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Evan Patterson (Jan 03 2024 at 02:02):Time for everyone's favorite activity: bike-shedding terminology. I have need of the following family of concepts:
as well as the variants where "multicategory" is replaced with "symmetric multicategory." AFAIK, none of these are standard concepts. The first have been called "double multicategories" but @Mike Shulman suggested in another thread that they be called "multi-double categories" instead.
I am hoping to find some terminology that is, as they say, "functorial with respect to the ideas" but also not too verbose. Any suggestions?
Nathanael Arkor (Jan 03 2024 at 07:43):If you're happy with the terminology "multi-double categories", then the obvious choices are "multi-double reflexive graphs" and "multi-double graphs", no? This is a little verbose, but also descriptive, and the verbosity seems proportional to the complexity of the structures.
Mike Shulman (Jan 03 2024 at 07:44):To my mind, there's nothing "double" about the latter two. To me the "double" in "double category" means that there are two category structures (vertical and horizontal). Although perhaps it means something different to others.
Nathanael Arkor (Jan 03 2024 at 08:42):But (reflexive) graphs internal to multicategories contain two graph structures, no? Considering internal categories rather than internal graphs only adds identities/composites, but it doesn't change the "shape" of the structure.
Nathanael Arkor (Jan 03 2024 at 08:43):(I could see an argument for using "multi-double graph" for "graphs internal to multigraphs", but I don't think this is the point you're making.)
Mike Shulman (Jan 03 2024 at 08:44):I said
To me the "double" in "double category" means that there are two category structures (vertical and horizontal).
I don't mean the category of objects-and-vertical-arrows and the category of horizontal-arrows-and-squares; I mean the category of objects-and-vertical-arrows and the category of objects-and-horizontal-arrows.
Mike Shulman (Jan 03 2024 at 08:45):A (multi)category internal to graphs does not have "two graphs" in an analogous sense to this.
Mike Shulman (Jan 03 2024 at 08:48):(Evan and I had this same conversation at the other thread that he mentioned!)
Nathanael Arkor (Jan 03 2024 at 09:49):Mike Shulman said:
I said
To me the "double" in "double category" means that there are two category structures (vertical and horizontal).
I don't mean the category of objects-and-vertical-arrows and the category of horizontal-arrows-and-squares; I mean the category of objects-and-vertical-arrows and the category of objects-and-horizontal-arrows.
Yes, I understand this (and agree with you). But you suggested "multi-double category" in the context of "categories internal to multicategories" rather than "double multicategory" specifically to address this point. Both terms have "double" in them. As far as I see, the terminology "multi-double graph" (as opposed to "double multigraphs") is consistent with the terminology you proposed for categories internal to multicategories.
Nathanael Arkor (Jan 03 2024 at 09:51):image.png
Graphs internal to multicategories (or multigraphs) should have 2-cells that look something like this. Thus, there is a horizontal graph and and a vertical "multigraph".
Mike Shulman (Jan 03 2024 at 09:51):Both terms have double in them because the structure in question is, in particular, a double category, with a vertical and a horizontal category. A graph internal to categories, by contrast, is not, in particular, a "double graph" in the sense of having a "vertical graph" and a "horizontal graph".
Mike Shulman (Jan 03 2024 at 09:52):I suppose the category direction has an underlying graph, but that's not the same thing.
Mike Shulman (Jan 03 2024 at 09:52):The word "double" doesn't include the word "category".
Nathanael Arkor (Jan 03 2024 at 09:53):Mike Shulman said:
I suppose the category direction has an underlying graph, but that's not the same thing.
Yes, that's exactly why I said:
(I could see an argument for using "multi-double graph" for "graphs internal to multigraphs", but I don't think this is the point you're making.)
But you said:
To my mind, there's nothing "double" about the latter two.
which seems misleading if you agree that there is a structure in both directions if we consider underlying graphs.
Nathanael Arkor (Jan 03 2024 at 09:53):But I think we're on the same page now.
Nathanael Arkor (Jan 03 2024 at 09:54):I do think "multi-double graph" would be better terminology for graphs internal to multigraphs.
Mike Shulman (Jan 03 2024 at 09:54):There is a structure in both directions, but it's not the same structure, so I think it doesn't make sense to call it "double".
Mike Shulman (Jan 03 2024 at 09:54):Yes, I agree that's what "multi-double graph" should mean.
Nathanael Arkor (Jan 03 2024 at 09:55):Right, I see. I agree. One would probably like some modifier to describe the additional compositional structure of graphs internal to multicategories.
Mike Shulman (Jan 03 2024 at 09:56):I think sometimes an internal X in the category C is called a "C-X". For instance, a Cat-group is an internal group in categories. So you could say Multicat-graph.
Evan Patterson (Jan 03 2024 at 19:47):Thanks Mike and Nathanael for your replies.
I like the idea of just saying Multicat-graph, which is both short and clear. But I also like to have "plain English" phrases for concepts that I care about, which would make me want to say "multicategorical graph" as a synonym for Multicat-graph.
One point in favor of this is that people apparently also say "categorical group" for groups internal to Cat.
Mike Shulman (Jan 03 2024 at 20:02):And "topological group" for groups internal to Top. Yeah, I can see "multicategorical graph".