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The two out of three property in ind-categories and a convenient model category of spaces. (English) Zbl 1376.55017

A weak fibration category is a category equipped with two subcategories of weak equivalences and fibrations, satisfying a set of axioms. This notion, introduced in [I. Barnea and T. M. Schlank, Adv. Math. 291, 784-858 (2016; Zbl 1333.18023)] is closely related to K. S. Brown’s notion of a “category of fibrant objects” [Trans. Am. Math. Soc. 186, 419-458 (1973; Zbl 0245.55007)] and H. J. Baues’ notion of a “fibration category” [Algebraic homotopy. Cambridge etc.: Cambridge University Press (1989; Zbl 0688.55001)].
Let Pro\((\mathcal{C})\) be the pro-category of inverse limits in \(\mathcal{C}\), whose objects are diagrams \(I\longrightarrow \mathcal{C}\), with \(I\) a cofiltered category.
Given a weak fibration category \(\mathcal{C}\), a class of weak equivalences in \(\mathrm{Pro}(\mathcal{C})\) can be defined. Weak fibration categories \(\mathcal{C}\) for which this class in Pro\((\mathcal{C})\) satisfies the two out of three property are called pro-admisible. A pro-admissible weak fibration category \(\mathcal{C}\) induces in a natural way a model structure on \(\mathrm{ Pro}(\mathcal{C})\), provided \(\mathcal{C}\) has colimits and satisfies a technical condition called homotopically small [Barnea and Schlank, loc. cit.]. As a consequence ([I. Barnea and T. M. Schlank, Homology Homotopy Appl. 17, No. 2, 235-260 (2015; Zbl 1334.18006)]), any small pro-admissible weak fibration category \(\mathcal{C}\) induces a model structure on \(\mathrm{Pro}(\mathcal{C})\).
Dually, the author defines the notion of a weak cofibration category, and deduces that a small ind-admissible (the class of weak equivalences satisfies the two out of three property) weak cofibration category induces a model structure on its ind-category (which is the dual notion of a pro-category). The main result in this paper gives sufficient intrinsic conditions on a small simplicial weak cofibration category from which one can deduce its ind-admissibility.
These results are applied to the category of compact metrizable spaces, CM, showing that this category is ind-admissible, and thus the category Ind(CM) has an induced model structure. This is also true if CM is replaced by any (essentially) small collection of compact Hausdorff spaces, that is closed under infinite limits and colimits and contains all realizations of finite simplicial sets.

MSC:

55U35 Abstract and axiomatic homotopy theory in algebraic topology
55P05 Homotopy extension properties, cofibrations in algebraic topology
55U10 Simplicial sets and complexes in algebraic topology
54B30 Categorical methods in general topology
18G55 Nonabelian homotopical algebra (MSC2010)
18B30 Categories of topological spaces and continuous mappings (MSC2010)
18C35 Accessible and locally presentable categories
18D20 Enriched categories (over closed or monoidal categories)
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References:

