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Model categories and their localizations. (English) Zbl 1017.55001
Mathematical Surveys and Monographs. 99. Providence, RI: American Mathematical Society (AMS). xv, 457 p. (2003).
This monograph consists of two parts. The first part is a definitive examination of the process of localizing a model category with respect to a morphism in that category. The second, and by far the longer, part is a thorough treatment of model categories and their homotopy categories. Central to this part is the introduction and exploration of the notion of a cellular model category. This is the sort of model category for which the localization arguments supplied in the first part will work.
While localization of simply connected spaces with respect to singular homology goes back to Dennis Sullivan, the idea of localizing the model category of spaces with respect to a morphism of spaces is essentially due to A. K. Bousfield in his article, “The localization of spaces with respect to homology” [Topology 14, 133-150 (1975; Zbl 0309.55013)]. Later, E. Dror-Farjoun [see, for example, “Cellular spaces, null spaces, and homotopy localizations”, Lect. Notes Math. 1622 (1996; Zbl 0842.55001)] realized the importance of a localizition with respect to an arbitrary map, and at that point a whole industry was born.
The key argument of Bousfield’s original paper, which remains central to the theory, relies on the fact that if \(X\) is a simplicial set, then the functor which assigns to a simplicial set \(Y\) the set of morphisms of simplicial sets \(X \to Y\) commutes with colimits over ordinal numbers whose cardinality is larger than the number of simplices of \(X\). This argument of Bousfield’s, suitably generalized, is featured in this book as the “Bousfield-Smith cardinality argument”. The Smith here is Jeff Smith, who noticed some of the wider implications of these kinds of thoughts. A great deal of the work here is finding a suitable axiomatic framework in which this argument works. This leads to cellular model categories, which, roughly speaking, are model categories in which one can define cell complexes and sub-complexes – and, central to the Bousfield-Smith cardinality argument, one can count the number of cells in a sub-complex.
Also reviewed in the second part of this book are some of the more standard aspects of model category theory, such as properness, derived function spaces, homotopy (co-)limits, and so on. Much of the material is available in the literature, but in a scattered form, and perhaps not as accessibly as here.
This book was many years in the writing, and it shows. It is very carefully written, exhaustively (even obsessively) cross-referenced, and precise in all its details. In short, it is an important reference for the subject.

55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
18-02 Research exposition (monographs, survey articles) pertaining to category theory
55U35 Abstract and axiomatic homotopy theory in algebraic topology
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
18E35 Localization of categories, calculus of fractions