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Algebraic models for homotopy types. (English) Zbl 1088.55006
Despite the slightly misleading title, this survey-style article deals with a computational model of homotopy theory developed by the authors in previous work. They discuss the fundamental problem, viz. to find a computable category equivalent to the homotopy category of simply connected spaces, and why it is not solvable in this full generality. They then advertize their solution of restricting the scope to a certain subcategory which they call simplicial sets with effective homology. In joint work with X. Dousson and Y. Siret, the authors have developed a computer program, Kenzo, to do homology computations for such simplicial sets with effective homology. They document this in the present paper with some sample runs of Kenzo on spaces whose homology would be quite laborious to compute by hand. Among the spaces that can be constructed with Kenzo, and which are thus amenable to computations, are (infinite) projective spaces, spaces obtained by attaching cells by degree-\(n\) maps, and, most powerfully, by taking loop spaces. In the final chapter, an alternative approach of representing homotopy types using \(E_\infty\)-structures along the lines of work of M. Mandell is mentioned as a project worth pursuing.
The paper gives a decent introduction to the problems around computational homotopy theory and the author’s Lisp-based approach Kenzo. Details are mostly omitted. The style is somewhat verbose and occasionally irritating.

55P15 Classification of homotopy type
18G40 Spectral sequences, hypercohomology
18G55 Nonabelian homotopical algebra (MSC2010)
55S45 Postnikov systems, \(k\)-invariants
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