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John Baez (Jul 30 2025 at 13:19):When I started out getting interested in categories, maybe the late 1980s and early 1990s, double categories seemed to be quite unpopular. Now they are all the rage. When did this first start to change, and why?
I know the answer must be quite complex, but I'm trying to write about this so I'd be interested in various takes on it.
Nathanael Arkor (Jul 30 2025 at 13:27):(A very short answer for now, as I'm on the move.) Weak double categories were (for all intents and purposes) introduced in Limits in double categories by Grandis and Paré in 1999. This was really the starting point for modern double category theory. Along with Marco and Bob's papers in the 2000s, Mike's 2007 paper Framed bicategories and monoidal fibrations was influential in demonstrating the application of double categories.
Bryce Clarke (Jul 30 2025 at 13:27):John Baez said:
I know the answer must be quite complex, but I'm trying to write about this so I'd be interested in various takes on it.
It may also be worthwhile to ask this question on the categories mailing list! I would be very interesting to hear the various perspectives.
John Baez (Jul 30 2025 at 13:29):I hadn't realized weak double categories were introduced so late! What does "for all intents and purposes" mean, @Nathanael Arkor? (It'd be dangerous for me to write something like that, even if it's true, because whoever had thought about them earlier would be offended.)
Nathanael Arkor (Jul 30 2025 at 13:58):They appear in passing in §8.11 of Street's Cosmoi of Internal Categories, and in a paper of the Ehresmanns, they claim that a French category theorist (I think Lair, but I will need to check when I am at a computer) defined such a notion, but the paper never appeared.
Nathanael Arkor (Jul 30 2025 at 13:59):Weak double categories can also be obtained by restricting Verity's double bicategories.
Nathanael Arkor (Jul 30 2025 at 14:00):However, as far as I am aware, Grandis and Paré's paper is the first to give the explicit description of precisely what is meant by a weak double category today.
David Michael Roberts (Jul 30 2025 at 14:24):https://mathoverflow.net/a/431929/4177 says Ehresmann defined double categories in print in 1963, and possibly in 1961, according to Guitart.
David Michael Roberts (Jul 30 2025 at 14:25):Not massively relevant, but prehistory perhaps. That Bénabou (as a student of Ehresmann) took double categories and defined 2-categories, and thence bicategories, is kinda fun.
John Baez (Jul 30 2025 at 14:36):But, just so everyone knows, Ehresmann only considered strict double categories. I think Nathanael is quite correct that the growth of popularity of double categories could only get going in earnest once people learned to work with weak double categories... since many of the most important examples 'in nature' are weak (in one direction).
And I think the crucial example for the rise of double categories was the double category of (enriched) categories, functors and profunctors.
I should look at the early work on 'proarrow equipments' and see to what extent they were formulated in terms of double categories. By the time we get to Mike's 'framed bicategories' and 'fibrant double categories' they clearly were, but that's later.
Bryce Clarke (Jul 30 2025 at 15:03):Section 0.5 of Verity's thesis that Nathanael linked provides some historical context for double categories, presumably written around 2011 when it was reprinted.
Mike Shulman (Jul 30 2025 at 15:40):My impression is that before my paper, Verity was the only one to formulate proarrow equipments using double categories, and his approach never caught on much, probably because double bicategories were so complicated.
For me, Prof was certainly only one example among many. My original motivation for studying double categories was the May-Sigurdsson double category Ex of parametrized spectra. I don't recall how central Prof was as an example for Grandis and Pare; I do think they certainly also considered examples like Span.
David Corfield (Jul 30 2025 at 16:07):Mike Shulman said:
My original motivation for studying double categories was the May-Sigurdsson double category Ex of parametrized spectra
I'd also be interested to hear more about any uptake of double categories in "mainstream" mathematics. E.g., did/do any algebraic topologists appeal to them?
Is it perhaps that for more telling use, there would need to be first the homotopification to [[double infinity categories]], which comes more recently?
Nathanael Arkor (Jul 30 2025 at 16:16):Bob Paré told me that his original motivation for caring about weak double categories was observing that the "hom functor" for a double category should land in Span, but this is merely a weak double category.
John Baez (Jul 30 2025 at 17:27):Thanks, Nathanael - that sort of from-the-horse's-mouth historical observation is very nice.
Siya Mthimkulu (Jul 30 2025 at 17:36):I think Evan Patterson wrote a nice note on his website on this.
John Baez (Jul 30 2025 at 17:46):Mike Shulman said:
My original motivation for studying double categories was the May-Sigurdsson double category Ex of parametrized spectra.
Do you know if any other algebraic topologists followed up on that kind of application of double categories?
David Corfield said:
I'd also be interested to hear more about any uptake of double categories in "mainstream" mathematics. E.g., did/do any algebraic topologists appeal to them?
Of course Ronnie Brown and coauthors have been using double categories in algebraic topology for a long time - here's an early paper:
But I have the feeling that this work never became 'mainstream' in algebraic topology. At least Brown always acted like an outsider, unappreciated... and Peter May, more of an 'insider', dismissed Brown's ideas in my presence.
