The realization space of a \(\Pi\)-algebra: a moduli problem in algebraic topology
Let \(\Pi\) denote the full subcategory of the pointed homotopy category on the finite wedges of spheres \(S^{n_{1}} \vee \dots \vee S^{n_{k}}\). A \(\Pi\)-algebra \(A\) is a contravariant set-valued functor defined on \(\Pi\) which takes wedges to products. Examples include the functors \(\pi_{\ast}X\) represented by pointed spaces \(X\) on the pointed homotopy category. The authors develop an obstruction theory for determining when a given \(\Pi\)-algebra \(A\) is realized by a space \(X\) in the sense that \(A \cong \pi_{\ast}X\). The main tool is a tower of partial moduli spaces of realizations which is effectively a Postnikov tower for the full moduli space of realizations of \(A\) in the category of simplicial spaces. The partial moduli are moduli of particular types of simplicial spaces. Simplicial spaces represent simplicial \(\Pi\)-algebras, and the Dwyer-Kan-Stover \(E_{2}\)-model structure for simplicial spaces is the natural context for manipulating their homotopy types, see [\textit{W.G. Dwyer, D.M. Kan} and \textit{C. R. Stover}, J. Pure Appl. Algebra 90, No. 2, 137--152 (1993; Zbl 0814.55008)].
scientific article; zbMATH DE number 2091696