# nLab (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

# Contents

## Idea

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.

## Definition

### Intrinsic definition

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 .

### Models

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

## Applications

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

## References

chapter 3 of

Last revised on November 19, 2012 at 23:23:38. See the history of this page for a list of all contributions to it.