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

- homotopy hypothesis-theorem
- delooping hypothesis-theorem
- periodic table
- stabilization hypothesis-theorem
- exactness hypothesis
- holographic principle

- (n,r)-category
- Theta-space
- ∞-category/∞-category
- (∞,n)-category
- (∞,2)-category
- (∞,1)-category
- (∞,0)-category/∞-groupoid
- n-category = (n,n)-category
- 2-category, (2,1)-category
- 1-category
- 0-category
- (?1)-category?
- (?2)-category?

- n-poset = (n-1,n)-category
- n-groupoid = (n,0)-category

- categorification/decategorification
- geometric definition of higher category
- algebraic definition of higher category
- stable homotopy 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

Last revised on November 26, 2018 at 03:15:45. See the history of this page for a list of all contributions to it.