Simplicial space
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In mathematics, a simplicial space is a simplicial object in the category of topological spaces. In other words, it is a contravariant functor from the simplex category Δ to the category of topological spaces.[1]
In mathematics, a Segal category is a model of an infinity category introduced by (Hirschowitz Simpson), based on work of Graeme Segal in 1974.
In mathematics, a Segal space is a simplicial space satisfying some pullback conditions, making it look like a homotopical version of a category. More precisely, a simplicial set, considered as a simplicial discrete space, satisfies the Segal conditions if and only if it is the nerve of a category. The condition for Segal spaces is a homotopical version of this. Complete Segal spaces were introduced by (Rezk 2001) as models for (∞, 1)-categories.
References
- Hirschowitz, André; Simpson, Carlos (1998). "Descente pour les n-champs" (in français). arXiv:math/9807049.
- Joyal, A. (2008), The theory of quasi-categories and its applications, lectures at CRM Barcelona, pp. 164–169, archived from the original on 2011-07-06, https://web.archive.org/web/20110706214636/http://www.crm.cat/HigherCategories/hc2.pdf
- Rezk, Charles (2001), "A model for the homotopy theory of homotopy theory", Transactions of the American Mathematical Society 353 (3): 973–1007, doi:10.1090/S0002-9947-00-02653-2, ISSN 0002-9947
- ↑ Baues, Hans Joachim (1995), "Homotopy types", Handbook of Algebraic Topology, Amsterdam: North-Holland, pp. 1–72, doi:10.1016/B978-044481779-2/50002-X, ISBN 9780444817792. See in particular p. 8.
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