Quasi-category
Generated by GPT-5-miniExpansion Funnel Raw 68 → Dedup 0 → NER 0 → Enqueued 0
| Quasi-category | |
|---|---|
| Name | Quasi-category |
| Type | Model of (∞,1)-category |
| Field | Algebraic topology; Category theory; Homotopy theory |
| Introduced | 2000s |
| Introduced by | Boardman; Vogt; Joyal; Lurie |
Quasi-category A quasi-category is a simplicial set satisfying inner horn-filling conditions that serves as a model for higher-categorical structures arising in Algebraic Topology, Category Theory, and Homotopy Theory. Originating in work by Boardman, Vogt, and substantially developed by Joyal and Lurie, quasi-categories connect with classical constructions by Grothendieck, Kan, Quillen, and Brown and underpin modern formulations in Derived Algebraic Geometry, Higher Topos Theory, and Stable Homotopy Theory.
Definition
A quasi-category is a simplicial set X for which every inner horn Λ^n_k → X (0 < k < n) admits a filler Δ^n → X; this condition parallels the Kan condition of Kan complex while relaxing outer horn fillers used in notions by Dwyer and Kan. The formalism uses simplicial language developed by Eilenberg, Mac Lane, and May and is typically presented in textbooks and monographs by Goerss, Jardine, Lurie, and Weibel.
Basic Properties and Examples
Basic properties include homotopy coherent compositions, mapping spaces given by Kan complexes à la Simplicial Hom, and notions of equivalence strongly related to Joyal’s model structure studied by Joyal and Tierney. Standard examples arise from nerves: the nerve of an ordinary category yields a quasi-category via Nerve (category), while nerves of simplicial categories studied by Cordier, Porter, and Bergner produce quasi-categories reflecting homotopy-coherent information. Other examples include simplicial nerves of model categories from Quillen and homotopy coherent nerves used in Lurie's work on Higher Topos Theory and Derived Algebraic Geometry.
Homotopy Theory and Model Structures
The homotopy theory of quasi-categories is organized by Joyal’s model structure on simplicial sets, compared and Quillen equivalent to the Bergner model structure on simplicial categories developed by Bergner. Quasi-categorical equivalences correspond to categorical equivalences studied by Dwyer and Kan in the context of simplicial localizations by Hammock localization and Dwyer–Kan localization. The interplay with Model category techniques from Quillen and the straightening–unstraightening equivalence in Lurie’s work ties quasi-categories to parametrized families and fibrations analogous to those in Grothendieck’s theory of fibrations and to the homotopical analyses of Hovey and Hirschhorn.
Constructions and Operations
Constructions include the homotopy coherent nerve functor connecting simplicial categories to quasi-categories introduced by Cordier and Porter and refined by Lurie; nerve and realization adjunctions mirror classical adjunctions by Adjoint functor theorem authors such as Freyd and Kelly. Operations such as limits and colimits inside a quasi-category are encoded by inner horn fillers and limit diagrams studied in Street’s and Bénabou’s works; mapping cones and suspensions appear in stable settings addressed by Neeman and Hovey in Triangulated category contexts. Monoidal structures on quasi-categories relate to work by Day and Kelly and feed into construction of Monoidal category-enriched higher categories used in Topological Quantum Field Theory research by Atiyah and Segal.
Relations to Other Models of (∞,1)-Categories
Quasi-categories are one of several equivalent models for (∞,1)-categories alongside simplicial categories (Bergner), Segal categories developed by Segal and Rezk’s complete Segal spaces, and relative categories studied by Barwick and Kan. Quillen equivalences between these models were established in work connecting Bergner’s model structure, Rezk’s complete Segal space model, and Joyal’s structure, with comparisons invoking tools from Dwyer–Kan theory, Simplicial localization, and methods used by Toën and Vezzosi in Homotopical Algebraic Geometry.
Applications and Further Developments
Quasi-categories play a central role in Lurie’s Higher Topos Theory and Derived Algebraic Geometry, underpinning modern formulations of Spectral Algebraic Geometry and constructions in Stable Homotopy Theory linked to Brown–Peterson spectrum and Morava K-theory. They appear in the categorical foundations of Factorization Homology, higher Morita categories studied by Lurie and Baez, and in categorical approaches to Topological Field Theories pursued by Freed and Costello. Ongoing developments tie quasi-categories to computational homotopy theory work by Hatcher and May, to categorical semantics in Homotopy Type Theory and the Univalence axiom championed by Voevodsky, and to interactions with Arithmetic Geometry via higher categorical methods used by Kontsevich and Soibelman.