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Amar Hadzihasanovic (Nov 13 2020 at 09:55):I am wondering if anyone knows any references about this problem:
Given a category presented by generators and relations (essentially, a graph together with an equivalence relation on finite paths with the same endpoints), when can I “contract” a set of generating morphisms (identify their endpoints and remove them from the graph) in such a way that the category obtained is equivalent to the original one?
This is like the problem of discrete Morse theory but for presented categories.
(Even better if you know about it for monoidal categories and monoidal equivalence...)
Simon Burton (Nov 13 2020 at 15:53):I've thought about this in relation to locally presentable categories, although this is all still a pipe-dream right now. (1) algebraic morse theory "is" just gaussian elimination (2) the grobner basis method "is" the same idea as gaussian elimination (3) categorifying ring theory (or algebraic geometry) you get something like symmetric monoidal locally presentable categories, and the claim is that this "same" algorithm should also categorify.
Part of me wonders if this is what is going on with the catlab julia library, but i have no idea about that.