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Comparison of the coherent pro-homotopy theories of Edwards-Hastings, Lisica-Mardešić and Günther. (English) Zbl 0795.55010

The coherent prohomotopy category was introduced by the reviewer [Math. Z. 140, 1-21 (1974; Zbl 0275.55022)] and shortly after by D. A. Edwards and H. M. Hastings [Čech and Steenrod homotopy theories with applications to geometric topology, Lect. Notes Math. 542 (1976; Zbl 0334.55001)]. (The two definitions were shown to be equivalent by the reviewer in 1988 [Topology Appl. 28, No. 3, 289-293 (1988; Zbl 0647.18008)].) These definitions inverted the level homotopy equivalences in the procategory of spaces to obtain the coherent prohomotopy category. In 1982-1984, Yu. T. Lisitsa and S. Mardešić [Glas. Mat., III. Ser. 19(39), 335-399 (1984; Zbl 0553.55009)] published a detailed geometrically constructed version of coherent prohomotopy much on the lines of R. M. Vogt’s paper [Math. Z. 134, 11-52 (1973; Zbl 0276.55006)] which had dealt with coherent homotopy for diagrams. Although Vogt had proved a result linking his coherent homotopy category with a category of fractions of a diagram category, no detailed direct verification of the equivalence for the procategory case was given until 1989 when J.-M. Cordier published a proof [Cah. Topologie Géom. Différ. Catégoriques 30, No. 3, 257-275 (1989; Zbl 0679.55006)]. This paper puts forward an interesting new approach to homotopy coherence and proves that the resulting theory yields a coherent prohomotopy category equivalent to both the Edwards-Hastings and Lisitsa-Mardešić models. The methods used are powerful and are likely to have consequences outside the narrow range of application indicated here. (Reviewers remark: In particular it would be interesting to tie them in with the theories of \(A_ \infty\)-algebras and homotopy coherent algebraic theories pioneered by Stasheff, Boardman, Vogt and others).
Reviewer: T.Porter (Bangor)

MSC:

55P55 Shape theory
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