Dear Jean-Pierre, Yes, someone has. You can look here https://arxiv.org/abs/1612.02885 for a discussion about an extension of category theory along the lines of your suggestion Y != U. There these are called casting categories. (Caveat - this paper is being ...rather drastically revised prior to publication, and should be updated soon.) Home sets are disjoint there, but there is an ordering on the objects (which could be arrows...it’s a little like higher category theory, which I know little about.) The upshot is that there is a “Pythonic” downcast when there is a mismatch between two maps. The intuition is that a clever coercion algorithm is running behind the scenes. If you have suggestions/ideas/questions about applications/uses of such things, you should contact the author (me). Best Wishes, Lucius
On Feb 28, 2018, at 18:27, Marquis Jean-Pierre <jean-pierre.marquis@umontreal.ca> wrote:
Dear all,
I was recently asked a historical question on the set-based definition of categories and I could not find any reference one way or another. I am turning to the community in the hope that someone will be able to come up with a reference.
Here goes.
First, context. In the usual definition, composition of morphisms f: X -> Y and g: U -> V is restricted to the case where Y = U and it is required that the hom-sets be disjoints. Has anyone every considered the possibility of having a more 'relaxed' definition where disjointness is not required? One possibility is to let the codomain of f: X -> Y be included in Y without making the latter a unique attribute of f.
The question: has anyone considered a variant of CT where composition does not require Y = U and/or hom-sets need not be disjoint? Mac Lane does say in his book (p. 27) that some people omit the condition about disjointness but he does not say who and whether it could be ignored. His remark suggests that he believes that these authors are just sloppy.
You can reply directly to me.
Thanks,
Jean-Pierre
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