Meyer, Jean-Pierre Bar and cobar constructions. I. (English) Zbl 0542.57036 J. Pure Appl. Algebra 33, 163-207 (1984). In der Arbeit wird eine allgemeine Theorie von Bar- und Cobarkonstruktionen entwickelt, der sich eine Vielzahl von Konstruktionen der Homotopietheorie, insbesondere von Homotopielimites und Homotopiecolimites, unterordnet. Der Autor orientiert sich bei der Definition seiner Begriffe an einer von A. D. Elemendorf in [Trans. Am. Math. Soc. 277, 275-284 (1983; 521.57027)] gegebenen Umformulierung eines Konzeptes einer Barkonstruktion, das auf J. P. May [Classifying spaces and fibrations, Mem. Am. Math. Soc. 155 (1975; Zbl 0321.55033)] zurückgeht. Die dort zugrundeliegende Kategorie der topologischen Räume wird hier durch eine geeignete Kategorie mit Homotopie ersetzt, was speziell Anwendungen etwa auf die Kategorie der G- Räume und der Spektren gestattet. Der Begriff Barkonstruktion wird wie üblich in zwei Stufen aus dem Begriff einer simplizialen Barkonstruction mittels geometrischer Realisierung gewonnen. Dementsprechend wird auch für den Begriff geometrische Realisierung eine allgemeine Theorie bis zu einer abstrakten Fassung des Satzes entwickelt, daß die geometrische Realisierung einer Abbildung zwischen simplizialen Räumen, die in jeder Dimension eine Homotopieäquivalenz ist, eine Homotopieäquivalenz ist. Reviewer: K.H.Kamps Cited in 3 ReviewsCited in 19 Documents MSC: 57T30 Bar and cobar constructions 55U35 Abstract and axiomatic homotopy theory in algebraic topology 18G30 Simplicial sets; simplicial objects in a category (MSC2010) 18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.) 18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.) Keywords:bar construction; cobar construction; geometric realization; homotopy limits; homotopy colimits; categories with homotopy; 521.57027 Citations:Zbl 0321.55033 PDFBibTeX XMLCite \textit{J.-P. Meyer}, J. Pure Appl. Algebra 33, 163--207 (1984; Zbl 0542.57036) Full Text: DOI References: [1] Adams, J. F., Stable Homotopy and Generalized Homology (1974), Univ. of Chicago Press: Univ. of Chicago Press Chicago · Zbl 0309.55016 [2] Adams, J. F., On the cobar construction, Colloque de Topologie Algébrique, 81-87 (1956), Louvain · Zbl 0071.16404 [3] Anderson, D. W., Fibrations and geometric realizations, Bull. Amer. Math. Soc., 84, 765-788 (1978) · Zbl 0408.55002 [4] Boardman, J. M., Stable homotopy theory (1970), The Johns Hopkins University, Mimeo. Notes · Zbl 0198.56102 [5] Bousfield, A. K.; Kan, D. M., Homotopy Limits, (Completions and Localizations. Completions and Localizations, Lecture Notes in Math., 304 (1972), Springer: Springer Berlin) · Zbl 0259.55004 [6] Brown, R., Elements of Modern Topology (1968), McGraw-Hill: McGraw-Hill New York · Zbl 0159.52201 [7] Brown, R.; Heath, P. P., Coglueing homotopy equivalences, Math. Z., 113, 313-325 (1970) · Zbl 0185.51101 [8] Dubuc, E. J., Kan extensions in enriched category theory, (Lecture Notes in Math., 145 (1970), Springer: Springer Berlin) · Zbl 0228.18002 [9] Eilenberg, S.; Kelly, G. M., Closed categories, (Proc. Conf. on Categorical Algebra. Proc. Conf. on Categorical Algebra, La Jolla, 1965 (1966), Springer: Springer New York), 421-562 · Zbl 0192.10604 [10] Eilenberg, S.; MacLane, S., On the groups \(H\)(π, \(n)\), I. Ann. of Math., 58, 2, 55-106 (1953) · Zbl 0050.39304 [11] Elmendorf, A., Systems of fixed point sets, Trans. Amer. Math. Soc., 277, 275-284 (1983) · Zbl 0521.57027 [12] Fritsch, R., Remark on the simplicial — cosimplicial tensor product, Proc. Amer. Math. Soc., 87, 200-202 (1983) · Zbl 0511.18015 [13] Furey, R.; Heath, P., Notes on Toppair, \(Top_∗\) and regular fibrations, coglueing and duality, Canad. Math. Bull., 24, 317-329 (1981) · Zbl 0486.55012 [14] Gray, J. W., The categorical comprehension scheme, (Lecture Notes in Math., 99 (1969), Springer: Springer Berlin), 242-312 · Zbl 0211.03403 [15] Gugenheim, V. K.A. M.; May, J. P., On the theory and applications of differential torsion products, Mem. Amer. Math. Soc., 142 (1974) · Zbl 0292.55019 [16] Gugenheim, V. K.A. M.; Munkholm, H. J., On the extended functoriality of Tor and Cotor, J. Pure Appl. Algebra, 4, 9-29 (1974) · Zbl 0358.18015 [17] Hardie, K. A., Quasifibration and adjunction, Pacific J. Math., 35, 389-397 (1970) · Zbl 0207.53402 [18] Hilton, P. J.; Stammbach, U., A Course in Homological Algebra (1971), Springer: Springer New York · Zbl 0238.18006 [19] Husemoller, D.; Moore, J. C.; Stasheff, J., Differential homological algebra and homogeneous spaces, J. Pure Appl. Algebra, 5, 113-185 (1974) · Zbl 0364.18008 [20] Kamps, K. H., Fasterungen und Cofaserungen in Kategorien mit Homotopiesystem, Dissertation (1968), Saarbrücken [21] Kamps, K. H., Kan-Bedingungen und abstrakte Homotopie-theorie, Math. Z., 124, 215-236 (1972) · Zbl 0223.55020 [22] Kelly, G. M., Basic Concepts of Enriched Category Theory (1982), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0478.18005 [23] MacLane, S., Categories for the Working Mathematician (1971), Springer: Springer New York [24] May, J. P., The geometry of iterated loop spaces, (Lecture Notes in Math., 271 (1972), Springer: Springer Berlin) · Zbl 0244.55009 [25] May, J. P., Classifying spaces and fibrations, Mem. Amer. Math. Soc., 155 (1975) · Zbl 0321.55033 [26] May, J. P., \(E_∞\) spaces, group completions, and permutative categories, (Segal, G., New Developments in Topology (1974), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 61-93 · Zbl 0281.55003 [27] Meyer, J.-P., Acyclic models for multicomplexes, Duke Math. J., 67-85 (1978) · Zbl 0374.55020 [28] Mielke, M. V., The interval in algebraic topology, Illinois J. Math., 25, 51-62 (1981) · Zbl 0425.55010 [29] Mielke, M. V., Exact intervals, Illinois J. Math., 25, 593-598 (1981) · Zbl 0444.55017 [30] Milnor, J., The geometric realization of a semi-simplicial complex, Ann. of Math., 65, 2, 357-362 (1957) · Zbl 0078.36602 [31] Pareigis, B., Categories and Functors (1970), Academic Press: Academic Press New York · Zbl 0211.32402 [32] Puppe, V., A remark on “homotopy fibrations”, Manuscripta Math., 12, 113-120 (1974) · Zbl 0277.55015 [33] Quillen, D. G., Homotopical Algebra, (lecture Notes in Math., 43 (1967), Springer: Springer Berlin) · Zbl 0168.20903 [34] Quillen, D. G., Higher algebraic \(K\)-theory \(I\), (Lecture Notes in Math., 341 (1973), Springer: Springer Berlin), 85-147 · Zbl 0292.18004 [35] Rector, D. L., Steenrod operations in the Eilenberg-Moore spectral sequence, Comm. Math. Helv., 45, 540-552 (1970) · Zbl 0209.27501 [36] Ruiz, C.; Ruiz, R., Characterization of the set-theoretical geometric realization in the non-euclidean case, Proc. Amer. Math. Soc., 81, 321-324 (1981) · Zbl 0472.55016 [37] Ruiz, R., Change of models in Top and \(Δ∗\(S\), (Doctoral Thesis (1975), Temple University: Temple University Philadelphia) [38] Segal, G., Classifying spaces and spectral sequences, Pub. Math. I.H.E.S., 34, 105-112 (1968) · Zbl 0199.26404 [39] Segal, G., Categories and cohomology theories, Topology, 13, 293-312 (1974) · Zbl 0284.55016 [40] Stasheff, J. D., Parallel transport in fibrespaces, Bol. Soc. Mat. Mex., 11, 68-84 (1968) [41] Stasheff, J. D., Associated fibre spaces, Mich. Math. J., 15, 457-470 (1968) · Zbl 0177.25902 [42] Strøm, A., Note on confibrations II, Math. Scand., 22, 130-142 (1968) · Zbl 0181.26504 [43] R.W. Thomason, Algebraic \(k\); R.W. Thomason, Algebraic \(k\) · Zbl 0596.14012 [44] Thomason, R. W., Homotopy colimits in the category of small categories, Math. Proc. Camb. Phil. Soc., 85, 91-109 (1979) · Zbl 0392.18001 [45] R.W. Thomason, First quadrant spectral sequences in algebraic \(K\); R.W. Thomason, First quadrant spectral sequences in algebraic \(K\) · Zbl 0502.55012 [46] J. Tornehave, On BSG and the symmetric groups, Preprint.; J. Tornehave, On BSG and the symmetric groups, Preprint. [47] Vogt, R., Homotopy limits and colimits, Math. Z., 134, 11-52 (1973) · Zbl 0276.55006 [48] Vogt, R., Boardman’s stable homotopy category, (Lecture Note Series, 21 (1970), Aarhus University) · Zbl 0224.55014 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.