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 S. That is, i is a cofibration if for each topological space S, and for any continuous maps and g:X\to S with g\circ i=f, for any homotopy from f to f', there is a continuous map g':X \to S and a homotopy from g to g' such that for all a\in A and t\in I. (Here, I denotes the unit interval [0,1].) 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

In what follows, let I = [0,1] denote the unit interval. A map of topological spaces is called a cofibration pg 51 if for any map f:A \to S such that there is an extension to X (meaning: there is a map f':X \to S 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"Cofibration diagram.svg"where S^I is the path space of S equipped with the compact-open topology. For the notion of a cofibration in a model category, see model category.

Examples

In topology

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. In 1937, Borsuk proved that if X is a binormal space (X is normal, and its product with the unit interval X\times I is normal) then every closed subspace of X has the homotopy extension property with respect to any absolute neighborhood retract. Likewise, if A is a closed subspace of X and the subspace inclusion is an absolute neighborhood retract, then the inclusion of A into X 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. If f:X \to Y is a continuous map between topological spaces, there is an associated topological space Mf called the mapping cylinder of f. There is a canonical subspace embedding i: X\to Mf and a projection map r: Mf\to Y such that as pictured in the commutative diagram below. Moreover, i is a cofibration and r is a homotopy equivalence. 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 f:X \to Y factors as the composition of a cofibration and a homotopy equivalence which is also a fibration. A topological space X with distinguished basepoint x is said to be well-pointed if the inclusion map {x}\to X is a cofibration. The inclusion map of the boundary sphere of a solid disk is a cofibration for every n. A frequently used fact is that a cellular inclusion is a cofibration (so, for instance, if (X, A) is a CW pair, then A \to X is a cofibration). This follows from the previous fact and the fact that cofibrations are stable under pushout, because pushouts are the gluing maps to the n-1 skeleton.

In chain complexes

Let \mathcal{A} be an Abelian category with enough projectives. If we let be the category of chain complexes which are 0 in degrees q << 0, then there is a model category structure 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 is a complex of projective objects in \mathcal{A}. It follows that the cofibrant objects are the complexes whose objects are all projective.

Simplicial sets

The category of simplicial sets 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.

Properties

Constructions with cofibrations

Cofibrant replacement

Note that in a model category \mathcal{M} if i:* \to X is not a cofibration, then the mapping cylinder Mi 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 A \to X we define the cofiber to be the induced quotient space X/A. In general, for f:X \to Y, the cofiber pg 59 is defined as the quotient space""which is the mapping cone of f. Homotopically, the cofiber acts as a homotopy cokernel of the map f:X \to Y. 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.