Homotopy fiber
In 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 f : A → B {\displaystyle f:A\to B} . 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 groups ⋯ → π n + 1 ( B ) → π n ( Hofiber ( f ) ) → π n ( A ) → π n ( B ) → ⋯ {\displaystyle \cdots \to \pi _{n+1}(B)\to \pi _{n}({\text{Hofiber}}(f))\to \pi _{n}(A)\to \pi _{n}(B)\to \cdots } Moreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished triangle C ( f ) ∙ [ − 1 ] → A ∙ → B ∙ → [ + 1 ] {\displaystyle C(f)_{\bullet }[-1]\to A_{\bullet }\to B_{\bullet }\xrightarrow {[+1]} } gives a long exact sequence analogous to the long exact sequence of homotopy groups.