Mapping cylinderIn mathematics, specifically algebraic topology, the mapping cylinder of a continuous function between topological spaces and is the quotient where the denotes the disjoint union, and ∼ is the equivalence relation generated by That is, the mapping cylinder is obtained by gluing one end of to via the map . Notice that the "top" of the cylinder is homeomorphic to , while the "bottom" is the space . It is common to write for , and to use the notation or for the mapping cylinder construction.
CofibrationIn 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 .
Homotopy fiberIn mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces . It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of homotopy groupsMoreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished trianglegives a long exact sequence analogous to the long exact sequence of homotopy groups.
Homotopy colimit and limitIn mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of and colimit extended to the homotopy category . The main idea is this: if we have a diagramconsidered as an object in the , (where the homotopy equivalence of diagrams is considered pointwise), then the homotopy limit and colimits then correspond to the and coconewhich are objects in the homotopy category , where is the category with one object and one morphism.
Puppe sequenceIn mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built from the mapping fibre (a fibration), and a long coexact sequence, built from the mapping cone (which is a cofibration). Intuitively, the Puppe sequence allows us to think of homology theory as a functor that takes spaces to long-exact sequences of groups. It is also useful as a tool to build long exact sequences of relative homotopy groups.
FibrationThe notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in Postnikov systems or obstruction theory. In this article, all mappings are continuous mappings between topological spaces. A mapping satisfies the homotopy lifting property for a space if: for every homotopy and for every mapping (also called lift) lifting (i.e. ) there exists a (not necessarily unique) homotopy lifting (i.e.
