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  • Cited by 7
Publisher:
Cambridge University Press
Online publication date:
March 2018
Print publication year:
2018
Online ISBN:
9781316181874

Book description

The notion of an (∞,1)-category has become widely used in homotopy theory, category theory, and in a number of applications. There are many different approaches to this structure, all of them equivalent, and each with its corresponding homotopy theory. This book provides a relatively self-contained source of the definitions of the different models, the model structure (homotopy theory) of each, and the equivalences between the models. While most of the current literature focusses on how to extend category theory in this context, and centers in particular on the quasi-category model, this book offers a balanced treatment of the appropriate model structures for simplicial categories, Segal categories, complete Segal spaces, quasi-categories, and relative categories, all from a homotopy-theoretic perspective. Introductory chapters provide background in both homotopy and category theory and contain many references to the literature, thus making the book accessible to graduates and to researchers in related areas.

Reviews

'The writing is accessible, even for students, and the ideas are clear. The author gives references for every claim and definition, with the added advantage that some technical [lengthy] points can be left out to avoid burying the ideas.'

Najib Idrissi Source: zbMATH

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Contents

  • Introduction
    pp 1-3
References
[1] Ara, D., Higher quasi-categories vs higher Rezk spaces, J. K-Theory 14 (2014 CrossRef | Google Scholar), no. 3, 701–749.
[2] Ara, D., On the homotopy theory of Grothendieck ∞-groupoids, J. Pure Appl. Algebra 217 (2013 CrossRef | Google Scholar), 1237–1278.
[3] Ara, D., and Métayer, F., The Brown–Golasinski model structure on strict ∞-groupoids revisited, Homology, Homotopy Appl. 13 (2011 CrossRef | Google Scholar), no. 1, 121–142.
[4] Awodey, S., Category Theory, Oxford Logic Guides 49, Clarendon Press, 2006 Google Scholar.
[5] Ayala, D., Francis, J., and Rozenblyum, N., Factorization homology I: higher categories (2015 Google Scholar), available at arXiv:1504.04007.
[6] Ayala, D., Francis, J., and Rozenblyum, N., A stratified homotopy hypothesis (2015 Google Scholar), available at arXiv:1502.01713.
[7] Badzioch, B., Algebraic theories in homotopy theory, Ann. of Math. (2) 155 (2002 CrossRef | Google Scholar), no. 3, 895–913.
[8] Baez, J.C., and Dolan, J., Higher-dimensional algebra. III, n-categories and the algebra of opetopes, Adv. Math. 135 (1998 Google Scholar), no. 2, 145–206.
[9] Barwick, C., From operator categories to topological operads (2013 Google Scholar), available at arXiv:1302.5756.
[10] Barwick, C., On left and right model categories and left and right Bousfield localizations, Homology, Homotopy Appl. 12 (2010 CrossRef | Google Scholar), no. 2, 245–320.
[11] Barwick, C., and Kan, D.M., Relative categories: another model for the homotopy theory of homotopy theories, Indag. Math. (N.S.) 23 (2012 CrossRef | Google Scholar), no. 1–2, 42–68.
[12] Barwick, C. and Kan, D.M., n-relative categories: a model for the homotopy theory of n-fold homotopy theories, Homology, Homotopy Appl. 15 (2013 Google Scholar), no. 2, 281–300.
[13] Barwick, C., and Schommer-Pries, C., On the unicity of the homotopy theory of higher categories (2011 Google Scholar), available at arXiv:1112.0040.
[14] Basic, M., and Nikolaus, T., Dendroidal sets as models for connective spectra, J. K-Theory 14 (2014 CrossRef | Google Scholar), no. 3, 387–421.
[15] Batanin, M.A., Monoidal globular categories as a natural environment for the theory of weak n-categories, Adv. Math. 136 (1998 CrossRef | Google Scholar), 39–103.
[16] Baues, H.-J., and Blanc, D., Comparing cohomology obstructions, J. Pure Appl. Algebra 215 (2011 CrossRef | Google Scholar), 1420–1439.
[17] Baues, H.-J., and Blanc, D., Higher order derived functors and the Adams spectral sequence, J. Pure Appl. Algebra 219 (2015 CrossRef | Google Scholar), no. 2, 199–239.
