In category theory, the notion of a projective object generalizes the notion of a projective module.
An object
P
in a category
C
is projective if the hom functor
Hom
(
P
,
−
)
:
C
→
S
e
t
preserves epimorphisms. That is, every morphism
f
:
P
→
X
factors through every epi
Y
→
X
.
Let
C
be an abelian category. In this context, an object
P
∈
C
is called a projective object if
Hom
(
P
,
−
)
:
C
→
A
b
is an exact functor, where
A
b
is the category of abelian groups.
The dual notion of a projective object is that of an injective object: An object
Q
in an abelian category
C
is injective if the
Hom
(
−
,
Q
)
functor from
C
to
A
b
is exact.
Let
A
be an abelian category.
A
is said to have enough projectives if, for every object
A
of
A
, there is a projective object
P
of
A
and an exact sequence
P
⟶
A
⟶
0.
In other words, the map
p
:
P
→
A
is "epic", or an epimorphism.
Let
R
be a ring with 1. Consider the category of left
R
-modules
M
R
.
M
R
is an abelian category. The projective objects in
M
R
are precisely the projective left R-modules. So
R
is itself a projective object in
M
R
.
Dually, the injective objects in
M
R
are exactly the injective left R-modules.
The category of left (right)
R
-modules also has enough projectives. This is true since, for every left (right)
R
-module
M
, we can take
F
to be the free (and hence projective)
R
-module generated by a generating set
X
for
M
(we can in fact take
X
to be
M
). Then the canonical projection
π
:
F
→
M
is the required surjection.
