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Stream: theory: applied category theory

Topic: process categories

Spencer Breiner (Apr 27 2022 at 20:00):

Is everyone else tired of saying "symmetric monoidal category" all the time?

I move that we all should adopt the much friendlier term "process category", especially in expository writing.

Thoughts? Objections? Who will join me?

Mike Shulman (Apr 27 2022 at 20:24):

I think that's why some people say "SMC".

Todd Trimble (Apr 27 2022 at 21:23):

"Process category" sounds like a concept with an attitude.

I won't be joining you, sorry.

Mike Shulman (Apr 27 2022 at 21:51):

In addition to the virtue of consistency (as Jaap van Oosten said, the only thing worse than bad terminology is continually changing terminology), there's a virtue in terminology that's both systematic and directly connected to its meaning, especially in expository writing.

John Baez (Apr 27 2022 at 22:28):

Yeah, I wouldn't use "process category" as a substitute for "symmetric monoidal category" since it won't catch on anyway and it would just add yet another term for people to remember.

If you get tired of writing it, it's easy to create a LaTeX macro \s that expands out to "symmetric monoidal category". :upside_down:

Chad Nester (Apr 28 2022 at 06:30):

I like "permutative" for the strict version, but it seems to have fallen out of fashion.

Chad Nester (Apr 28 2022 at 06:30):

(if it was ever in fashion at all)

Jules Hedges (Apr 28 2022 at 09:40):

I can see the temptation, and I do really like "process category" as a phrase.... as long as you simultaneously rename compact closed categories to "network categories". It might be a good idea in writing targetted for non-category theorists, provided it comes with a warning label saying the standard name. SMC has the dual advantage that everyone knows it, and it locates the concept in a periodic table of generalisations

Spencer Breiner (Apr 28 2022 at 12:06):

Mike Shulman said:

In addition to the virtue of consistency (as Jaap van Oosten said, the only thing worse than bad terminology is continually changing terminology), there's a virtue in terminology that's both systematic and directly connected to its meaning, especially in expository writing.

This assumes your audience already knows the technical meaning of "symmetric" and "monoidal".

Jon Sterling (Apr 28 2022 at 12:13):

I think part of the problem with renamings like this it that there REALLY ARE a lot of applications of these tools that don't actually look or feel like processes or networks, and for which it is not actually helpful to think of the things as processes. I understand the impulse of people who are really excited about the applications of SMCs to processes, but this is not the only applications of SMCs.

It can definitely be a good terminology to introduce in specialized settings, such as those that ACT is currently emphasizing. But it would not be a good idea to try and popularize these as general-purpose terminologies.

Spencer Breiner (Apr 28 2022 at 12:25):

I'm curious about all the applications of SMCs that don't "look or feel" like processes. Obviously you can choose whatever intuition you want to think about a formal structure, but I'm having trouble thinking of any that strongly resist such an interpretation. Matrices? Proofs? Maybe cobordisms and other geometric categories.

Jules Hedges (Apr 28 2022 at 12:37):

A thing I repeat a lot is that in applied category theory most categories tend to have $\circ$ = "time-like" and $\otimes$ = "space-like", or both $\circ$ and $\otimes$ = "space-like". The former are "process theories" (they tend to be merely symmetric monoidal, a major exception being quantum processes), the latter are "network theories" (they tend to be compact closed, $\dag$-hypergraph categories)

Jules Hedges (Apr 28 2022 at 12:38):

For a specific example, I would not call the category of cospans of graphs a "process theory", because its morphisms are clearly networks and not processes

Jules Hedges (Apr 28 2022 at 12:38):

(This is of course informal interpretation. All of this can have holes poked in it if you try)

Spencer Breiner (Apr 28 2022 at 12:58):

Thanks, @Jules Hedges, that is a useful distinction, and seems to subsume the cobordism case as well.

Even for network categories, though, I find a process interpretation makes the input/output distinction a bit less artificial.

Joe Moeller (Apr 28 2022 at 13:07):

Jules Hedges said:

A thing I repeat a lot is that in applied category theory most categories tend to have $\circ$ = "time-like" and $\otimes$ = "space-like", or both $\circ$ and $\otimes$ = "space-like". The former are "process theories" (they tend to be merely symmetric monoidal, a major exception being quantum processes), the latter are "network theories" (they tend to be compact closed, $\dag$-hypergraph categories)

It would be nice to have more of a dictionary like this mapped out.

Todd Trimble (Apr 28 2022 at 14:49):

Spencer Breiner said:

I'm curious about all the applications of SMCs that don't "look or feel" like processes. Obviously you can choose whatever intuition you want to think about a formal structure, but I'm having trouble thinking of any that strongly resist such an interpretation. Matrices? Proofs? Maybe cobordisms and other geometric categories.

How does one consider a category of modules as a "process category"? Or the category of sup-lattices? Or just about any of the classical SMC's used as input data for infinite loop space machines.

There are plenty of uses of SMC's outside of ACT.

Jules Hedges (Apr 28 2022 at 15:30):

I think Spencer has a particular kind of audience in mind: he's at NIST so he might be writing for engineers, or policy people. It'd definitely be a bad idea to try to convince, say, algebraists to use terminology like that

Spencer Breiner (Apr 28 2022 at 15:32):

I'm not suggesting anyone needs to use this terminology. I'm just suggesting that we should adopt it as an synonym, so that we can use it unambiguously.

For thinking of modules in terms of processes, I would point to graphical linear algebra.