[18] Baues, H.-J., and Blanc, D., Stems and spectral sequences, Algebr. Geom. Topol. 10 (2010 CrossRef | Google Scholar), 2061–2078.
[19] Baues, H.-J., and Jibladze, M., Secondary derived functors and the Adams spectral sequence, Topology 45 (2006 CrossRef | Google Scholar), 295–324.
[20] Baues, H.-J., and Wirsching, G., Cohomology of small categories, J. Pure Appl. Algebra 38 (1985 CrossRef | Google Scholar), 187–211.
[21] Beke, T., Sheafifiable homotopy model categories, Math. Proc. Cambridge Philos. Soc. 129 (2000 CrossRef | Google Scholar), 447–475.
[22] Bénabou, J., Introduction to bicategories, Reports of the Midwest Category Seminar, Springer, 1967 CrossRef | Google Scholar, pp. 1–77.
[23] Berger, C., Iterated wreath product of the simplex category and iterated loop spaces, Adv. Math. 213 (2007 CrossRef | Google Scholar), 230–270.
[24] Bergner, J.E., A characterization of fibrant Segal categories, Proc. Amer. Math. Soc. 135 (2007 CrossRef | Google Scholar), 4031–4037.
[25] Bergner, J.E., Complete Segal spaces arising from simplicial categories, Trans. Amer. Math. Soc. 361 (2009 Google Scholar), 525–546.
[26] Bergner, J.E., Equivalence of models for equivariant (∞, 1)-categories, Glasg. Math. J. 59 (2017 CrossRef | Google Scholar), no. 1, 237–253.
[27] Bergner, J.E., A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007 CrossRef | Google Scholar), 2043–2058.
[28] Bergner, J.E., Rigidification of algebras over multi-sorted theories, Algebr. Geom. Topol. 6 (2006 CrossRef | Google Scholar), 1925–1955.
[29] Bergner, J.E., Simplicial monoids and Segal categories, Contemp. Math. 431 (2007 Google Scholar), 59–83.
[30] Bergner, J.E., Three models for the homotopy theory of homotopy theories, Topology 46 (2007 CrossRef | Google Scholar), 397–436.
[31] Bergner, J.E., Workshop on the homotopy theory of homotopy theories (2011 Google Scholar), available at arXiv:1108.2001.
[32] Bergner, J.E., and Rezk, C., Comparison of models for (∞, n)-categories, I, Geom. Topol. 17 (2013 Google Scholar), 2163–2202.
[33] Bergner, J.E., and Rezk, C., Comparison of models for (∞, n)-categories, II (2014 Google Scholar), available at arXiv:1406.4182.
[34] Blanc, D., and Paoli, S., Segal-type algebraic models of n-types, Algebr. Geom. Topol. 14 (2014 Google Scholar), no. 6, 3419–3491.
[35] Blanc, D., and Paoli, S., Two-track categories, J. K-Theory 8 (2011 CrossRef | Google Scholar), no. 1, 59–106.
[36] Boardman, J.M., and Vogt, R.M., Homotopy Invariant Algebraic Structures on Topological Spaces, Lecture Notes in Mathematics 347, Springer, 1973 CrossRef | Google Scholar.
[37] Bousfield, A.K., and Kan, D.M., Homotopy Limits, Completions, and Localizations, Lecture Notes in Mathematics 304, Springer, 1972 CrossRef | Google Scholar.
[38] Brown, R., Higgins, P.J., and Sivera, R., Nonabelian Algebraic Topology, with contributions by C.D., Wensley and S.V., Soloviev, EMS Tracts in Mathematics 15, European Mathematical Society, Zürich, 2011 Google Scholar.
[39] Bullejos, M., Cegarra, A.M., and Duskin, J., On catn-groups and homotopy types, J. Pure Appl. Algebra 86 (1993 CrossRef | Google Scholar), no. 2, 135–154.
[40] Cabello, J.G., and Garzón, A.R., Closed model structures for algebraic models of n-types, J. Pure Appl. Algebra 103 (1995 CrossRef | Google Scholar), no. 3, 287–302.
[41] Cheng, E., Weak n-categories: comparing opetopic foundations, J. Pure Appl. Algebra 186 (2004 Google Scholar), no. 3, 219–231.
[42] Cheng, E., Weak n-categories: opetopic and multitopic foundations, J. Pure Appl. Algebra 186 (2004 Google Scholar), no. 2, 109–137.