In 1993 Bullejos, Cegarra and Duskin used n-fold categories internal to to model connected homotopy (n+1)-types (see the nice review by Simona Paoli from 2010). But again, I wouldn't call this work 'mainstream' algebraic topology.
(By the way, I'm not saying mainstream = good. There is however some interest in seeing how much double categories have been adopted by the mainstream. The mainstream seems less interested in algebraic models of homotopy n-types than in studying various invariants of homotopy types - particularly generalized cohomology theories.)
Mike Shulman (Jul 30 2025 at 19:04):John Baez said:
Do you know if any other algebraic topologists followed up on that kind of application of double categories?
That's a good question, I don't really know. I think certainly the May-Sigurdsson Ex has found uses, but I don't know if anyone has used a double-categorical perspective. May-Sigurdsson described it basically as a proarrow equipment, without using that language (a "bicategory with base change objects").
David Corfield (Jul 31 2025 at 08:11):As I suggested above, there is some ongoing work on double -categories with eventual applications somewhat in mind. For instance,
Adrian Clough (Jul 31 2025 at 09:07):Nathanael Arkor said:
Bob Paré told me that his original motivation for caring about weak double categories was observing that the "hom functor" for a double category should land in Span, but this is merely a weak double category.
Interesting! Could you elaborate?
David Corfield (Jul 31 2025 at 09:18):Continuing my puzzlement, can there be any good reason, intellectual rather than cultural, for double categories being all the rage in ACT, but not in pure mathematics, mathematical physics, etc. Looking at their centrality to ACT in grand frameworks such as Towards a double operadic theory of systems, it's hard to see any intrinsic reason why double categories wouldn't be valuable elsewhere in mathematics and its applications.
Nathanael Arkor (Jul 31 2025 at 09:28):Adrian Clough said:
Interesting! Could you elaborate?
Paré's paper Yoneda theory for double categories develops the theory (though he doesn't mention here that this was one of his original motivations; that comes from personal conversation).
Nathanael Arkor (Jul 31 2025 at 09:30):David Corfield said:
Continuing my puzzlement, can there be any good reason, intellectual rather than cultural, for double categories being all the rage in ACT, but not in pure mathematics, mathematical physics, etc.
I think that it's inaccurate to say that double categories are less popular in pure category theory than applied category theory. Double category theories have been increasingly popular in pure category theory in the last 15 or so years. It may only be more visible in ACT because it's a smaller field.
Nathanael Arkor (Jul 31 2025 at 09:31):I think it's clear, for instance, to anyone who has been attending the Category Theory conference in the last few years, that a rising proportion of mathematicians are acknowleding the usefulness of double categories in their work.
fosco (Jul 31 2025 at 09:58):My limited understanding of the development of the subject is that a breakthrough was @Mike Shulman and @Geoff Cruttwell paper "A unified framework for generalized multicategories", providing precisely what the title says, in relation to Leinster's fc-multicategories and Wood's equipments. Especially the latter notion feels a powerful one, but written in an unwieldy way. With virtual double categories, everything is elegant and easy to bookkeep, and a proarrow equipment is just a special kind of vdc with nice properties. This was a breakthrough because it neatly put into perspective an old idea that (I suppose, because it happened to me) never really clicked.
Part of the reason why I became friend with @Nathanael Arkor was our joint fascination for the expressive power of vdcs; after some time Nathanael really did something with them, I was more content with just learning without a "concrete" way to test the language, I found it beautiful but for me it was always all a bit up in the air.
Yet, to this day, it remains one of my favourite papers in @Mike Shulman long list of works :-) every time I go back reading it, I learn something new.
And, I certainly feel awkward reviewing Mike's work when he's in the room, so perhaps I'll leave to you to dis/agree with my retrospective reading of how things went...
John Baez (Jul 31 2025 at 11:54):David Corfield said:
Continuing my puzzlement, can there be any good reason, intellectual rather than cultural, for double categories being all the rage in ACT, but not in pure mathematics, mathematical physics, etc. Looking at their centrality to ACT in grand frameworks such as Towards a double operadic theory of systems, it's hard to see any intrinsic reason why double categories wouldn't be valuable elsewhere in mathematics and its applications.
Things don't happen merely because they could. I believe often the limiting factor is not any intrinsic reason but just that nobody has tried yet. I believe the fraction of interesting things we could do that actually get done is quite small. It's much easier to do less interesting things that are almost like things that people have already done.
In applied category theory, back around 2014, I was very excited about using bicategories to describe composition of open systems (1-morphisms) and also morphisms between open systems (2-morphisms). But by 2016, after Kenny Courser had managed to build symmetric monoidal bicategories of open systems, we realized that symmetric monoidal double categories were better. Not only are they technically simpler, they more clearly reflect what's really going on.
I had hoped that this bicategorical or double categorical framework for open systems would be useful in science (e.g. physics and chemistry). So far that hasn't really happened.
