Correction: I had mistakenly written

> ... it is NOT the case for varieties IN GENERAL that
> the epimorphisms coincide with the mappings whose underlying functions
> are one-to-one -- ...

where I had intended the last phrase to be

> are surjective -- ...

Apologies. A corrected version appears below.

-- Fred

------ Original Message ------

Received: Sun, 12 Feb 2006 07:57:50 PM EST
From: "Fred E.J.Linton" <flinton@...>
To: <univalg@yahoogroups.com>
Subject: Re: [univalg] M-mapping

> To answer Sara Raja's questions about the category of
> M-sets and M-mappings in relatively general terms,
> let me observe first that this category is both:
>
> (i) a variety (= equationally definable class) of algebras; and
> (ii) a topos.
>
> Now, because it is a variety, the mappings that are monomorphisms
> coincide with the mappings which, as functions, are one-to-one.
>
> (This one observation settles the first question.)
>
> Next, because it is a topos, the mappings that are epimorphisms
> coincide with the mappings which happen to be coequalizers.
>
> At the same time, because it is still a variety, the coequalizers
> coincide with the mappings which, as functions, are surjective.
>
> (The last two observations jointly settle the other question.)
>
> For citations establishing the two variety assertions, one may take
> my 1965 La Jolla Conference paper (in Springer's Proc. Conf. Categ. Alg.)
> or Manes' Springer volume on universal algebra, or Barr's "TTT".
>
> For the topos assertion, see Peter Johnstone's Cambridge U. Press book,
> "Topos Theory".
>
> Of course, there are many concrete categories other than varieties
> in which the monomorphisms coincide with the maps whose underlying
> functions are one-to-one -- a sufficient condition for this on a
> concrete category X with "underlying set functor" U: X --> Sets is,
> I believe, that U be both faithful and representable.
>
> At the same time, it is NOT the case for varieties IN GENERAL that
> the epimorphisms coincide with the mappings whose underlying functions
> are surjective -- just think, e.g., of unital rings, or monoids;
> and something very special indeed is needed for the epimorphisms
> of a general concrete category to be surjective -- think, say, of
> Hausdorff topological spaces, where "epimorphic" means "with dense
> image."
>
> Hope this helps.
>
> -- Fred Linton
>
> ------ Original Message (in part) ------
>
> Received: Sun, 12 Feb 2006 08:48:50 AM EST
> From: "Sara Raja" <sara_raja1979@...>
> To: univalg@yahoogroups.com
> Subject: [univalg] M-mapping
>
> > Hello every body , I have a question in categories. Please help me
> > to solve it.
> >
> > Let f:X --> Y be a M-mapping then show that :
> > 1) f is injective <--> f is monic,
> > 2) f is surjective <--> f is epic.
> >
> > ...[Two definitions suppressed]...