In mathematics, in particular homotopy theory, a continuous mapping


where and are topological spaces, is a cofibration if it lets homotopy classes of maps be extended to homotopy classes of maps whenever a map can be extended to a map where , hence their associated homotopy classes are equal .

This type of structure can be encoded with the technical condition of having the homotopy extension property with respect to all spaces . 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. Because of the generality this technical condition is stated, it can be used in model categories.


Homotopy theoryEdit

In what follows, let   denote the unit interval.

A map   of topological spaces is called a cofibration[1]pg 51 if for any map   such that there is an extension to  , meaning there is a map   such that  , we can extend a homotopy of maps   to a homotopy of maps  , where


We can encode this condition in the following commutative diagram


where   is the path space of  .

Cofibrant objectsEdit

For a model category  , such as for pointed topological spaces, an object   is called cofibrant if the map   is a cofibration. Note that in the category of pointed topological spaces, the notion of cofibration coincides with the previous definition assuming the maps are pointed maps of topological spaces.


In topologyEdit

Cofibrations are an awkward class of maps from a computational perspective because they are more easily seen as a formal technical tool which enables one to "do" homotopy theoretic constructions with topological spaces. Fortunately, for any map


of topological spaces, there is an associated cofibration to a space   called the mapping cylinder (where   is a deformation retract of, hence homotopy equivalent to it) which has an induced cofibration called replacing a map with a cofibration


and a map   through which   factors through, meaning there is a commutative diagram


where   is a homotopy equivalence.

In addition to this class of examples, there are

  • 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  , and pushouts are the gluing maps to the   skeleton.
  • Cofibrations are preserved under pushouts and composition, which is stated precisely below.

In chain complexesEdit

If we let   be the category of chain complexes which are   in degrees  , then there is a model category structure[2]pg 1.2 where the weak equivalences are quasi-isomorphisms, so maps of chain complexes which are isomorphisms after taking cohomology, fibrations are just epimorphisms, and cofibrations are given by maps


which are injective and the cokernel complex   is a complex of projective objects in  . In addition, the cofibrant objects are the complexes whose objects are all projective objects in  .

Semi-simplicial setsEdit

For the category   of semi-simplicial sets[2]pg 1.3 (meaning there are no co-degeneracy maps going up in degree), there is a model category structure with fibrations given by Kan-fibrations, cofibrations injective maps, and weak equivalences given by weak equivalences after geometric realization.


  • 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   is any (continuous) map (between compactly generated spaces), and   is a cofibration, then the induced map   is a cofibration.
  • The mapping cylinder can be understood as the pushout of   and the embedding (at one end of the unit interval)  . That is, the mapping cylinder can be defined as  . By the universal property of the pushout,   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   (between compactly generated spaces), one defines the mapping cylinder
One then decomposes   into the composite of a cofibration and a homotopy equivalence. That is,   can be written as the map
with  , when   is the inclusion, and   on   and   on  .
  • There is a cofibration (A, X), if and only if there is a retraction from   to  , 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.

Constructions with cofibrationsEdit

Cofibrant replacementEdit

Note that in a model category   if   is not a cofibration, then the mapping cylinder   forms a cofibrant replacement. In fact, if we work in just the category of topological spaces, the cofibrant replacement for any map from a point to a space forms a cofibrant replacement.


For a cofibration   we define the cofiber to be the induced quotient space  . In general, for  , the cofiber[1]pg 59 is defined as the quotient space


which is the mapping cone of  . Homotopically, the cofiber acts as a homotopy cokernel of the map  . In fact, for pointed topological spaces, the homotopy colimit of


In fact, the sequence of maps   comes equipped with the cofiber sequence which acts like a distinguished triangle in triangulated categories.

See alsoEdit


  1. ^ a b May, J. Peter. (1999). A concise course in algebraic topology. Chicago: University of Chicago Press. ISBN 0-226-51182-0. OCLC 41266205.
  2. ^ a b Quillen, Daniel G. (1967). Homotopical algebra. Berlin: Springer-Verlag. ISBN 978-3-540-03914-3. OCLC 294862881.