Simplicial space

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.

A Segal category is a kind of simplicial space, this is a model of an infinity category introduced by Hirschowitz & Simpson (1998), based on work of Graeme Segal in 1974.

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

• Baues, Hans Joachim (1995), "Homotopy types", in James, I. M. (ed.), Handbook of Algebraic Topology, Amsterdam: North-Holland, pp. 1–72, doi:10.1016/B978-044481779-2/50002-X, ISBN 9780444817792, MR 1361886.
• Hirschowitz, André; Simpson, Carlos (1998). "Descente pour les n-champs" (in French). arXiv:math/9807049.
• Bergner, Julia E. (2007). "Three models for the homotopy theory of homotopy theories". Topology. 46 (4): 397–436. doi:10.1016/j.top.2007.03.002.
• 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

External links

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