[43] Chu, H., Haugseng, R., and Heuts, G., Two models for the homotopy theory of ∞-operads (2016 Google Scholar), available at arXiv:1606.03826.
[44] Cisinski, D.-C., Batanin higher groupoids and homotopy types, Categories in Algebra, Geometry and Mathematical Physics, Contemporary Mathematics, 431, American Mathematical Society, 2007 Google Scholar, pp. 171–186.
[45] Cisinski, D.-C., and Moerdijk, I., Dendroidal Segal spaces and ∞-operads, J. Topol. 6 (2013 Google Scholar), no. 3, 675–704.
[46] Cisinski, D.-C., and Moerdijk, I., Dendroidal sets and simplicial operads, J. Topol. 6 (2013 Google Scholar), no. 3, 705–756.
[47] Cisinski, D.-C., and Moerdijk, I., Dendroidal sets as models for homotopy operads, J. Topol. 4 (2011 CrossRef | Google Scholar), no. 2, 257–299.
[48] Cordier, J.M., and Porter, T., Vogt's theorem on categories of homotopy coherent diagrams, Math. Proc. Cambridge Philos. Soc. 100 (1986 CrossRef | Google Scholar), 65–90.
[49] Dugger, D., Combinatorial model categories have presentations, Adv. Math. 164 (2001 CrossRef | Google Scholar), no. 1, 177–201.
[50] Dugger, D. Google Scholar, A Primer on Homotopy Colimits, available at pages.uoregon.edu/ ddugger/hocolim.pdf.
[51] Dugger, D., and Spivak, D.I., Mapping spaces in quasicategories, Algebr. Geom. Topol. 11 (2011 CrossRef | Google Scholar), 263–325.
[52] Dugger, D., and Spivak, D.I., Rigidification of quasicategories, Algebr. Geom. Topol. 11 (2011 CrossRef | Google Scholar), 225–261.
[53] Dwyer, W.G., and Kan, D.M., Calculating simplicial localizations, J. Pure Appl. Algebra 18 (1980 CrossRef | Google Scholar), 17–35.
[54] Dwyer, W.G., and Kan, D.M., A classification theorem for diagrams of simplicial sets, Topology 23 (1984 CrossRef | Google Scholar), 139–155.
[55] Dwyer, W.G., and Kan, D.M., Equivalences between homotopy theories of diagrams, Algebraic Topology and Algebraic K-Theory, Annals of Mathematics Studies 113, Princeton University Press, 1987 Google Scholar, pp. 180–205.
[56] Dwyer, W.G., and Kan, D.M., Function complexes in homotopical algebra, Topology 19 (1980 CrossRef | Google Scholar), 427–440.
[57] Dwyer, W.G., and Kan, D.M., Simplicial localizations of categories, J. Pure Appl. Algebra 17 (1980 CrossRef | Google Scholar), no. 3, 267–284.
[58] Dwyer, W.G., Kan, D.M., and Smith, J.H., Homotopy commutative diagrams and their realizations, J. Pure Appl. Algebra 57 (1989 CrossRef | Google Scholar), 5–24.
[59] Dwyer, W.G., and Spalinski, J., Homotopy theories and model categories, Handbook of Algebraic Topology, Elsevier, 1995 Google Scholar.
[60] Friedman, G., An elementary illustrated introduction to simplicial sets, Rocky Mountain J. Math. 42 (2012 CrossRef | Google Scholar), no. 2, 353–423.
[61] Gabriel, P., and Zisman, M., Calculus of Fractions and Homotopy Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 35, Springer, 1967 CrossRef | Google Scholar.
[62] Goerss, P.G., and Jardine, J.F., Simplicial Homotopy Theory, Progress in Mathematics, 174, Birkhäuser, 1999 CrossRef | Google Scholar.
[63] Gordon, R., Power, A.J., and Street, R., Coherence for tricategories, Mem. Amer. Math. Soc. 117 (1995 Google Scholar), no. 558.
[64] Gurski, M.N., An algebraic theory of tricategories, Ph.D. Thesis, University of Chicago, 2006 Google Scholar.
[65] Gurski, N., Nerves of bicategories as stratified simplicial sets, J. Pure Appl. Algebra 213 (2009 CrossRef | Google Scholar), no. 6, 927–946.
[66] Hatcher, A., Algebraic Topology, Cambridge University Press, 2002 Google Scholar.
