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Modeling homotopy theories. (English) Zbl 1432.18019

From the text: Our aim in this article is to give an overview of the development of the notion of homotopy theory, starting with these classical examples and leading up to more modern perspectives. As we discuss briefly at the end, we have chosen one of several possible entry points into this subject and in particular do not say much about the higher categorical aspects of the theory. We simply remark that the “homotopy theories” that we discuss here coincide with the “\((\infty, 1)\)-categories” that are being used widely in a number of related fields.

MSC:

18N40 Homotopical algebra, Quillen model categories, derivators
18N60 \((\infty,1)\)-categories (quasi-categories, Segal spaces, etc.); \(\infty\)-topoi, stable \(\infty\)-categories
18-03 History of category theory
01A65 Development of contemporary mathematics
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