Jules Hedges (Apr 28 2022 at 15:33):

Please don't take this as criticism, but mathematicians tend to heavily underestimate just how difficult it is communicating category-theoretic stuff to non-mathematicians. This is a huge part of the work of applied category theory, for those of us on the more applied end of it

Jules Hedges (Apr 28 2022 at 15:34):

Replacing a scary term with a not-scary term can be really quite effective

Spencer Breiner (Apr 28 2022 at 15:35):

And while it's a more pressing when writing for those who don't know what a monoid is, I also just think it sounds nicer, and I would prefer to use that terminology, even if I'm talking to someone from model theory or differential equations.

Reid Barton (Apr 28 2022 at 15:36):

You could also consider the term "tensor category"--that's already used in different ways by different people but it always includes at least a monoidal structure.

Jules Hedges (Apr 28 2022 at 15:39):

I've personally in both writing and talks used the term "process theory" when I mean "symmetric monoidal category" when I'm targetting groups where there's no risk of confusing them, like economists. I've been quietly doing that for years

Spencer Breiner (Apr 28 2022 at 15:40):

Thanks @Reid Barton. I agree that this is a better turn of phrase, although "tensor" is already a heavy lift for many.

Part of the problem is the variety of usage on "tensor category", which is why I brought the issue up here in addition to writing it into my paper.

Jules Hedges (Apr 28 2022 at 15:40):

Jules Hedges said:

I've personally in both writing and talks used the term "process theory" when I mean "symmetric monoidal category" when I'm targetting groups where there's no risk of confusing them, like economists. I've been quietly doing that for years

For this purpose "process theory" is better than "process category", because "category" itself is an unnecessarily technical word

Jules Hedges (Apr 28 2022 at 15:43):

I used to think big changes of terminology were hopeless to even attempt, until I realised at some point that the majority of people I know have started writing $f;g$ instead of $g \circ f$, when a few years ago that change itself seemed hopeless

Spencer Breiner (Apr 28 2022 at 15:46):

"Process theory" is a good suggestion if you want suppress the definition of category all together, but otherwise I think the change of noun is a bit confusing, especially if I also want to talk about functors, etc.

Joe Moeller (Apr 28 2022 at 15:49):

I like process theory. And I think this bumps right up against Coecke--Fritz--Spekkens saying a "resource theory" is an SMC where we call the objects resources and the morphisms processes. It's often the case that it makes more sense to name a category after its morphisms rather than its objects, doubly so in ACT.

Joe Moeller (Apr 28 2022 at 15:51):

To turn Todd's point around, I think it's fun to take one of these non-processy categories and stretch the mind a bit to think of its morphisms as processes of some absurdly general sort.

Joe Moeller (Apr 28 2022 at 15:53):

Sticking to the applied side, is there some word that feels like a common generalization of "process" and "network" that maybe could be used?

Jules Hedges (Apr 28 2022 at 15:54):

I think my own take is this kind of thing should be done informally on a per-author and per-audience basis, and there's no real use in arguing for a community-agreed thing

Spencer Breiner (Apr 28 2022 at 15:55):

Agreed, but talking about it advertises the idea to other people.

Todd Trimble (Apr 28 2022 at 15:55):

Jules Hedges said:

I think my own take is this kind of thing should be done informally on a per-author and per-audience basis, and there's no real use in arguing for a community-agreed thing

Obviously!!!

Todd Trimble (Apr 28 2022 at 15:56):

I mean, who is "we"? Are we being sufficiently inclusive here? ;-) I, for one, begin to feel marginalized.

Spencer Breiner (Apr 28 2022 at 15:57):

Plus, I think there may be a game-theoretic element, where the common knowledge of the practice lowers the cost to using the term.

Spencer Breiner (Apr 28 2022 at 15:59):

This is clearer in the long-run, where a critical mass in the community using the term would allow us to skip over a reference to the older term. Most people don't mention "triples" any more when they define monads.

Jules Hedges (Apr 28 2022 at 16:01):

This is #practice: applied ct and "we" is ACTists: we're (trying our best to be) a welcoming community but we are quite noticeably a separate community with our own journal, our own conference and to some extent our own norms

Todd Trimble (Apr 28 2022 at 16:05):

Okay. But FWIW, I think you're right, Jules: such things should be voluntary (per-author, per-audience).

Bye, then.

Jules Hedges (Apr 28 2022 at 16:21):

Joe Moeller said:

I like process theory. And I think this bumps right up against Coecke--Fritz--Spekkens saying a "resource theory" is an SMC where we call the objects resources and the morphisms processes. It's often the case that it makes more sense to name a category after its morphisms rather than its objects, doubly so in ACT.

Funnily enough, I find that paper a bit strange because, having been brought up in the Bob Coecke School of Categories, I find "SMC = resource theory" to be almost a tautology

Jules Hedges (Apr 28 2022 at 16:22):

I'd argue that "resource theory" is an exact synonym of "linear process theory"

Jules Hedges (Apr 28 2022 at 16:24):

The category of traces of a Petri net is a canonical such thing

Simon Burton (Apr 28 2022 at 16:36):

It's a good question.. we already have the synonyms arrow/morphism/map.. And "process" seems reasonable to add to this list. I already have used this trick when talking to physicists about category theory, to call it process theory.

John Baez (Apr 28 2022 at 18:11):

If I said that Hopf algebras form a "process category" with respect to the usual tensor product of Hopf algebras it would feel strange - and no mathematician would understand me until I said "I mean a symmetric monoidal category".

John Baez (Apr 28 2022 at 18:17):

If someone thinks "process category" will help beginners or outsiders get to like the idea, then of course they should try it and see if it helps. But if those beginners or outsiders ever want to read some books or papers that go into more detail, they'll need to learn the term "symmetric monoidal category" - until the hoped-for revolution is complete and there are lots of textbooks that say "process category".

John Baez (Apr 28 2022 at 18:18):

I will be dead by then, so it's hard for me to get really interested.