I think the main reason is that it would take someone who really understands some science also double categories well about a year of work to develop a good application. I don't think anyone qualified has put that much time into it. I know I haven't! I probably should. But it will require being frustrated and 'stuck' for weeks or months, which is not much fun.
It turned out to be much easier to take structured cospan ideas and implement them in software, using CatLab. But even here, the double categorical aspect has not yet been implemented.
CatColab, on the other hand, really does use double categories - but in a completely different way! It uses double categorical doctrines as a framework for expressing theories, which are themselves a framework for building models.
David Jaz Myers' double categorical systems theory goes in yet another direction. This may have an impact on the real world, because he's been put in charge of the ARIA Safeguarded AI project as chief cat herder. That is, he's been told to organize all the applied category theorists working for this project and get them to produce some working software.
David and Sophie Libkind's Towards a double operadic theory of systems can be seen as part of that push.
Kevin Carlson (Jul 31 2025 at 17:16):To be sure, while David is now starting to cat-herd the theorists on the project, he is not responsible for causing cool software to be made! There are two other chunks of the project launching led by software development shops, and then a whole other phase not yet extant where something like a new public benefit company implementing the mathematical and software ideas from the early phase is meant to get fully rolled out.
John Baez (Jul 31 2025 at 19:09):Okay, I'm glad David's burden is less than I thought.
Jonas Frey (Jul 31 2025 at 20:46):@fosco I share the sentiment about the central role played by Mike's work on double categories over the last almost 20 years, and I'm also a fan of the Cruttwell-Shulman paper. Virtual double categories are really great!
Mike Shulman (Jul 31 2025 at 21:35):Aw, shucks. (-:
It's nice to have our work appreciated. I do feel I should point out, for anyone listening who may not know, that virtual double categories are the same as Leinster's fc-multicategories, and I would say that Tom recognized their importance in a general way before we did. In particular, he observed that they are the natural context for the monoids-and-modules construction.
Mike Shulman (Jul 31 2025 at 21:44):As long as we're talking about the double-categorical renaissance, there's another kind of double category that I feel deserve more attention than they've gotten: those whose two kinds of morphisms are distinct but dual, in particular with neither being stricter than the other.
The paradigmatic double category of this sort is for a 2-monad , where the two classes of morphisms are the lax and colax -morphisms. I think this double category ought to be a nice place for studying 2-monad theory, e.g. its companion pairs are (up to equivalence) pseudo -morphisms, while its conjoint pairs are "doctrinal adjunctions". Sometimes it may be useful to generalize it to a triple category where the transverse morphisms are the strict -morphisms. In particular, I have a suspicion that these double or triple categories would be a good context in which to talk about limits of -algebras that involve both lax and colax morphisms, such as the comma object where is colax and is lax. But I don't think anyone has gone this direction.
The other example of a double category I know of like this is the double category of model categories with left and right Quillen functors, which I showed can be used to study coherence problems involving composites of left and right derived functors. But I don't know of any applications that have been made of this idea either.
David Corfield (Aug 01 2025 at 07:54):Mike Shulman said:
there's another kind of double category that I feel deserve more attention than they've gotten: those whose two kinds of morphisms are distinct but dual, in particular with neither being stricter than the other.
There's work relating them to orthogonal factorization systems:
Nathanael Arkor (Aug 01 2025 at 08:23):This is based on earlier work by @Miloslav Štěpán: Factorization systems and double categories.
Joe Moeller (Aug 08 2025 at 06:08):People have recently used double categories in K-theory, for instance in understanding the Waldhausen construction: https://arxiv.org/abs/2107.07701v3
John Baez (Aug 08 2025 at 08:07):Interesting!
The main appeal of these double categories is that they generalize the structure of exact sequences to key non-additive settings such as finite sets and varieties, where the notion of complements replaces that of kernels and cokernels. This makes it possible to study finite sets and varieties as if they were the objects of an exact category, for the purposes of K-theory.
I have a paper on biochemistry with one of the coauthors, Maru Saruzola. We never submitted it for publication. :crying_cat:
Nathanael Arkor (Aug 08 2025 at 08:10):Why "never" and not "haven't yet"?
John Baez (Aug 08 2025 at 08:14):At what point does "haven't yet" turn into "never"? One of life's great questions...
Nathanael Arkor (Aug 08 2025 at 08:30):Presumably when there is no longer an intention.
John Baez (Aug 08 2025 at 08:46):We wrote the paper Biochemical coupling through emergent conservation laws with Blake Pollard and Jonathan Lorand, who are either out of academia or leaving it. Nobody but me had a sustained interest in biochemistry. We all felt that the paper would need to be changed substantially to get it accepted by any sort of journal that has biochemists as referees. Right now it's supposed to be readable by mathematicians and scientists of all sorts. So to publish it, I'd have to do a lot of work on my own now.
It's a fun paper, so I hope y'all read it! It addresses the mystery of "coupling": how chemical processes that "run uphill" (require energy) can be driven by other processes that "run downhill" (release energy).
Maybe I should think about this using double categories. :wink: