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 . That is, is a cofibration if for each topological space , and for any continuous maps and with , for any homotopy from to , there is a continuous map and a homotopy from to such that for all and . (Here, denotes the unit interval .)

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

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Homotopy theory

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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   equipped with the compact-open topology.

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

Examples

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In topology

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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   is a binormal space (  is normal, and its product with the unit interval   is normal) then every closed subspace of   has the homotopy extension property with respect to any absolute neighborhood retract. Likewise, if   is a closed subspace of   and the subspace inclusion   is an absolute neighborhood retract, then the inclusion of   into   is a cofibration.[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.

If   is a continuous map between topological spaces, there is an associated topological space   called the mapping cylinder of  . There is a canonical subspace embedding   and a projection map   such that   as pictured in the commutative diagram below. Moreover,   is a cofibration and   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   factors as the composition of a cofibration and a homotopy equivalence which is also a fibration.[4]

A topological space   with distinguished basepoint   is said to be well-pointed if the inclusion map   is a cofibration.

The inclusion map   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   is a CW pair, then   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   skeleton.

In chain complexes

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Let   be an Abelian category with enough projectives.

If we let   be the category of chain complexes which are   in degrees  , 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   is a complex of projective objects in  . It follows that the cofibrant objects are the complexes whose objects are all projective.

Simplicial sets

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The category   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.

Properties

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  • 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.
  • 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 cofibrations

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Cofibrant replacement

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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.

Cofiber

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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 also

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References

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  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. ^ Edwin Spanier, Algebraic Topology, 1966, p. 57.
  3. ^ Garth Warner, Topics in Topology and Homotopy Theory, section 6.
  4. ^ Arne Strøm, The homotopy category is a homotopy category
  5. ^ a b Quillen, Daniel G. (1967). Homotopical algebra. Berlin: Springer-Verlag. ISBN 978-3-540-03914-3. OCLC 294862881.