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 one-to-one -- 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]...