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Fibrant diagrams, rectifications and a construction of Loday. (English) Zbl 0715.55012

The rectification of a homotopy coherent diagram is a special case of the coherent Kan extension [the authors, Math. Proc. Camb. Philos. Soc. 100, 65-90 (1986; Zbl 0603.55017)]. The authors show that rectified diagrams are fibrant [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)]. Then useful applications to a construction of cat\({}^ n\)-groups from homotopy coherent n-cubes are presented.
Reviewer: M.Golasiński

MSC:

55U10 Simplicial sets and complexes in algebraic topology
18D20 Enriched categories (over closed or monoidal categories)
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
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References:

[1] Bourn, D.; Cordier, J.-M., A general formulation of homotopy limits, J. Pure Appl. Algebra, 29, 129-141 (1983) · Zbl 0575.55006
[2] Bousfield, A. K.; Kan, D. M., Homotopy Limits, Completions and Localizations, (Lecture Notes in Mathematics, 304 (1972), Springer: Springer Berlin) · Zbl 0259.55004
[3] Cordier, J.-M., Sur la notion de diagramme homotopiquement cohérent, Cahiers Topologie Géom. Différentielle Catégoriques, 23, 93-112 (1982) · Zbl 0493.55009
[4] Cordier, J.-M., Sur les limites homotopiques de diagrammes homotopiquement cohérents, Compositio Math., 62, 367-388 (1987) · Zbl 0622.55008
[5] Cordier, J.-M., Extension de Kan simplicialement cohérente (1985), Amiens, Preprint
[6] Cordier, J.-M., Une représentation de \(Ho(Top^A) (1985)\), Preprint
[7] Cordier, J.-M.; Porter, T., Vogt’s theorem on categories of homotopy coherent diagrams, Math. Proc. Cambridge Philos. Soc., 100, 65-90 (1986) · Zbl 0603.55017
[8] Cordier, J.-M.; Porter, T., Maps between homotopy coherent diagrams, Topology Appl., 28, 255-275 (1988) · Zbl 0655.55008
[9] Curtis, E. B., Simplicial Homotopy Theory, Adv. in Math., 6, 107-209 (1971) · Zbl 0225.55002
[10] Edwards, D. A.; Hastings, H. M., Čech and Steenrod Homotopy Theories with Applications to Geometric Topology, (Lecture Notes in Mathematics, 542 (1976), Springer: Springer Berlin) · Zbl 0334.55001
[11] Gilbert, N. D., On the fundamental \(cat^n\)-group of an \(n\)-cube of spaces, (Algebraic Topology, Barcelona, 1986. Algebraic Topology, Barcelona, 1986, Proceedings, Lecture Notes in Mathematics, 1298 (1987), Springer: Springer Berlin) · Zbl 0642.55005
[12] K.H. Kamps and T. Porter, Abstract Homotopy and Simple Homotopy Theory, U.C.N.W. Pure Maths Preprint 86.9 (further sections in preparation).; K.H. Kamps and T. Porter, Abstract Homotopy and Simple Homotopy Theory, U.C.N.W. Pure Maths Preprint 86.9 (further sections in preparation).
[13] Kelly, G. M., The Basic Concepts of Enriched Category Theory, (London Mathematical Society Lecture Note Series, 64 (1983), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 0709.18501
[14] Loday, J.-L., Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Algebra, 24, 179-202 (1982) · Zbl 0491.55004
[15] Mac Lane, S., Categories for the Working Mathematician, (Graduate Texts in Mathematics, 5 (1971), Springer: Springer Berlin) · Zbl 0906.18001
[16] Steiner, R., Resolutions of spaces by cubes of fibrations, J. London Math. Soc., 34, 2, 169-176 (1986) · Zbl 0576.55007
[17] Vogt, R. M., Homotopy limits and colimits, Math. Z., 134, 11-52 (1973) · Zbl 0276.55006
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