Direct products/sums: when the associated morphisms are epi/mono
Exercise 7 (Chapter 2, “Categories”):
In the category of sets, the two morphisms (
and
) in a direct product are monomorphisms and the two morphisms in a direct sum are epimorphisms. Is this true in every category?
There are two mistakes in this wording.
- The author has exchanged “monomorphisms” and “epimorphisms”. In a direct product of sets, the morphisms are generally not monomorphisms (injective), and in a direct sum of sets, they are generally not epimorphisms (surjective).
- In a direct product of sets, the morphisms are almost always epimorphisms. If one of the sets is empty, the product is empty, and the morphism associated with the other set (an empty mapping) is not an epimorphism; unless this other set is empty too.
The general theorem I will prove is this:
Let
and
be objects of a category
, and
a
-direct product of
and
. If
is not empty then
is an epimorphism.
We consider the above conditions satisfied. There exists a morphism .
Let be the identity morphism
and
the morphism
. Applying the universal property of the direct product
to the triplet
, we obtain that there exists a unique
such that both
and
.
The first of these two relations says that is the identity on
. If we have an object
and two morphisms
and
from
to
such that
, then, composing this relation on the right with
and eliminating the resulting identity morphisms, we get
. Hence
is an epimorphism.
We obtain, of course, a similar result exchanging the roles of and
.
Reversing the arrows, we get the corresponding result for direct sums:
Let
and
be objects of a category
, and
a
-direct sum of
and
. If
is not empty then
is a monomorphism.
Plus, of course, the similar result obtained by exchanging and
.
It so happens that in the category of sets, in the case of a direct product, both and
are always monomorphisms, even if
, respectively
, are empty.