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

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