Concrete categories: injective/surjective implies mono/epi; functor generalization
A category is by definition concrete if there exists a faithful functor
from
to the category of sets
. (A functor is faithful if it is one-to-one (injective) regarding morphisms.)
It happens to be the case that in the category , monomorphisms and one-to-one mappings are the same thing; similarly for epimorphisms and surjections.
We will show that for any concrete category , with faithful functor
from
to
, a sufficient (though not always necessary) condition for a morphism
in
to be mono is that its image by
be itself mono, that is, one-to-one.
More generally, we will show that if categories and
are such that there exists a faithful functor
from
to
, a sufficient condition for a morphism
in
to be mono is that its image in
by
be itself mono. The same will hold for epimorphisms.
Proof
I will first prove the general case last mentioned.
Assume categories and
with a faithful functor
from the former to the latter.
Let and
be two objects in
and
a morphism from
to
such that
is a monomorphism.
Let be an object in
and
and
two morphisms from
to
such that
.
Since is a functor, we can write
, and similarly for
; hence:
Since is a monomorphism, it follows that
.
Since is faithful, this implies
.
This being the case for any object in
and
and
morphisms from
to
such that
, it follows that
is a monomorphism.
We have thus shown that if is mono, then
itself is mono.
A similar reasoning shows that if is epi, then
itself is epi.
In the case of concrete categories, is the set category, in which injective maps are mono and surjective maps are epi. Typically, the
-morphisms are particular mappings between the underlying sets, that is the sets obtained through the functor
. In this case, one can say that injective implies mono and surjective implies epi.