The condition that aa quasicategory has “no non-trivial cells” above degree (which makes it a particularly strict model of an (n,1)-category) is not invariant under categorical equivalence. Hence there is no intrinsic characterization of the class of the simplicial sets which are “(n+1)-coskeletal” in this sense.
However there is such a description of the class of quasi-categories which are equivalent to such -coskeletal quasicategories. To make this more concrete the notion of a minimal inner fibration can be used (a quasi-categorical analog of minimal Kan fibrations). THis is an inner fibration of simplicial sets satisfying a relative homotopy condition and that of a minimal quasi-category .
Every quasi-category is equivalent to a minimal quasi-category.
denote a lifting problem. Then putative solutions of this lifting problem are called homotopic relative over if they are equivalent as objects in the fiber of the map
Equivalently are homotopic relative over if there is a map
and is an equivalence in the -category for every vertex of .
Let be an inner ﬁbration of simplicial sets. is called minimal inner fibration if for every pair of maps which are homotopic relative to over .
An -category is called minimal -category if is minimal.
Let be an -category and let . The the following statements are equivalent:
There exists a minimal model such that is an -coskeletal quasi-category.
There exists a categorical equivalence , where is an -coskeletal quasi-category.
For every pair of objects , the mapping space is -truncated.
Section 2.3.3 and section 2.3.4 of