related by the Dold-Kan correspondence
Depending on the chosen model category structure, the category sSet of simplicial sets may model ∞-groupoids (for the standard model structure on simplicial sets) or (∞,1)-categories in the form of quasi-categories (for the Joyal model structure) on SSet.
Both are special cases of a model structure on enriched categories.
it is an -full and faithful functor in that for all objects the morphism
is a weak equivalence in the standard model structure on simplicial sets.
Such a morphism is also called a Dwyer-Kan weak equivalence after the work by Dwyer-Kan on simplicial localization.
weak equivalences the Dwyer-Kan equivalences;
The entries of the following table display models, model categories, and Quillen equivalences between these that present the (∞,1)-category of (∞,1)-categories (second table), of (∞,1)-operads (third table) and of -monoidal (∞,1)-categories (fourth table).
|enriched (∞,1)-category||internal (∞,1)-category|
|SimplicialCategories||homotopy coherent nerve||SimplicialSets/quasi-categories||RelativeSimplicialSets|
|SimplicialOperads||homotopy coherent dendroidal nerve||DendroidalSets||RelativeDendroidalSets|
A model category structure on the category of -categories with a fixed set of objects was first given in
Dywer, Spalinski and later Rezk then pointed out that there ought to exist a model category structure on the collection of all -categories that models the (∞,1)-category of (∞,1)-categories. This was then constructed in
A survey is in section 3 of
See also section A.3.2 of
for the moment see (∞,2)-category for more on this
Section A.3 of