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John Baez (Dec 27 2024 at 21:24):Double categories are pretty fashionable in category theory nowadays, what with a series of online conferences about them, and people using them to study open systems, and now even as a key part of the software system Catcolab.
But my impression is that 20-50 years ago double categories were quite unpopular, even in the category theory community, and even among people interested in 2-categories. I felt that Grandis and Pare were "voices in the wilderness" preaching the importance of double categories but largely going unheard (with some obvious exceptions).
Does anyone have thoughts about this - or much better, written evidence of people discussing the (un)popularity of double categories back then?
Evan Patterson (Dec 27 2024 at 21:43):Good topic. One thing that comes to mind is this abstract posted on the n-Cat Cafe as late as 2008, which begins:
Ross Street, Double categories for better or worse, Australian Category Seminar, Macquarie University, Wednesday May 28, 2008.
Abstract: I suspect no-one actually thinks of double categories as warm and cuddly, or wants to study them for their own sakes. Usually they are studied as a means to an end... Much interest in double categories has been to use them to study 2-categories.
So here is a statement in plain terms by a famous 2-category theorist that such interest as there is in double categories is mostly as a means to understand 2-categories.
Mike Shulman (Dec 27 2024 at 22:52):For what it's worth, that was also my impression. Around 20 years ago I was a graduate student, and it definitely seemed to me that all the interest and momentum was around -categories. (But I expect you know that better than I do, having been one of the organizers of the 2004 IMA conference on -categories that was my introduction to the field.) I remember being surprised by the realization during my thesis work that the structures I was looking at weren't really bicategories but double categories, and making an intentional effort to popularize double categories and other higher-categorical structures.
Mike Shulman (Dec 27 2024 at 22:52):As for written evidence, it's hard to find evidence of a negative, but we could point to writings that could have mentioned double categories but didn't. For instance, Steve Lack's 2007 2-categories companion only mentions double categories as "a related notion" that "will be less important". (Of course the title declares that it will be about 2-categories rather than double categories, but I happen to know that that paper was titled after it was written, so the title didn't influence the choice of topics.)
Mike Shulman (Dec 27 2024 at 22:54):We could also look at things that could have been done earlier but weren't, or were done in a different way. For instance, there was a substantial amount of work that went into studying the bicategory Prof and characterizing bicategories like it. I feel like if double categories had been more fashionable, some of that work would have considered Prof as a double category.
John Baez (Dec 28 2024 at 00:23):Thanks, @Evan Patterson and @Mike Shulman ! That abstract by Street is very good evidence: little did we suspect that Street favored "warm and cuddly" math. :smirk: The remark by Lack is also good.
I can probably get more evidence by talking to Grandis and Pare.
What I'm planning to do, btw, is write a short paper about the rise of double category theory, especially in "applied" category theory. Both of you have played key roles in that rise.
Mike Shulman (Dec 28 2024 at 00:46):To be precise, what Lack said is that double categories "will be less important in this companion". I didn't mean to give the impression he was stating they were less important in general (although he may have felt that they were).
Graham Manuell (Dec 28 2024 at 07:08):My impression is that internal categories are less studied than enriched categories more generally, though I do not know a reference discussing this.
David Corfield (Dec 28 2024 at 08:35):My first introduction to the wider world of higher category theoretic structures was @Mike Shulman speaking of proarrow equipments as belonging to a 'zoo of “higher categorical structures”'.
Mike's introduction at the nCafe includes:
One theme that seems to keeps popping up in my research is that -categories are really just one corner of a largely-unexplored zoo of higher categorical structures. In particular, I’ve become a big proponent of double categories, equipments, and related structures.
John Baez (Dec 28 2024 at 16:17):Yes, Mike really helped popularize double categories as well as proving a lot of theorems that show how we can use them effectively to do various things.