Axiomatic stable homotopy theory.

*(English)*Zbl 0881.55001
Mem. Am. Math. Soc. 610, 114 p. (1997).

A stable homotopy category is, in this paper, defined to be a triangulated closed symmetric monoidal category with some structures which are adopted as axioms, keeping the homotopy category of spectra in mind. Starting with this definition, the authors construct cellular towers, Bousfield localization, Brown representability and so on in the category. They even prove analogous theorems to the nilpotence theorem and the thick subcategory theorem of Devinatz, Hopkins and Smith, while the situation is more general and the results are weaker. There are various examples that satisfy the axioms in other fields of mathematics, which naturally gives the applications there. This book consists of ten sections and two appendices. In the first two sections, the definition is given, and fundamental properties are studied including a construction of cellular towers. Sections 3 and 4 are devoted to Bousfield localization and Brown representability. Nilpotence and thick subcategory theorems are discussed in section 5. In section 6, the authors introduce a Noetherian category, which is simpler and behaves better than the homotopy category of spectra. In the next two sections, they state basic properties of special cases of connective and semisimple stable homotopy categories, which are similar to the homotopy category of spectra and is equivalent to the category of graded rational vector spaces, respectively. Section 9 contains several examples. This paper ends with some suggestions for further study in the last section. The appendices present triangulated categories and closed symmetric monoidal categories, which is useful for and reader unfamiliar with these categories.

Reviewer: Katsumi Shimomura (Kochi)

##### MSC:

55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |

55P42 | Stable homotopy theory, spectra |

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

55U15 | Chain complexes in algebraic topology |

55N20 | Generalized (extraordinary) homology and cohomology theories in algebraic topology |

18G35 | Chain complexes (category-theoretic aspects), dg categories |