pseudo) functors, but may be viewed equivalently as (pseudo) double categories.

Thanks for your comments, @Nathanael Arkor . I probably should have written "neglected" rather than "forgotten". The example you give characterising one-object double categories is a good one.

Bryce Clarke (Feb 29 2024 at 16:56):

Perhaps with sufficient time and motivation, I will add some content to the [[double categories]] nLab page which discusses this perspective to help bring it out of relative obscurity.

Benjamin Merlin Bumpus (he/him) (Feb 29 2024 at 18:28):

Bryce Clarke said:

The standard definition of a (strict) double category is as an internal category in $\mathbf{Cat}$. Therefore a double category $\mathbb{D}$ has a category $D_{0}$ of objects and tight morphisms, and a category $D_{1}$ of loose morphisms and squares together with domain, codomain, unit, and composition functors which satisfy suitable axioms. This definition was given by Charles Ehresmann in Catégories structurées, Annales scientifiques de l'École Normale Supérieure 80.4 (1963): 349-426.

However, Ehresmann also gave another "naive" definition of double category which I learnt yesterday while reading Andrée Bastiani and Charles Ehresmann, Multiple functors. I. Limits relative to double categories, Cahiers de topologie et géométrie différentielle, Volume 15 (1974) no. 3, pp. 215-292. On page 25 of that paper, a double category is defined as a pair of categories $(\Sigma^{\circ}, \Sigma^{\bullet})$ with the same set of morphisms satisfying two axioms (the idea being that the shared set of morphisms is the set of squares of the double category). This definition is also attributed to the 1963 paper of Ehresmann.

It seems like this "naive" definition has been largely forgotten by the community in favour of the standard definition as an internal category. I wonder if there are constructions in double category theory which are more naturally stated in the non-standard definition? Or if there are interesting generalisations using this other definition of a double category. It is also not really clear to me which of the two definitions "came first".

I haven't read the paper, but I'm intrigued and I'm wondering if you can help me get an intuitive feeling for what's going on: when you/they say "with the same set of morphisms", do you mean as a span in $\mathsf{Cat}$?

Bryce Clarke (Feb 29 2024 at 18:36):

There are many possible categories associated a double category. Given a double category $\mathbb{D}$, this definition uses the categories $D^{\circ}$ of whose objects are the tight morphisms and whose morphisms are the squares, and the category $D^{\bullet}$ whose objects are the loose morphisms and whose morphisms are the squares.

Bryce Clarke (Feb 29 2024 at 18:36):

Notice that both $D^{\circ}$ and $D^{\bullet}$ both have the same morphisms: namely, the squares/cells of the double category $\mathbb{D}$.

Benjamin Merlin Bumpus (he/him) (Feb 29 2024 at 18:37):

ah, I see, thanks!

Benjamin Merlin Bumpus (he/him) (Feb 29 2024 at 18:40):

I get it now, but it still seems somewhat hacky though to have to specify that the two categories share the same collection of morphisms (since they we're implicitly assuming some universe these collections live in). That's why my first guess was that there ought to be span somewhere..
Anyway, it's still a neat idea :)

Bryce Clarke (Feb 29 2024 at 18:40):

It is kind of like the "arrows only" definition of a category, as the double category is defined using only the tight + loose morphism and squares; the objects are not mentioned at all.

Kevin Arlin (Feb 29 2024 at 18:53):

Yeah, it would probably feel less hacky to write it all out at once as sets $S,H,V,$ with source and target maps from $S$ to $H$ and $V$ etc. Maybe this is what you meant by a span, Ben.

Nathanael Arkor (Feb 29 2024 at 19:32):

Nathanael Arkor said:

For instance, it came up in this Zulip thread, and plays a role in this characterisation of one-object double categories. Arguably, it is also the perspective that is taken with [[proarrow equipments]] and [[F-categories]], which are presented as certain identity-on-objects (pseudo) functors, but may be viewed equivalently as (pseudo) double categories.

Reflecting on this a little more, I think the example in the "double bicategory" thread, and the F-categorical examples aren't really an instance of the same phenomenon (since they involve two coincident 2-categories/bicategories, rather than two coincident categories), though they are still closer to the "coincident categories" presentation than the usual presentation of a double category.

Nathanael Arkor (Feb 29 2024 at 19:33):

So I think it's only the example of the one-object double category that comes to mind.

Nathanael Arkor (Feb 29 2024 at 19:37):

It very much feels like a "string-diagrammatic" presentation of a double category to me.

Nathanael Arkor (Feb 29 2024 at 19:43):

It seems, then, that there are (at least) three natural definitions of a double category in terms of a pair of categories (plus extra structure/properties):

• A category of objects and tight morphisms; together with a category of loose morphisms and 2-cells.
• A category of tight morphisms and 2-cells; together with a category of loose morphisms and 2-cells.
• A category of objects and tight morphisms; together with a category of objects and loose morphisms; and data specifying the 2-cells in a frame.

Evan Patterson (Feb 29 2024 at 21:22):

It seems to me that there are two flavors of double category that show up in examples. The first kind are strict double categories where neither of the classes of arrows is privileged over the other. Examples are squares in a category; more generally, "quintets" in a 2-category; monoidal categories with lax and colax monoidal functors; more generally, (pseudo) double categories with lax and colax double functors. For this kind of double category, it may well be more natural to start from a pair of category structures on the same objects.

The other kind of double category, which tend to be pseudo, have an obvious distinction between tight and loose morphisms, e.g., double categories of spans, cospans, matrices, relations, or profunctors. For these examples, you really want the asymmetrical description of a double category as a pseudocategory in Cat. That these examples are so important explains why this axiomatization has become the standard one.

Tim Hosgood (Mar 01 2024 at 01:42):

Nathanael Arkor said:

It seems, then, that there are (at least) three natural definitions of a double category in terms of a pair of categories (plus extra structure/properties):

• A category of objects and tight morphisms; together with a category of loose morphisms and 2-cells.
• A category of tight morphisms and 2-cells; together with a category of loose morphisms and 2-cells.
• A category of objects and tight morphisms; together with a category of objects and loose morphisms; and data specifying the 2-cells in a frame.

just as an indication of what happens outside CT, the last of these three is the one "usually" used in K-theory where the notion of squares category is useful (a simple double category with chosen initial basepoint).

Matteo Capucci (he/him) (Mar 04 2024 at 14:29):

I realize now this might be what Grandis and Parè do in some of their papers, where they see a double category as presented by a span of spans of set. This is also what you get when you think of double categories as algebraic structures on cubical sets.
In both cases you have a span of spans:
image.png

Matteo Capucci (he/him) (Mar 04 2024 at 14:30):

where the two spans in the square present the two categories Ehresmann gives