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Homotopy coherent category theory and \(A_\infty\)-structures in monoidal categories. (English) Zbl 0892.18003

As was pointed out by R. M. Vogt [Math. Z. 134, 11-52 (1973; Zbl 0276.55006)], the natural notion of homotopy coherent map between homotopy coherent diagrams, is a homotopy coherent diagram of type a cylinder on the original indexing category. Even with a rich structure on the receiving category as is found with \(\mathcal T op\) or \(\mathcal K an\), composition of such maps requires some choices and is even then not associative. It is however, associative up to (specified) homotopy and so the “caterory” of homotopy coherent diagrams of given type and these maps between them is not a category. Usually the receiving category is assumed to be, for instance, simplicially enriched and then the result ends up being simplicially and on taking \(\pi_0\) of each “hom” it yields a category [cf. J.-M. Cordier and T. Porter, Math. Proc. Camb. Philos. Soc. 100, 65-90 (1986; Zbl 0603.55017) or M. A. Batanin, Cah. Topologie Géom. Différ. Catégoriques 34, No. 4, 279-304 (1993; Zbl 0793.18005)].
Such a passage to the quotient category hinders the development of adequate analogues of categorical methods (here motivated by homotopy coherent categorical versions of strong shape theory, but also of importance when considering homotopy coherent algebras over operads, monads, etc.). To handle the homotopy coherent analogue of the category of diagrams, the author develops the notion of simplicial \(A_\infty\)-graph. This requires the setting-up of considerable technical machinery on \(A_\infty\)-structures in general. One of the main results then is that the ‘category’ of homotopy coherent diagrams of type a given simplically enriched category in a locally Kan simplicial category is itself a locally Kan simplicial \(A_\infty\)-graph. This impressive technical result includes a large number of special cases that should be of great significance in a wide variety if applications.
Reviewer: T.Porter (Bangor)

MSC:

18D20 Enriched categories (over closed or monoidal categories)
55U35 Abstract and axiomatic homotopy theory in algebraic topology
18D35 Structured objects in a category (MSC2010)
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
18A10 Graphs, diagram schemes, precategories
18G55 Nonabelian homotopical algebra (MSC2010)
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References:

