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Wikipedia / Cofibration

In mathematics, in particular homotopy theory, a continuous mapping between topological spaces

is a cofibration if it has the homotopy extension property, for any homotopy.)

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.

Definition

Homotopy theory

denote the unit interval.

of topological spaces is called a cofibrationpg 51, whereWe can encode this condition in the following commutative diagram is the path space equipped with the compact-open topology.

For the notion of a cofibration in a model category, see model category.

Examples

In topology

is a binormal space is a cofibration. 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 is a homotopy equivalence. This result can be summarized by saying that "every map is equivalent in the homotopy category to a cofibration."

factors as the composition of a cofibration and a homotopy equivalence which is also a fibration.

is said to be well-pointed is a cofibration.

.

is a CW pair skeleton.

In chain complexes

be an Abelian category, then there is a model category structurepg 1.2 where the weak equivalences are the quasi-isomorphisms, the fibrations are the epimorphisms, and the cofibrations are maps is a complex of projective objects. It follows that the cofibrant objects are the complexes whose objects are all projective.

Simplicial sets

of simplicial setspg 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.

Properties

  • For Hausdorff spaces, every cofibration is a closed inclusion (injective with closed image); the result also generalizes to weak Hausdorff spaces.

  • The pushout is a cofibration.

  • The mapping cylinder. By the universal property is a cofibration precisely when a mapping cylinder can be constructed for every space X.

  • There is a cofibration (A, X), if and only if there is a retraction, 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 cofibrations

Cofibrant replacement

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

we define the cofiber, the cofiberpg 59 is defined as the quotient space. 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.

See also

References