[67] Hermida, C., Makkai, M., and Power, J., On weak higher dimensional categories. I, J. Pure Appl. Algebra 154 (2000 CrossRef | Google Scholar), no. 1–3, 221–246.
[68] Heuts, G., Hinich, V., and Moerdijk, I., On the equivalence between Lurie's model and the dendroidal model for infinity-operads, Adv. Math. 302 (2016 CrossRef | Google Scholar), 869–1043.
[69] Hirschhorn, P. S., Model Categories and Their Localizations, Mathematical Surveys and Monographs 99, American Mathematical Society, 2003 Google Scholar.
[70] Hirschowitz, A., and Simpson, C., Descente pour les n-champs (1998 Google Scholar), available at arXiv:9807049.
[71] Hovey, M., Model Categories, Mathematical Surveys and Monographs 63, American Mathematical Society, 1999 Google Scholar.
[72] Ilias, A., Model structure on the category of small topological categories, J. Homotopy Relat. Struct. 10 (2015 CrossRef | Google Scholar), no. 1, 63–70.
[73] Joyal, A., Quasi-categories and Kan complexes, J. Pure Appl. Algebra 175 (2002 CrossRef | Google Scholar), 207–222.
[74] Joyal, A., and Tierney, M., Quasi-categories vs Segal spaces, Contemp. Math. 431 (2007 Google Scholar), 277–326.
[75] Kachour, C., Operadic definition of non-strict cells, Cah. Topol. Géom. Différ. Catég. 52 (2011 Google Scholar), no. 4, 269–316.
[76] Kazhdan, D., and Varshavskiĭ, Ya., The Yoneda lemma for complete Segal spaces, Funktsional. Anal. i Prilozhen. 48 (2014), no. 2, 3–38; translation in Funct. Anal. Appl. 48 (2014 CrossRef | Google Scholar), no. 2, 81–106.
[77] Kock, J., Joyal, A., Batanin, M., and Mascari, J.-F., Polynomial functors and opetopes, Adv. Math. 224 (2010 CrossRef | Google Scholar), 2690–2737.
[78] Lack, S., A Quillen model structure for 2-categories, K-Theory 26 (2002 CrossRef | Google Scholar), 171–205.
[79] Lack, S., A Quillen model structure for bicategories, K-Theory 33 (2004 CrossRef | Google Scholar), no. 3, 185–197.
[80] Lack, S., A Quillen model structure for Gray-categories, J. K-Theory 8 (2011 CrossRef | Google Scholar), 183–221.
[81] Lafont, Y., Métayer, F., and Worytkiewicz, K., A folk model structure on omegacat, Adv. Math. 224 (2010 CrossRef | Google Scholar), 1183–1231.
[82] Lawvere, F.W., Functorial semantics of algebraic theories, Proc. Natl. Acad. Sci. USA 50 (1963 CrossRef | Google Scholar | PubMed), 869–872.
[83] Leinster, T., Higher Operads, Higher Categories, London Mathematical Society Lecture Note Series 298, Cambridge University Press, 2004 CrossRef | Google Scholar.
[84] Leinster, T., Operads in higher-dimensional category theory, Theory Appl. Categ. 12 (2004 Google Scholar), no. 3, 73–194.
[85] Leinster, T., A survey of definitions of n-category, Theory Appl. Categ. 10 (2002 Google Scholar), 1–70.
[86]Loday, J.-L., Spaces with finitely many nontrivial homotopy groups, J. Pure Appl. Algebra 24 (1982 CrossRef | Google Scholar), no. 2, 179–202.
[87] Lurie, J. Google Scholar, Higher Algebra, available at www.math.harvard.edu/∼1lurie/papers/ HA.pdf.
[88] Lurie, J., Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press, 2009 Google Scholar.
[89] Lurie, J., On the classification of topological field theories, Current Developments in Mathematics, 2008, International Press, 2009 Google Scholar, pp. 129–280.
[90] Mac Lane, S., Categories for theWorking Mathematician, Second Edition, Graduate Texts in Mathematics 5, Springer, 1997 Google Scholar.
[91] May, J.P., Simplicial Objects in Algebraic Topology, University of Chicago Press, 1967 Google Scholar.
