(infinity,1)-category of (infinity,1)-categories

**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/∞-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 ∞-categories?
* complicial set
* weak complicial set
* algebraic definition of higher category
* bicategory
* bigroupoid
* tricategory
* tetracategory
* strict ∞-category
* Batanin ∞-category?
* Trimble ∞-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

The collection of all (∞,1)-categories forms naturally the (∞,2)-category (∞,1)Cat.

But for many purposes it is quite sufficient to regard only invertible natural transformations between (∞,1)-functor, which means that one needs just the maximal (∞,1)-category inside that $(\infty,2)$-category of all $(\infty,1)$-categories.

Given that an $(\infty,1)$-category is a context for abstract homotopy theory, the $(\infty,1)$-category of $(\infty,1)$-categories is also called the the **homotopy theory of homotopy theories**.

The full SSet-enriched-subcategory of SSet on those simplicial sets which are quasi-categories is – by the properties discussed at (∞,1)-category of (∞,1)-functors – itself a quasi-category-enriched category. This is the (∞,2)-category of (∞,1)-categories.

The sSet-subcategory of that obtained by picking of each hom-object the core, i.e. the maximal ∞-groupoid/Kan complex yields an ∞-groupoid/Kan complex-enriched category. This is the **$(\infty,1)$-category of $(\infty,1)$-categories** in its incarnation as a simplicially enriched category. Forming its homotopy coherent nerve produces the **quasi-category of quasi-categories** .

The Joyal-model structure for quasi-categories is an $sSet_{Joyal}$-enriched model category and hence its full SSet-subcategory on cofibrant-fibrant objects is the $(\infty,2)$-category of $(\infty,1)$-categories.

An $SSet_{Quillen}$-enriched model category (i.e. enriched over the ordinary model structure on simplicial sets) whose full subcategory of fibrant-cofibrant objects is the $(\infty,1)$-category $(\infty,1)Cat$ is the model structure on marked simplicial sets (over the terminal set). Its underlying plain model category is Quillen equivalent to the Joyal-model structure, but it is indeed $sSet_{Quillen}$-enriched.

Other model structures that present the $(\infty,1)$-category of all $(\infty,1)$-categories are

- of particular interest is the $(\infty,1)$-subcategory $(\infty,1)PresCat_1 \hookrightarrow (\infty,1)Cat_1$ of presentable (∞,1)-categories.

chapter 3 of

Revised on November 19, 2012 23:23:38
by Ingo Blechschmidt
(79.219.158.237)