P.J. Higgins' "Categories and Groupoids" question ....

P.J. Higgins' "Categories and Groupoids" question ....

[Vasili I. Galchin]

Hello Cat Group,

      I have been reading
http://www.tac.mta.ca/tac/reprints/articles/7/tr7abs.html as well as
other works on groupoids .... I really like Higgins' book but in
Chapter 1 I keep seeing allusions to "injections" and
"surjections"instead of more general  "monomorphisms" and
"epimorphisms", respectively. Is this because of his assumption (and
others ..) that starting point is a "small" graph, category, etc.,
i.e. sets not classes? If so, has anybody expanded treatment of
groupoids beyond "sets" to "classes"?

        Also IMO "subgraphs" should be presented as as "monos" in the
"graph category"....

Kind regards,

Vasili


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Re: P.J. Higgins' "Categories and Groupoids" question ....

[Steve Vickers]

Dear Vasili,

I would say it is nothing to do with size issues, but stems from his stated aim of treating categories as algebraic structures. If you look at the algebra of monoids, or of rings, you find that surjections are different from epis. Examples: if you take a monoid and freely adjoin inverses, then you get an  epi from the first to the second that is not in general a surjection. The ring homomorphism from Z to Q is epi. Most algebraists would be more interested in the surjections than the epis, and I think Higgins is just extending that preference to particular categories such as C and G.

Steve Vickers.

> On 19 Feb 2014, at 07:29, "Vasili I. Galchin" <vigalchin@gmail.com> wrote:
> 
> Hello Cat Group,
> 
>      I have been reading
> http://www.tac.mta.ca/tac/reprints/articles/7/tr7abs.html as well as
> other works on groupoids .... I really like Higgins' book but in
> Chapter 1 I keep seeing allusions to "injections" and
> "surjections"instead of more general  "monomorphisms" and
> "epimorphisms", respectively. Is this because of his assumption (and
> others ..) that starting point is a "small" graph, category, etc.,
> i.e. sets not classes? If so, has anybody expanded treatment of
> groupoids beyond "sets" to "classes"?
> 
>        Also IMO "subgraphs" should be presented as as "monos" in the
> "graph category"....
> 
> Kind regards,
> 
> Vasili

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: P.J. Higgins' "Categories and Groupoids" question ....

[Luiz Henrique]

Hello Vasili.
I don't know if this is what you are looking for but I have some (very
short) notes on arxiv about groupoid objects in categories. There, the
idea is to avoid set theoretical considerations as far as possible.
The link is
http://arxiv.org/abs/1207.3694

Best regards,

Luiz


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