A short question about the set-based definition of categories
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Hi Jean-Pierre: Your statement of the disjointness condition puzzled me a little, because it would seem to rule out the composition of morphisms X-->X-->X, but Mac Lane makes it clear that the case (X,Y) = (U,V) is an exception. That said, I don't interpret the disjointness axiom as an axiom for composition, but for hom-sets. If f ?? hom(X,Y) ??? hom(U,V), X???U or Y???V, then domain and/or codomain are not functions or don't play well with hom(-,-) . This would be counterintuitive even in a directed multigraph; composition doesn't come into it. Bob Par??, Dorette Pronk, and I have discussed a definition of composition for which codomain-domain matching is not required. We're primarily interested in the higher category case; the ordinary category case seems straightforward but unexciting. The idea is that the composition of f:X-->Y and g:U-->V should be an equivalence class of hom(X,Y) x hom(U,V) that has well-defined compositions with other arrows /in a certain syntax. /That last is important: we don't define "the composition of the set {f,g}" without reference to order, even where both are well defined.Here, given f,f':X-->Y and g,g':U-->V, the operation acts as a placeholder for a "test" morphism in the gap. f*g = f'*g' if fkg =f'kg' for all k:Y-->U. (Nontrivial example: in the category of groups, f*g = -f*-g. ) Hope this helps, Robert , just as ordinary composition depends on syntax (ie, order), this generalized composition would On 2/28/2018 8:27 PM, Marquis Jean-Pierre 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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
-
Lucius Schoenbaum -
Marquis Jean-Pierre -
Robert Dawson