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 (1998), 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.

• Hirschowitz, André; Simpson, Carlos (1998). "Descente pour les n-champs" (in French). arXiv:math/9807049.
• Joyal, A. (2008), The theory of quasi-categories and its applications, lectures at CRM Barcelona (PDF), pp. 164–169, archived from the original (PDF) on 2011-07-06
• 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, MR 1804411

• Segal space at the nLab
• Complete Segal space at the nLab
• Segal category at the nLab

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