Algebraic models for homotopy types

Abstract

The classical problem of algebraic models for homotopy types is precisely stated here in terms of our ability to compute with the models. Two different natural statements for this problem are produced, the simplest one being entirely solved by the notion of $\mathcal{SS}_{EH}$-structure, due to the authors. Other tentative solutions, Postnikov towers and $E_\infty$-chain complexes, are considered and compared with the $\mathcal{SS}_{EH}$-structures. In particular, an imprecision in the usual definition of the k-"invariants" is explained, which implies we seem far from a solution for the ideal statement of our problem. On the positive side, the problem stated below in the framed quotation is solved.