nLab
algebraic quasi-category

**higher category theory** * category theory * homotopy theory ## Basic concepts * k-morphism, coherence * looping and delooping * looping and suspension ## Basic theorems * homotopy hypothesis-theorem * delooping hypothesis-theorem * periodic table * stabilization hypothesis-theorem * exactness hypothesis * holographic principle ## Applications * applications of (higher) category theory * higher category theory and physics ## Models * (n,r)-category * Theta-space * ∞-category/ω-category * (∞,n)-category * n-fold complete Segal space * (∞,2)-category * (∞,1)-category * quasi-category * algebraic quasi-category * simplicially enriched category * complete Segal space * model category * (∞,0)-category/∞-groupoid * Kan complex * algebraic Kan complex * simplicial T-complex * n-category = (n,n)-category * 2-category, (2,1)-category * 1-category * 0-category * (−1)-category * (−2)-category * n-poset = (n-1,n)-category * poset = (0,1)-category * 2-poset = (1,2)-category * n-groupoid = (n,0)-category * 2-groupoid, 3-groupoid * categorification/decategorification * geometric definition of higher category * Kan complex * quasi-category * simplicial model for weak ω-categories * complicial set * weak complicial set * algebraic definition of higher category * bicategory * bigroupoid * tricategory * tetracategory * strict ω-category * Batanin ω-category * Trimble ω-category * Grothendieck-Maltsiniotis ∞-categories * stable homotopy theory * symmetric monoidal category * symmetric monoidal (∞,1)-category * stable (∞,1)-category * dg-category * A-∞ category * triangulated category ## Morphisms * k-morphism * 2-morphism * transfor * natural transformation * modification ## Functors * functor * 2-functor * pseudofunctor * lax functor * (∞,1)-functor ## Universal constructions * 2-limit * (∞,1)-adjunction * (∞,1)-Kan extension * (∞,1)-limit * (∞,1)-Grothendieck construction ## Extra properties and structure * cosmic cube * k-tuply monoidal n-category * strict ∞-category, strict ∞-groupoid * stable (∞,1)-category * (∞,1)-topos ## 1-categorical presentations * homotopical category * model category theory * enriched category theory

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*** **(∞,1)-category theory** ## Background * category theory * higher category theory * (n,r)-category ## Basic concepts * (∞,1)-category * hom-objects * equivalences in/of $(\infty,1)$-categories * sub-(∞,1)-category * reflective sub-(∞,1)-category * reflective localization * opposite (∞,1)-category * over (∞,1)-category * join of quasi-categories * (∞,1)-functor * exact (∞,1)-functor * (∞,1)-category of (∞,1)-functors * (∞,1)-category of (∞,1)-presheaves * **fibrations** * inner fibration * left/right fibration * Cartesian fibration * Cartesian morphism ## Universal constructions * limit * terminal object * adjoint functors ## Local presentation * locally presentable * essentially small * locally small * accessible * idempotent-complete ## Theorems * (∞,1)-Yoneda lemma * (∞,1)-Grothendieck construction * adjoint (∞,1)-functor theorem * (∞,1)-monadicity theorem ## Extra stuff, structure, properties * stable (∞,1)-category * (∞,1)-topos ## Models * category with weak equivalences * model category * derivator * quasi-category * model structure for quasi-categories * model structure for Cartesian fibrations * relation to simplicial categories * homotopy coherent nerve * simplicial model category * presentable quasi-category * Kan complex * model structure for Kan complexes

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An algebraic quasi-category is a quasi-category equipped with a choice of (inner) horn fillers.

Algebraic quasi-categories give a algebraic definition of (∞,1)-categories.

For more see the section Algebraic fibrant models for higher categories at model structure on algebraic fibrant objects.

Revised on July 8, 2010 20:19:21 by Toby Bartels (173.60.119.197)