Homotopy fiber

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In mathematics, especially homotopy theory, the homotopy fiber is part of a construction of associating to an arbitrary continuous function of topological spaces $f \colon A \to B$ a fibration.

In particular, given such a map, define $E f$ to be the set of pairs $(a, p)$ where $a \in A$ and $p \colon [0,1] \to B$ is a path such that $p (0) = f (a)$ . We give $E f$ a topology by giving it the subspace topology as a subset of $A \times B^I$ (where $B I$ is the space of paths in $B$ which as a function space has the compact-open topology). Then the map $E_f \to B$ given by $(a,p) \mapsto p(1)$ is a fibration. Furthermore, $E f$ is homotopy equivalent to $A$ as follows: Embed $A$ as a subspace of $E f$ by $a \mapsto (a, p_a)$ where $p a$ is the constant path at $f (a)$ . Then $E f$ deformation retracts to this subspace by contracting the paths.

The fiber of this fibration (which is only well-defined up to homotopy equivalence) is the homotopy fiber $F f$ , which can be defined as the set of all $(a, p)$ with $a \in A$ and $p \colon [0,1] \to B$ a path such that $p (0) = f (a)$ and $p (1) = b 0$ , where $b_0 \in B$ is some fixed basepoint of $B$ .