I am a linguistics major with background in math. I was wondering about categories arising in fields apart from mathematics, Especially linguistics but I would appreciate examples from other disciplines as well.
Note : my background so far in category theory has been first 2 chapters of "Basic Category theory" by Tom Leinster
Everyone of us does category theory, sometimes without even knowing it.
Suppose that A is the brother of B, who is the husband of D, and A is the husband of C, who is the sister of D. Then A, B, C, D and their relationships form a category and the situation is elegantly summarised with a commutative diagram: Click here. This is a nice example from an (introductory) talk on category theory https://www.youtube.com/watch?v=ho7oagHeqNc&t=348s which I definitely recommend.
For an accessible review of applications of category theory in language or natural language processing, I would recommend having a look at p35-48 of Tai-Danae Bradley's "What is Applied Category Theory?", available at https://arxiv.org/pdf/1809.05923.pdf.
Hope this helps.