The coherent homotopy category over a fixed space is a category of fractions (Q1187113)

Let 1) \(\text{Ho}(\text{Top}_ B)\) be the category \(\text{Top}_ B(\Sigma^{-1})\), where \(\Sigma\) denotes the class of all \(\phi: X\to Y\) over \(B\) which are homotopy equivalences (not necessarily over \(B\)), and 2) \({\mathcal H}_ B\) be the category with the same objects as \(\text{Top}_ B\) having equivalence classes of homotopies \(h_t: f\simeq g\phi\), \(f: X\to B\), \(g: Y\to B\), \(\phi: X\to Y\) as morphisms. The objective of this paper is to provide an isomorphism between both categories. This is accomplished by making excessive use of the explicit construction of a quotient category (resp. its characterization) \({\mathcal C}(\Sigma^{-1})\) (\(\mathcal C\) = a given category, \(\Sigma\) = class of morphisms) due to \textit{J. Dugundji} and the reviewer [Trans. Am. Math. Soc. 140, 239--256 (1969; Zbl 0182.25902)].