Reduced product objects in model categories (Q558391)

The authors introduce the concept of reduced product object in abstract model categories and generalize the classical James theorem: \(X_\infty \sim \Omega\Sigma X\) (see \textit{I.M.~James} [Ann. Math. (2) 62, 170--197 (1955; Zbl 0064.41505)]). This generalization is subject to several conditions imposed on a model category. The most important assumption is the Cube axiom: if the bottom face of a cubical diagram in a model category is a homotopy pushout and the four vertical faces are homotopy pullbacks, then the top face is a homotopy pushout. This axiom was proven for topological spaces by \textit{M.~Mather} [Can. J. Math. 28, 225--263 (1976; Zbl 0351.55005)]. In the framework of abstract model categories the Cube axiom was used for the first time by \textit{J.-P.~Doeraene} [J. Pure Appl. Algebra 84, No.~3, 215--261 (1993; Zbl 0777.55007)] in order to generalize the concept of Lusternik-Schnirelmann category to abstract model categories. The same condition appeared in the work of \textit{D.~Chataur} and \textit{J.~Scherer} [Fibrewise nullification and the cube theorem. arXiv:math.AT/0303062] on fibrewise localization in an abstract model category. The reduced powers \(X_n\) of an object \(X\), or the relative version \((X,A)_n\), are defined generalizing the approach of \textit{B.~Gray} [Proc. Lond. Math. Soc. (3) 26, 497--520 (1973; Zbl 0263.55012)]. The James space \(X_\infty\) is the colimit of \(X_n\). It is shown that for any cofibration \(A\to X\) there exists a map \((X,A)_\infty \to X/A\) having \(A_\infty\) as its homotopy fibre. Substituting \((CA,A)\) the authors obtain that \(A_\infty\) is weakly equivalent to the homotopy fibre of the map \(\ast \to CA/A \sim \Sigma A\), or \(A_\infty \sim \Omega\Sigma A\).