[1] Adamek J., Rosicky J. Locally Presentable and Accessible Categories, Cambridge Univer sity Press, Cambridge, 1994. · Zbl 0795.18007
[2] Anderson D. W. Fibrations and geometric realizations, Bulletin of the American Mathe matical Society 84.5, 1978, p. 765-788. · Zbl 0408.55002
[3] Artin M., Mazur B. ´Etale Homotopy, Lecture Notes in Mathematics, Vol. 100, Springer Verlag, Berlin, 1969. · Zbl 0182.26001
[4] Barnea I., Harpaz Y., Horel G. Pro-categories in homotopy theory, Algebraic and Geo metric Topology 17.1, 2017, p. 567-643. · Zbl 1360.18025
[5] Barnea I., Joachim M., Mahanta S. Model structure on projective systems of C∗-algebras and bivariant homology theories, New York Journal of Mathematics 23, 2017, p. 383- 439. · Zbl 1380.46053
[6] Barnea I., Schlank T. M. A new model for pro-categories, Journal of Pure and Applied Algebra 219.4, 2015, p. 1175-1210. · Zbl 1304.18035
[7] Barnea I., Schlank T. M. Model structures on ind-categories and the accessibility rank of weak equivalences, Homology, Homotopy and Applications 17.2, 2015, p. 235-260. · Zbl 1334.18006
[8] 650 ILAN BARNEA
[9] Barnea I., Schlank T. M. A projective model structure on pro simplicial sheaves, and the relative ´etale homotopy type, Advances in Mathematics 291, 2016, p. 784-858. · Zbl 1333.18023
[10] Barnea I., Schlank T. M. From weak cofibration categories to model categories, preprint available at http://arxiv.org/abs/1610.08068, 2017.
[11] Barwick C., Kan D. M. Relative categories: Another model for the homotopy theory of homotopy theories, Indag. Math. 23.1-2, 2012, p. 42-68. · Zbl 1245.18006
[12] Baues H. J. Algebraic Homotopy, Cambridge University Press, Cambridge, 1988. · Zbl 0673.55007
[13] Brown K. S. Abstract homotopy theory and generalized sheaf cohomology, Trans. Amer. Math. Soc. 186, 1973, p. 419-458. · Zbl 0245.55007
[14] Cisinski D. C. Cat´egories d´erivables, Bulletin de la soci´et´e math´ematique de France 138.3, 2010, p. 317-393. · Zbl 1203.18013
[15] Dwyer W., Kan D. Simplicial localizations of categories, J. Pure Appl. Algebra 17, 1980, p. 267-284. · Zbl 0485.18012
[16] Edwards D. A., Hastings H. M. ˇCech and Steenrod Homotopy Theories with Applications to Geometric Topology, Lecture Notes in Mathematics, Vol. 542, Springer-Verlag, Berlin, 1976. · Zbl 0334.55001
[17] Fajstrup L., Rosicky J. A convenient category for directed homotopy, Theory and Appli cations of Categories 21.1, 2008, p. 7-20. · Zbl 1157.18003
[18] Goerss P. G., Jardine J. F. Simplicial Homotopy Theory, Progress in Mathematics, Vol. 174, Birkh¨auser, Basel, 1999. · Zbl 0949.55001
[19] Hovey M. Model Categories, Mathematical Survays and Monographs, Vol. 63, Amer. Math. Soc., Providence, RI, 1991. · Zbl 0909.55001
[20] Isaksen D. C. Strict model structures for pro-categories, Categorical Factorization Techniques in Algebraic Topology, 179-198, Progress in Mathematics, Vol. 215, Birkh¨auser, Basel, 2004. · Zbl 1049.18008
[21] Lurie J. Higher Topos Theory, Annals of Mathematics Studies, Vol. 170, Princeton Uni versity Press, Princeton, NJ, 2009. · Zbl 1175.18001
[22] Radulescu-Banu A. Cofibrations in homotopy theory, preprint available at http://arxiv. org/abs/math/0610009, 2009.
[23] Strøm A. Note on cofibrations II, Math. Scand. 22, 1968, p. 130-142. · Zbl 0181.26504
[24] Artin M., Grothendieck A., Verdier J. L. Th´eorie des Topos et Cohomologie ´Etale des Sch´emas-Tome 1, Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin, 1972. THE TWO OUT OF THREE PROPERTY 651 · Zbl 0234.00007
[25] Szumi lo K. Two Models for the Homotopy Theory of Cocomplete Homotopy Theories, preprint available at http://arxiv.org/abs/1411.0303, 2014.
[26] Whitehead G. W. Elements of Homotopy Theory, Graduate Texts in Mathematics, Vol. 61, Springer-Verlag, New York, 1978. · Zbl 0406.55001
[27] Department of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem,
[28] 9190401, Israel.
[29] Email: ilanbarnea770@gmail.com
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