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**Homotopy theories.**
*(English)*
Zbl 0643.55015

Mem. Am. Math. Soc. 71, No. 383, 78 p. (1988).

Attempts to formulate a notion of “homotopy” outside the framework of topology probably go back to the Eckmann-Hilton approach to homotopy of modules [see P. Hilton, Homotopy theory and duality (1965; Zbl 0155.508)]. Influenced by this work, P. Huber gave an early method for introducing homotopy into general categories [Math. Ann. 144, 361-385 (1961; Zbl 0099.179)]. Later, in the spirit of Eilenberg-Steenrod, an axiomatization of homotopy theory based on the notion of closed model category was given by D. G. Quillen [Homotopical algebra, Lect. Notes Math. 43 (1967; Zbl 0168.209)] and applied, with great effect, to the study of rational homotopy theory [see D. Quillen, Ann. Math., II. Ser. 90, 205-295 (1969; Zbl 0191.537) and also A. K. Bousfield and V. K. A. M. Gugenheim, Mem. Am. Math. Soc. 179 (1976; Zbl 0338.55008)]. D. W. Anderson noted that, although quite successful, the closed model category idea did not apply to such standard working frameworks as finite CW complexes and equivariant theory. To remedy this situation, Anderson proposed his own axiomatization for categories with a fraction functor [Lect. Notes Math. 741, 196-205 (1979; Zbl 0417.57013)].

In the present work, the author approaches axiomatization from a rather different viewpoint. As he states, “what is considered is ‘homotopy theory’ as opposed to ‘the homotopy theory of...’.” With this in mind, the author describes the notion of hyperfunctor, a function T which takes each small category C to the category TC, each functor \(F: C\to D\) to one TF: TD\(\to TC\) and each natural transformation \(\phi\) : \(F\to G\) to one \(T\phi\) : TF\(\to TG\). A “homotopy theory” is then a hyperfunctor which satisfies certain properties abstracted from the following standard example: Let K denote the category of simplicial sets and suppose C is a small category. A “homotopy theory” \(\Pi\) is defined by letting \(\Pi\) C denote the homotopy category of the functor category K C. Indeed, the case \(C=1\) is ordinary homotopy theory. After showing that \(\Pi\) is, in fact, a homotopy theory, the author proves various other facts about homotopy theory hyperfunctors including an interesting density theorem (Theorem III 4.2). He proceeds to describe homotopical notions such as localization within his framework. It must be said that misprints crop up now and then. In particular, they even arise in the fundamental definitions of comma category (where Fb should be Gb) and left semi- direct product composition (where Ff should be Ff’), not to mention the Library of Congress Data (where the title is listed as “homogopy theories”).

In the present work, the author approaches axiomatization from a rather different viewpoint. As he states, “what is considered is ‘homotopy theory’ as opposed to ‘the homotopy theory of...’.” With this in mind, the author describes the notion of hyperfunctor, a function T which takes each small category C to the category TC, each functor \(F: C\to D\) to one TF: TD\(\to TC\) and each natural transformation \(\phi\) : \(F\to G\) to one \(T\phi\) : TF\(\to TG\). A “homotopy theory” is then a hyperfunctor which satisfies certain properties abstracted from the following standard example: Let K denote the category of simplicial sets and suppose C is a small category. A “homotopy theory” \(\Pi\) is defined by letting \(\Pi\) C denote the homotopy category of the functor category K C. Indeed, the case \(C=1\) is ordinary homotopy theory. After showing that \(\Pi\) is, in fact, a homotopy theory, the author proves various other facts about homotopy theory hyperfunctors including an interesting density theorem (Theorem III 4.2). He proceeds to describe homotopical notions such as localization within his framework. It must be said that misprints crop up now and then. In particular, they even arise in the fundamental definitions of comma category (where Fb should be Gb) and left semi- direct product composition (where Ff should be Ff’), not to mention the Library of Congress Data (where the title is listed as “homogopy theories”).

Reviewer: J.Oprea

### MSC:

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |

18G55 | Nonabelian homotopical algebra (MSC2010) |

18A25 | Functor categories, comma categories |