In mathematics, in particular homotopy theory, a continuous mapping
i
:
A
→
X
,
where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. This definition is dual to that of a fibration, which is required to satisfy the homotopy lifting property with respect to all spaces. This duality is informally referred to as Eckmann–Hilton duality.
A more general notion of cofibration is developed in the theory of model categories.
For Hausdorff spaces, every cofibration is a closed inclusion (injective with closed image); the result also generalizes to weak Hausdorff spaces.
The pushout of a cofibration is a cofibration. That is, if
g
:
A
→
B
is any (continuous) map (between compactly generated spaces), and
i
:
A
→
X
is a cofibration, then the induced map
B
→
B
∪
g
X
is a cofibration.
The mapping cylinder can be understood as the pushout of
i
:
A
→
X
and the embedding (at one end of the unit interval)
i
0
:
A
→
A
×
I
. That is, the mapping cylinder can be defined as
M
i
=
X
∪
i
(
A
×
I
)
. By the universal property of the pushout,
i
is a cofibration precisely when a mapping cylinder can be constructed for every space X.
Every map can be replaced by a cofibration via the mapping cylinder construction. That is, given an arbitrary (continuous) map
f
:
X
→
Y
(between compactly generated spaces), one defines the mapping cylinder
One then decomposes
f
into the composite of a cofibration and a homotopy equivalence. That is,
f
can be written as the map
X
→
j
M
f
→
r
Y
with
f
=
r
j
, when
j
:
x
↦
(
x
,
0
)
is the inclusion, and
r
:
y
↦
y
on
Y
and
r
:
(
x
,
s
)
↦
f
(
x
)
on
X
×
I
.
There is a cofibration (A, X), if and only if there is a retraction from
X
×
I
to
(
A
×
I
)
∪
(
X
×
{
0
}
)
, since this is the pushout and thus induces maps to every space sensible in the diagram.
Similar equivalences can be stated for deformation-retract pairs, and for neighborhood deformation-retract pairs.
Cofibrations are preserved under push-outs and composition, as one sees from the definition via diagram-chasing.
A frequently used fact is that a cellular inclusion is a cofibration (so, for instance, if
(
X
,
A
)
is a CW pair, then
A
→
X
is a cofibration). This follows from the previous fact since
S
n
−
1
→
D
n
is a cofibration for every
n
, and pushouts are the gluing maps to the
n
−
1
skeleton.
The homotopy colimit generalizes the notion of a cofibration.
