Homotopy extension propertyIn mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations. Let be a topological space, and let . We say that the pair has the homotopy extension property if, given a homotopy and a map such that then there exists an extension of to a homotopy such that .
Mapping cone (topology)In mathematics, especially homotopy theory, the mapping cone is a construction of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated . Its dual, a fibration, is called the mapping fibre. The mapping cone can be understood to be a mapping cylinder , with one end of the cylinder collapsed to a point. Thus, mapping cones are frequently applied in the homotopy theory of pointed spaces. Given a map , the mapping cone is defined to be the quotient space of the mapping cylinder with respect to the equivalence relation , .
Homotopy categoryIn mathematics, the homotopy category is a built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below. More generally, instead of starting with the category of topological spaces, one may start with any and define its associated homotopy category, with a construction introduced by Quillen in 1967. In this way, homotopy theory can be applied to many other categories in geometry and algebra.
Section (fiber bundle)In the mathematical field of topology, a section (or cross section) of a fiber bundle is a continuous right inverse of the projection function . In other words, if is a fiber bundle over a base space, : then a section of that fiber bundle is a continuous map, such that for all . A section is an abstract characterization of what it means to be a graph. The graph of a function can be identified with a function taking its values in the Cartesian product , of and : Let be the projection onto the first factor: .