[1] Artin, M.; Mazur, B., On the van Kampen theorem, Topology, 5, 179-189 (1966) · Zbl 0138.18301
[2] Batanin, M. A., Coherent categories with respect to monads and coherent prohomotopy theory, Cahiers Topologie Géom. Cahiers Topologie Géom, Différentielle Catégoriques, XXXIV-4, 279-304 (1993) · Zbl 0793.18005
[3] M.A. Batanin, Categorical strong shape theory, Cahiers Topologie Géom. Différentielle Catégoriques, to appear.; M.A. Batanin, Categorical strong shape theory, Cahiers Topologie Géom. Différentielle Catégoriques, to appear.
[4] Benabou, J., Les distributeurs, (Rapport No.33, Inst. Math. Pures and Appl. (1973), Univ. Louvain-la-Neuve: Univ. Louvain-la-Neuve Chichester) · Zbl 0162.32602
[5] Boardman, J. M.; Vogt, R. M., Homotopy Invariant Algebraic Structures on Topological Spaces, (Lecture Notes in Mathematics, Vol. 347 (1973), Springer) · Zbl 0285.55012
[6] Boum, D.; Cordier, J.-M., Distributeurs et théorie de la forme, Cahiers Topologie Geom.. Cahiers Topologie Geom., Différentielle Catégoriques, 21, 161-189 (1980) · Zbl 0439.55014
[7] Bousfield, A. K.; Kan, D. M., Homotopy Limits, Completions and Localizations, (Lecture Notes in Mathematics, Vol. 304 (1972), Springer: Springer Berlin) · Zbl 0259.55004
[8] Cordier, J. M., Extension de Kan simplicialement coherent (1985), Prepublication, Amiens
[9] Cordier, J. M., Comparaison de deux catégories d’homotopie de morphismes cohérents, Cahiers Topologie Géom. Cahiers Topologie Géom, Différentielle Catégoriques, 30-2, 257-275 (1989) · Zbl 0679.55006
[10] Cordier, J. M.; Porter, T., Vogt’s theorem on categories of homotopy coherent diagrams, (Math. Proc. Cambridge Phil. Soc., 100 (1986)), 65-90 · Zbl 0603.55017
[11] Cordier, J.-M.; Porter, T., Shape Theory: Categorical Methods of Approximation (1989), Ellis Horwood: Ellis Horwood Berlin · Zbl 0663.18001
[12] J.M. Cordier and T. Porter, Homotopy coherent category theory, Trans. Amer. Math. Soc. to appear.; J.M. Cordier and T. Porter, Homotopy coherent category theory, Trans. Amer. Math. Soc. to appear. · Zbl 0865.18006
[13] Eilenberg, S.; Kelly, G. M., Closed categories, (Proc. Conf. on Categorical Algebra. Proc. Conf. on Categorical Algebra, La Jolla, 1965 (1966), Springer: Springer Chichester), 421-562 · Zbl 0192.10604
[14] Ginzburg, V.; Kapranov, M. M., Kozsul duality for operads, Duke Math. J., 76, 203-273 (1994)
[15] Günther, B., The use of semisimplicial complexes in strong shape theory, Glasnik Mat., 27, 101-144 (1992) · Zbl 0781.55006
[16] Kelly, G. M., The Basic Concepts of Enriched Category Theory, (London Mathematical Society Lecture Notes Series, Vol. 64 (1983), Cambridge University Press: Cambridge University Press Berlin) · Zbl 0709.18501
[17] Lada, T. J., Strong homotopy algebras over monads, (Lecture Notes in Mathematics, Vol. 533 (1976), Springer: Springer Cambridge), 399-479
[18] Lisica, Y. T.; Mardesic, S., Coherent prohomotopy and strong shape theory, Glasnik Mat., 19, 335-399 (1984) · Zbl 0553.55009
[19] MacLane, S., Categories for the Working Mathematician, (Graduate Texts in Mathematics, Vol. 5 (1971), Springer: Springer Berlin) · Zbl 0705.18001
[20] May, J. P., The Geometry of Iterated Loop Spaces, (Lecture Notes in Mathematics, Vol. 271 (1972), Springer: Springer Berlin) · Zbl 0244.55009
[21] Meyer, J.-P., Bar and cobar constructions, J. Pure Appl. Algebra, 33, 163-207 (1984) · Zbl 0542.57036
[22] Quillen, D. G., Homotopical Algebra, (Lecture Notes in Mathematics, Vol. 43 (1967), Springer: Springer Berlin) · Zbl 0168.20903
[23] Schwänzl, R.; Vogt, R., Homotopy homomorphisms and the Hammock localization, Boletin Soc. Mat. Mexicana, 37, 431-449 (1992) · Zbl 0853.55010
[24] Smirnov, V. A., Homotopy theory of coalgebras, Math. USSR Izv., 27, 575-592 (1986) · Zbl 0612.55012
[25] Stasheff, J. D., Homotopy associativity of H-spaces, I, Trans. Amer. Math. Soc., 108, 275-292 (1963) · Zbl 0114.39402
[26] Steenrod, N. E., A convenient category of topological spaces, Michigan Math. J., 14, 133-152 (1967) · Zbl 0145.43002
[27] Thomason, R. W., Uniqueness of delooping machines, Duke Math. J., 46, 217-252 (1979) · Zbl 0413.55012
[28] Vogt, R. M., A note on homotopy equivalences, (Proc. Amer. Math. Soc., 32 (1972)), 627-629 · Zbl 0241.55009
[29] Vogt, R. M., Homotopy limits and colimits, Math. Z., 134, 11-52 (1973) · Zbl 0276.55006
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