[spekcular]

It may the case that combinatorics is large, with mathematicians focused on their own particular sub-specialities, but I still don't believe category theory can properly be construed as "mainstream combinatorics" in any real sense.

Regarding your claim about that paper being cited by papers in top journals, I checked the first five pages of citations in Google Scholar for combinatorics and probability journal papers.

It's cited once in an offhand way in the concluding discussion of "The Cycle Structure of Random Permutations." No categorical concepts are used there. Ditto for the "Independent process approximations" paper (except now cited in the introduction). Another one-sentence mention in the context of background literature appears in "Tree-valued Markov chains derived from Galton-Watson processes." Same for "A Combinatorial Proof of the Multivariable Lagrange Inversion Formula" and "Bijections for Cayley Trees, Spanning Trees, and Their g-Analogues."

There's a one-sentence mention with actual (slight) mathematical content in "Limit Distributions and Random Trees Derived from the Birthday Problem with Unequal Probabilities." But you don't need categories to prove the bijection they're referring to, from what I understand.

In none of these instances is the work used in a substantive fashion, and unless I missed something no paper features the word "category." It's getting cited because authors have a duty to survey any potentially related background literature.

(On page 6 I found "Commutative combinatorial Hopf algebras," which does use a functor form that paper. It was published in a journal that is decent, but very far from the top.)

So, I think we can reject the notion that Joyal's paper has seriously influenced the fields of combinatorics or probability.

I'm sorry to harp on this, but I see claims like yours about the importance of category theory thrown around a lot on here, and often I feel that they're clearly wrong. So I thought it would be good to provide some details this time around.

[bitdizzy]

I should have demonstrated my picks. I didn't just look for citation but also looked for at least some nontrivial review of the results of the paper^.

I pick these two papers. The first is literally about adjunctions pertaining to combinatorial species whereas the second devotes an entire section to reviewing the theory and stating results that it uses in a way that I don't really understand. I'm going to read the first paper though because it's relevant to my interests :)

Rajan. The adjoints to the derivative functor on species

https://www.sciencedirect.com/science/article/pii/0097316593...

Panagiotou, Konstantinos; Stufler, Benedikt; Weller, Kerstin. Scaling limits of random graphs from subcritical classes.

https://projecteuclid.org/euclid.aop/1474462098

The fact that I have to go rummaging around for these examples kind of proves your point, doesn't it? I don't think category theory will lead to any fantastic new results in the fields we're discussing, but the bar is quite low for it to be useful as an organizational tool.

^ I searched for the word functor of course! :)

[spekcular]

OK. The first paper is indeed literally about species and in a good but not top journal. But it's not connected to the mainstream of combinatorics in any way, in the sense that if you aren't a priori interested in species, there's no reason to be interested in the paper.

The second paper is interesting because it's in an excellent journal and about a problem not obviously connected with species. So I agree that it counts as a good example for your case. I also agree with your conclusion – it's an exception that proves the rule, so to speak. Most probabilists have no need for category theory, and most AoP papers don't use categories. (I feel that may literally be the only one?)

I also agree that, to the extent it's useful, it's useful as an organizing principle and not "substantively." I suppose this explains why I feel it's overhyped: mathematicians care about solving problems, and the insights that solve the problems ultimately have to come from some problem-specific observations.