Cofibration

In mathematics, in particular homotopy theory, a continuous mapping
where formula_2 and formula_3 are topological spaces, is a cofibration if it lets homotopy classes of maps formula_4 be extended to homotopy classes of maps formula_5 whenever a map formula_6 can be extended to a map formula_7 where formula_8, hence their associated homotopy classes are equal formula_9.
This type of structure can be encoded with the technical condition of having the homotopy extension property with respect to all spaces formula_10. 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.
Definition.
Homotopy theory.
A map formula_11 of topological spaces is called a cofibrationpg 51 if for any map formula_12 such that there is an extension to formula_3, meaning there is a map formula_14 such that formula_8, we can extend a homotopy of maps formula_16 to a homotopy of maps formula_17, whereformula_18We can encode this condition in the following commutative diagramwhere formula_19 is the path space of formula_10.
Cofibrant objects.
For a model category formula_21, such as for pointed topological spaces, an object formula_3 is called cofibrant if the map formula_23 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.
Examples.
In topology.
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 formula_25 called the mapping cylinder (where formula_26 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 formula_28 through which formula_29 factors through, meaning there is a commutative diagram
where formula_30 is a homotopy equivalence.
In addition to this class of examples, there are
In chain complexes.
If we let formula_36 be the category of chain complexes which are formula_37 in degrees formula_38, then there is a model category structurepg 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 mapsformula_39which are injective and the cokernel complex formula_40 is a complex of projective objects in formula_41. In addition, the cofibrant objects are the complexes whose objects are all projective objects in formula_41.
Semi-simplicial sets.
For the category formula_43 of semi-simplicial setspg 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.
Constructions with cofibrations.
Cofibrant replacement.
Note that in a model category formula_21 if formula_65 is not a cofibration, then the mapping cylinder formula_66 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.
Cofiber.
For a cofibration formula_32 we define the cofiber to be the induced quotient space formula_68. In general, for formula_24, the cofiberpg 59 is defined as the quotient spaceformula_70which is the mapping cone of formula_29. Homotopically, the cofiber acts as a homotopy cokernel of the map formula_24. In fact, for pointed topological spaces, the homotopy colimit offormula_73In fact, the sequence of maps formula_74 comes equipped with the cofiber sequence which acts like a distinguished triangle in triangulated categories.