Categorical homotopy theory (Q2879235)

This is a book with a contemporary viewpoint on homotopy theory, offering to a larger audience (than that of research articles, preprints, and mathematical blogs) the opportunity to learn - among other subjects - about how one thinks nowadays about homotopy limits and colimits, and how enriched category theory have entered into play.NEWLINENEWLINEQuillen's introduction of model categories [\textit{D. G. Quillen}, Homotopical algebra. Lecture Notes in Mathematics. 43. Berlin-Heidelberg-York: Springer-Verlag (1967; Zbl 0168.20903)], changed homotopy theory durably, it highlighted the basic structure that is necessary to do homotopy theory, thus unifying homological algebra and classical unstable or stable homotopy theory, to which one added later equivariant and motivic homotopy, etc. However the role of cofibrations and fibrations is less intrinsic than that of the weak equivalences. They are used in specific computations, such as the well-known process of replacing the maps in a push-out diagram by cofibrations so as to construct the homotopy push-out. The point of view here is that of Dwyer, Hirschhorn, Kan, and Smith [\textit{W. G. Dwyer} et al., Homotopy limit functors on model categories and homotopical categories. Mathematical Surveys and Monographs 113. Providence, RI: American Mathematical Society (AMS) (2004; Zbl 1072.18012)], where a weaker set of axioms is introduced, still allowing one to study total derived functors. The bar and cobar constructions are the tools that provide deformations to construct homotopy limits and colimits in simplicial model categories. Part~I of the book ends with the observation that the first historical definitions of homotopy (co)limits by \textit{A. K. Bousfield} and \textit{D. M. Kan} [Homotopy limits, completions and localizations. Lecture Notes in Mathematics. 304. Berlin-Heidelberg-York: Springer-Verlag (1972; Zbl 0259.55004)], are special cases of \textit{weighted} (co)limits which are identified with (co)bar constructions. So, having compared all available constructions of homotopy limits and colimits, the second part of the book about enriched homotopy theory follows naturally.NEWLINENEWLINEThe author relies in Part II on a preprint by \textit{M. Shulman} [``Homotopy limits and colimits and enriched homotopy theory'', \url{arXiv:math/0610194}]. The theory of weights is first explained in the unenriched case, i.e. the enrichment is over the category of sets, before going to the general case. The Bousfield-Kan formula is understood in terms of conical limits and colimits and the necessary modifications to obtain a homotopy colimit from a colimit consist in replacing the trivial weight, encoding the universal property of cones under a fixed object, to a weight encoding homotopy coherent cones. This motivates a thorough study of general weighted colimits and limits, enriched cobar and bar constructions. The last section of this part is largely devoted to the comparison between the notion of equivalence arising from the enrichment and the one coming from the homotopical structure (in a so-called \(\mathcal V\)-enriched homotopical category), which generalizes the classical comparison between weak and homotopy equivalences of spaces.NEWLINENEWLINEPart III deals with model categories in a similar way to \textit{J. P. May} and \textit{K. Ponto} [More concise algebraic topology. Localization, completion, and model categories. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press (2012; Zbl 1249.55001)], where the emphasis is put on the pair of weak factorization systems present in a model category structure. In this context, Quillen's small object argument is revisited as in [\textit{R. Garner}, Appl. Categ. Struct. 17, No. 3, 247--285 (2009; Zbl 1173.55009)]. The main benefit of the small object argument is the functoriality of the model categorical factorizations. As can be expected, this is readily generalized to an enriched small object argument, producing \textit{enriched} weak factorization systems. An interesting observation is that the Hurewicz model structure on chain complexes is not cofibrantly generated in the classical sense, but it is so in the enriched sense.NEWLINENEWLINEThe final part of the book is devoted to quasi-categories, as introduced by Joyal to model \((\infty, 1)\)-categories (see, e.g., [\textit{A. Joyal}, The theory of quasi-categories and its applications, Lectures at CRM Barcelona, 2008, \url{http://www.crm.cat/HigherCategories/hc2.pdf}]). The Joyal model structure on simplicial sets is defined in such a way that the fibrant objects are precisely the quasi-categories. The aim of the author is not to give a complete account of the theory, but rather to explain why quasi-categories model \(\infty\)-categories, thus focusing on mapping spaces between two objects of a given quasi-category. The treatment follows [\textit{D. Dugger} and \textit{D. I. Spivak}, Algebr. Geom. Topol. 11, No. 1, 263--325 (2011; Zbl 1214.55013)]. The last section is an introduction to the work of the author and Verity about the \(2\)-categorical aspects of the theory of quasi-categories [\textit{E. Riehl} and \textit{D. Verity}, Adv. Math. 280, 549--642 (2015; Zbl 1319.18005)].NEWLINENEWLINEThis monograph is not a complete textbook on homotopy theory and does not present either a detailed account on a big theory converging to a central theorem. It rather highlights various aspects of homotopy theory where category theory sheds light on not so well understood phenomena, it provides motivation and explanations, and redirects the reader to key references.