A short question about the set-based definition of categories

A short question about the set-based definition of categories

[Marquis Jean-Pierre]

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/ ]

Re: A short question about the set-based definition of categories

[Lucius Schoenbaum]

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/ ]

Re: A short question about the set-based definition of categories

[Robert Dawson]

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/ ]