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Maps between homotopy coherent diagrams. (English) Zbl 0655.55008

In “Homotopy invariant algebraic structures on topological spaces” (Lect. Notes Math. 347) (1973; Zbl 0285.55012)] J. M. Boardman and the reviewer proved the following results: Let \({\mathcal C}\) be a small topologically enriched category and X a homotopy coherent diagram of topological spaces. (1) Given a family f(C): X(C)\(\to Y(C)\), \(C\in ob {\mathcal C}\) of homotopy equivalences, then the Y(C) extend to a homotopy coherent diagram Y and the f(C) to a homotopy coherent homomorphism \(X\to Y\). (2) If f: \(X\to Y\) is a homotopy coherent homomorphism of homotopy coherent \({\mathcal C}\)-diagrams and H(C): f(C)\(\simeq g(C)\), \(C\in ob {\mathcal C}\), is a family of homotopies, g extends to a homotopy coherent homomorphism \(X\to Y\) and H to a coherent homotopy \(f\simeq g.\)
For discrete indexing categories, the authors extend these results to coherent diagrams in simplicial categories \({\mathcal B}\), for which each morphism space \({\mathcal B}(A,B)\) is a Kan complex. The proofs use induction on the nerve of the indexing category and the coherent nerve of the target category. They illustrate their constructions in low dimensions.
As applications they have group actions up to homotopy and strong shape theory in mind. The former case has been studied in detail by M. Fuchs [Proc. Am. Math. Soc. 58, 347-352 (1976; Zbl 0343.55010)] and by R. Schwänzl and the reviewer [Lect. Notes Math. 1217, 364-390 (1986; Zbl 0618.55007)].
Reviewer: R.Vogt

MSC:

55P99 Homotopy theory
55U35 Abstract and axiomatic homotopy theory in algebraic topology
57S17 Finite transformation groups
55U10 Simplicial sets and complexes in algebraic topology
55P91 Equivariant homotopy theory in algebraic topology
18G55 Nonabelian homotopical algebra (MSC2010)
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References:

[1] Boardman, J. M.; Vogt, R. M., Homotopy Invariant algebraic structures on topological spaces, (Lecture Notes Math., 347 (1973), Springer: Springer Berlin) · Zbl 0285.55012
[2] Cordier, J.-M., Sur la notion de diagramme homotopiquement cohérent, Proc. 3ème Colloque Catégories. Proc. 3ème Colloque Catégories, Cashiers Top. Geom. Diff., 23, 93-112 (1982), Amiens, 1980 · Zbl 0493.55009
[3] Cordier, J.-M.; Porter, T., Vogt’s theorem on categories of coherent diagrams, Math. Proc. Cambridge Phil. Soc., 100, 65-90 (1986) · Zbl 0603.55017
[4] Cordier, J.-M.; Porter, T., Homotopy Limits and Homotopy Coherence: A report on joint work, (Notes on Lectures given at Perugia (Sept.-Oct. 1984), Università di Perugia)
[5] Dwyer, W. G.; Kan, D. M., Simplicial localizations of categories, J. Pure Appl. Alg., 17, 267-284 (1980) · Zbl 0485.18012
[6] Porter, T., Coherent prohomotopy theory, Cashiers Topologie Géom. Différentielle, 19, 3-46 (1978) · Zbl 0387.55013
[7] Vogt, R. M., Homotopy Limits and colimits, Math. Z., 134, 11-52 (1973) · Zbl 0276.55006
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