n-category = (n,n)-category
n-groupoid = (n,0)-category
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 -category of all -categories.
Given that an -category is a context for abstract homotopy theory, the -category of -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 -category of -categories in its incarnation as a simplicially enriched category. Forming its homotopy coherent nerve produces the quasi-category of quasi-categories .
An -enriched model category (i.e. enriched over the ordinary model structure on simplicial sets) whose full subcategory of fibrant-cofibrant objects is the -category 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 -enriched.
Other model structures that present the -category of all -categories are
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