In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in particular homotopy theory, a continuous mapping
,

where A and X are topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

s, is a cofibration if it satisfies the homotopy extension property

Homotopy extension property

In mathematics, in the area of algebraic topology, the homotopy extension property indicates when a homotopy defined on a subspace can be extended to a homotopy defined on a larger space.-Definition:Let X\,\! be a topological space, and let A \subset X....

 with respect to all spaces Y. The name is because the dual condition, the homotopy lifting property

Homotopy lifting property

In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property is a technical condition on a continuous function from a topological space E to another one, B...

, defines fibration

Fibration

In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...

s. For a more general notion of cofibration see the article about model categories

Model category

In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes...

.

Basic theorems

• For Hausdorff space
  Hausdorff space
  In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
  
  s a cofibration is a closed inclusion (injective with closed image); for suitable spaces, a converse holds
• Every map can be replaced by a cofibration via the mapping cylinder
  Mapping cylinder
  In mathematics, specifically algebraic topology, the mapping cylinder of a function f between topological spaces X and Y is the quotientM_f = \,/\,\sim...
  
   construction
• There is a cofibration (A, X), if and only if there is a retraction
  Deformation retract
  In topology, a branch of mathematics, a retraction , as the name suggests, "retracts" an entire space into a subspace. A deformation retraction is a map which captures the idea of continuously shrinking a space into a subspace.- Retract :...
  
   from

to

,

since this is the pushout

Pushout (category theory)

In category theory, a branch of mathematics, a pushout is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain: it is the colimit of the span X \leftarrow Z \rightarrow Y.The pushout is the...

and thus induces maps to every space sensible in the diagram.

Examples

• 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 is a CW pair, then is a cofibration). This follows from the previous fact since is a cofibration for every .