In mathematics, in particular homotopy theory, a continuous mapping between topological spaces is a cofibration if it has the homotopy extension property with respect to all topological spaces
This definition is formally dual to that of a fibration, which is required to satisfy the homotopy lifting property with respect to all spaces; this is one instance of the broader Eckmann–Hilton duality in topology.
Cofibrations are a fundamental concept of homotopy theory.
Quillen has proposed the notion of model category as a formal framework for doing homotopy theory in more general categories; a model category is endowed with three distinguished classes of morphisms called fibrations, cofibrations and weak equivalences satisfying certain lifting and factorization axioms.
of topological spaces is called a cofibration[1]pg 51 if for any map
We can encode this condition in the following commutative diagramwhere
Topologists have long studied notions of "good subspace embedding", many of which imply that the map is a cofibration, or the converse, or have similar formal properties with regards to homology.
is normal, and its product with the unit interval
has the homotopy extension property with respect to any absolute neighborhood retract.
is an absolute neighborhood retract, then the inclusion of
[2][3] Hatcher's introductory textbook Algebraic Topology uses a technical notion of good pair which has the same long exact sequence in singular homology associated to a cofibration, but it is not equivalent.
The notion of cofibration is distinguished from these because its homotopy-theoretic definition is more amenable to formal analysis and generalization.
called the mapping cylinder of
There is a canonical subspace embedding
This result can be summarized by saying that "every map is equivalent in the homotopy category to a cofibration."
Arne Strøm has proved a strengthening of this result, that every map
factors as the composition of a cofibration and a homotopy equivalence which is also a fibration.
of the boundary sphere of a solid disk is a cofibration for every
A frequently used fact is that a cellular inclusion is a cofibration (so, for instance, if
This follows from the previous fact and the fact that cofibrations are stable under pushout, because pushouts are the gluing maps to the
be the category of chain complexes which are
, then there is a model category structure[5]pg 1.2 where the weak equivalences are the quasi-isomorphisms, the fibrations are the epimorphisms, and the cofibrations are maps
which are degreewise monic and the cokernel complex
of simplicial sets[5]pg 1.3 there is a model category structure where the fibrations are precisely the Kan fibrations, cofibrations are all injective maps, and weak equivalences are simplicial maps which become homotopy equivalences after applying the geometric realization functor.
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.
we define the cofiber to be the induced quotient space
, the cofiber[1]pg 59 is defined as the quotient space
Homotopically, the cofiber acts as a homotopy cokernel of the map
In fact, for pointed topological spaces, the homotopy colimit of
comes equipped with the cofiber sequence which acts like a distinguished triangle in triangulated categories.