[92] Moerdijk, I., Lectures on dendroidal sets, Simplicial Methods for Operads and Algebraic Geometry, Advanced Courses in Mathematics – CRM, Barcelona, Birkhäuser, 2010 CrossRef | Google Scholar, pp. 1–118.
[93] Moerdijk, I., and Weiss, I., Dendroidal sets, Algebr. Geom. Topol. 7 (2007 CrossRef | Google Scholar), 1441–1470.
[94] Moerdijk, I., andWeiss, I., On inner Kan complexes in the category of dendroidal sets, Adv. Math. 221 (2009 CrossRef | Google Scholar), no. 2, 343–389.
[95] Morrison, S., andWalker, K., Higher categories, colimits, and the blob complex, Proc. Natl. Acad. Sci. USA 108 (2011 CrossRef | Google Scholar | PubMed), no. 20, 8139–8145.
[96] Paoli, S., Weakly globular cat n-groups and Tamsamani's model, Adv. Math. 222 (2009 CrossRef | Google Scholar), no. 2, 621–727.
[97] Pellissier, R., Catégories enrichies faibles (2003 Google Scholar), available at arXiv:0308246.
[98] Porter, T., n-types of simplicial groups and crossed n-cubes, Topology 32 (1993 CrossRef | Google Scholar), no. 1, 5–24.
[99] Quillen, D., Higher algebraic K-theory. I, Algebraic K-Theory, I: Higher KTheories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Mathematics 341, Springer, 1973 Google Scholar.
[100] Quillen, D., Homotopical Algebra, Lecture Notes in Mathematics 43, Springer, 1967 CrossRef | Google Scholar.
[101] Reedy, C.L. Google Scholar, Homotopy Theory of Model Categories, available at www-math .mit.edu/˜psh.
[102] Rezk, C., A cartesian presentation of weak n-categories, Geom. Topol. 14 (2010 Google Scholar), 521–571.
[103] Rezk, C., A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001 CrossRef | Google Scholar), no. 3, 973–1007.
[104] Riehl, E., Categorical Homotopy Theory, New Mathematical Monographs 24, Cambridge University Press, 2014 CrossRef | Google Scholar.
[105] Riehl, E., and Verity, D., The 2-category theory of quasi-categories, Adv. Math. 280 (2015 CrossRef | Google Scholar), 549–642.
[106] Riehl, E., and Verity, D., Fibrations and Yoneda's lemma in an ∞-cosmos, J. Pure Appl. Algebra 221 (2017 CrossRef | Google Scholar), no. 3, 499–564.
[107] Riehl, E., and Verity, D., Kan extensions and the calculus of modules for ∞-categories, Algebr. Geom. Topol. 17 (2017 CrossRef | Google Scholar), no. 1, 189–271.
[108] Robertson, M., The homotopy theory of simplicially enriched multicategories (2011 Google Scholar), available at arXiv:1111.4146.
[109] Segal, G., Categories and cohomology theories, Topology 13 (1974 CrossRef | Google Scholar), 293–312.
[110] Simpson, C., Homotopy Theory of Higher Categories, NewMathematical Monographs 19, Cambridge University Press, 2012 Google Scholar.
[111] Street, R., The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987 CrossRef | Google Scholar), no. 3, 283–335.
[112] Strøm, A., The homotopy category is a homotopy category, Arch. Math. (Basel) 23 (1972 CrossRef | Google Scholar), 435–441.
[113] Tamsamani, Z., Sur les notions de n-categorie et n-groupóıde non-stricte via des ensembles multi-simpliciaux, K-Theory 16 (1999 CrossRef | Google Scholar), no. 1, 51–99.
[114] Thomason, R.W., Cat as a closed model category, Cah. Topol. Géom. Différ. Catég. 21 (1980 Google Scholar), no. 3, 305–324.
[115] Töen, B., Homotopical and higher categorical structures in algebraic geometry (a view towards homotopical algebraic geometry) (2003 Google Scholar), available at arXiv:0312262.
[116] Töen, B., Vers une axiomatisation de la théorie des catégories supérieures, K-Theory 34 (2005 CrossRef | Google Scholar), no. 3, 233–263.
[117] Verity, D.R.B., Weak complicial sets I: basic homotopy theory, Adv. Math. 219 (2008 CrossRef | Google Scholar), no. 4, 1081–1149.
[118] Weibel, C.A., An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, 1994 CrossRef | Google Scholar.

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