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I’m not at all sure that this should be a page, because there is some controversy about trying to fix a general definition of “applied category theory”. Either way, for the time being, I copied some extra context from Tai-Danae’s notes to make it clear that applications of category theory in Maths, CS and Physics are applied category theory from her perspective. And also to give the context that these notes were amassed during the ACT school, as she writes.
added a lead-in paragraph that points back to category theory.
Also made the intended pointer to the references be a pointer to the references. See the source code for how this works, for more see at HowTo the section How to cite and record references
Yeah, consonant with #4, I’m pretty uneasy with the way this phrase has been appropriated recently.
All the more would it make sense to have our entry provide clarification.
I had tried to go in this direction with the lines that I had added as of #5. For the record, here is what I had added:
The whole point of category theory is to study fundamental general abstract patterns and phenomena that (re-)appear throughout mathematics. Hence applications of theorems of category theory are ubiquituous in mathematics and in subjects with a mathematical basis, such as physics and computer science. Often this goes without saying.
I was thinking of further adding something to the extent that category theory is strongest when its applications are not announced with much aplomb, but just happen, much like results of, say, complex function theory are not announced much but just happen. There is no “complex function theory approach to physics” not because complex function theory wouldn’t have such applications but, on the contrary, its applications are so ubiquituous and the tools so fundamental to mathematics, that invoking them goes without saying. Same should apply to category theory, in fact even more so.
The problem that we are all sensing is with people who, despite their fondness of category theory, maybe don’t appreciate yet the sheer scope of methods of category theory, who will talk about the “topos approach to physics” meaning by it but the one, tiny aspect of Bohr toposes and being ignorant of the all-pervading role that topos theory plays in setting up even the foundations of classical physics; or even the “category approach to quantum physics” (as in the references quoted in the entry) meaning by it but the one tiny aspect of finite quantum mechanics in terms of dagger-compact categories. Incidentally, it is to a large extent the utterly immense success of the latter in attracting grants and public attention which has been driving the phenomenon we are seeing here.
There is no “complex function theory approach to physics” not because complex function theory wouldn’t have such applications but, on the contrary, its applications are so ubiquituous and the tools so fundamental to mathematics, that invoking them goes without saying.
Mind you, I’d still like to know if there’s anything general we can say about its appearance, as we began discussing back here.
As for “applied category theory”, such branding was always going to be contentious. You can see why people who see an opportunity to develop large tracks of terrain that haven’t been touched yet by category theory (engineering, biology, neuroscience) want to make common cause. But then they can’t help but come close to applications in computer science, which are wrapped up in some prominent centres with a certain small patch of physics. So then people come together to form a journal Compositionality, treating applications in computation, logic, physics, chemistry, engineering, linguistics, and cognition. If taken at its word this should now include all the work that Urs has been pointing us to in M-theory, which as far as I can see touches a vast portion of whatever interesting is going on in pure mathematics. But then what is the domain one is trying to carve out?
Maybe my point is most easily seen when considering the would-be analogs “applied set theory” or “applied logic”.
(alas, as Google knows, there is both a “Journal of applied Logic” and a “Jornal of applied Set Theory”.)
But then, I wonder how many laymen are aware that the most heavy-duty application of enriched category theory to date is algebraic topology. Maybe that subject should be renamed to “Applied enriched category theory”! ;-)
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