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.
This we discuss below.
For more on this see elsewhere
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.
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
Recall the slight but crucial difference between the two notions of “simplicial categories”, the other being an internal category in sSet. But also for this latter concept there is a model category structure which presents (infinity,1)-categories, see