Complete Segal space

Complete Segal space A Complete Segal space is a homotopical enhancement of Segal space designed to model (∞,1)-categories within the framework of Charles Rezk's model structures on simplicial spaces. It refines constructions appearing in the work of Graeme Segal, André Joyal, Jacob Lurie, and Curtis Rezk to provide a presentation amenable to comparisons with simplicial category, quasi-categorys, and model category theory in the tradition of Daniel Quillen and Quillen adjunctions.

Definition

A Complete Segal space is a fibrant object in the Rezk model structure on the category of simplicial spaces (bisimplicial sets) satisfying the Segal condition and the completeness condition introduced by Curtis Rezk. The Segal condition imposes that for each integer n≥2 the canonical map from the space of n-simplices to an iterated homotopy pullback of 1-simplices is a weak equivalence in the sense of Quillen model category theory, reflecting composition laws analogous to those in Henri Poincaré-inspired categorical constructions. The completeness condition requires that the space of homotopy equivalences of the underlying Segal space is equivalent to the space of objects, a criterion closely related to notions appearing in the work of André Joyal and later used by Jacob Lurie in higher topos theory.

Examples and basic properties

Basic examples include the nerves of ordinary small categorys viewed as discrete simplicial spaces, which yield Complete Segal space models when considered under appropriate fibrant replacement functors used by Curtis Rezk. Simplicial nerves of simplicial categorys and hammock localization constructions introduced by Dwyer and Kan produce Complete Segal spaces representing homotopy coherent categories studied alongside Boardman–Vogt and May-style loop space machines. Important properties include invariance under Dwyer–Kan equivalence, homotopy limits and colimits behavior governed by Bousfield localization techniques, and compatibility with Brown representability-style phenomena in stable homotopy theory contexts related to Adams spectral sequence considerations.

Model category and homotopy theory

The theory of Complete Segal spaces is formalized by the Rezk model structure on simplicial spaces, a left proper, combinatorial Quillen model category where weak equivalences are Dwyer–Kan equivalences between underlying homotopy theories. This model structure admits Quillen equivalences to other presentations of (∞,1)-categories, leveraging Bousfield localization, cofibrantly generated model structures, and left properness results found in the literature of Mark Hovey and Paul Goerss. The homotopy category of Complete Segal spaces captures mapping spaces and homotopy coherent composition, enabling comparisons with homotopical constructions used by Daniel Quillen, William G. Dwyer, Dan Kan, and techniques from Homotopy coherence developed by Boardman and Vogt.

Relationship to other models of (∞,1)-categories

Complete Segal spaces are Quillen equivalent to quasi-categorys (aka ∞-categorys) as developed in the work of André Joyal and systematized by Jacob Lurie in Higher Topos Theory, and to simplicial categories via the homotopy coherent nerve functor of Cordier and Porter. These equivalences situate Complete Segal spaces alongside other models such as Segal categorys, relative categories studied by Barwick and Kan, and model structures used in Toen and Vezzosi's homotopical algebraic geometry. Comparisons often employ explicit Quillen adjunctions and derived equivalences appearing in the literature of Dwyer–Kan equivalences, and are used to translate constructions between frameworks used by Lurie, Rezk, Joyal, and Bertrand Toën.

Constructions and applications

Constructions of Complete Segal spaces include taking simplicial nerves of model categorys, applying hammock localization, and performing fibrant replacement in the Rezk model structure; these constructions are used to encode derived mapping spaces, homotopy limits, and monoidal structures in contexts appearing in Derived algebraic geometry of Jacob Lurie and Bertrand Toën. Applications appear in the study of homotopy coherent diagrams, (∞,1)-topos theory, factorization homology in the work of Ayala and Francis, and categorical approaches to Topological quantum field theory influenced by Graeme Segal and Jacob Lurie. Complete Segal spaces also play a role in categorical semantics for stable homotopy theory and in comparisons with spectral categories and enrichment techniques appearing in research by B. Keller and Stefan Schwede.

Category:Higher category theory