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
***
**(∞,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

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)