Homotopy fiber

In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber)^[1] is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces ${\displaystyle f:A\to B}$. 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

${\displaystyle \cdots \to \pi _{n+1}(B)\to \pi _{n}({\text{Hofiber}}(f))\to \pi _{n}(A)\to \pi _{n}(B)\to \cdots }$

Moreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished triangle

${\displaystyle C(f)_{\bullet }[-1]\to A_{\bullet }\to B_{\bullet }\xrightarrow {[+1]} }$

gives a long exact sequence analogous to the long exact sequence of homotopy groups. There is a dual construction called the homotopy cofiber.