Segal space

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 iff 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.

• 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