Cofibration

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 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. In what follows, let denote the unit interval. A map of topological spaces is called a cofibrationpg 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 , whereWe can encode this condition in the following commutative diagramwhere is the path space of equipped with the compact-open topology. For the notion of a cofibration in a model category, see . 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.