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A survey of models for \((\infty, n)\)-categories. (English) Zbl 1476.55043

Miller, Haynes (ed.), Handbook of homotopy theory. Boca Raton, FL: CRC Press. CRC Press/Chapman Hall Handb. Math. Ser., 263-295 (2020).
The Handbook of homotopy theory has been reviewed in [Zbl 1468.55001].
Summary: A model category is a category possessing all small limits and colimits, together with three distinguished classes of morphisms, called weak equivalences, fibrations, and cofibrations, satisfying several axioms. For example, the category of topological spaces has a model structure Top in which the weak equivalences are the weak homotopy equivalences, all objects are fibrant, and the cofibrant objects are retracts of CW-complexes. The structure of a model category enables one to have a well-defined homotopy category without running into set-theoretic obstacles. A bisimplicial space is a double Segal space if and only if it is a Segal object in Segal spaces. A 2-category can be defined to be a category enriched in categories. There is a model structure \(\mathcal{CSS}\) on the category of bisimplical spaces in which the fibrant objects are precisely the 2-fold complete Segal spaces.
For the entire collection see [Zbl 1468.55001].

MSC:

55U35 Abstract and axiomatic homotopy theory in algebraic topology
55U40 Topological categories, foundations of homotopy theory
18N40 Homotopical algebra, Quillen model categories, derivators
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
18D20 Enriched categories (over closed or monoidal categories)
18N50 Simplicial sets, simplicial objects
18N65 \((\infty, n)\)-categories and \((\infty,\infty)\)-categories
18C10 Theories (e.g., algebraic theories), structure, and semantics
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology

Citations:

Zbl 1468.55001
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