nLab
equivalence of quasi-categories

Idea

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Definition

… quasi-category … Joyal’s model structure on simplicial sets …

Properties

Lemma

An (∞,1)-functor $f:C\to D$ is an equivalence if the following equivalent conditions hold

• On the underlying simplicial sets it is a weak categorical equivalence in Joyal’s model structure on simplicial sets.

• For every simplicial set $K$ the induced morphism $f_{*}:\mathrm{SSet}(K,C)\to \mathrm{SSet}(K,D)$ is a weak categorical equivalence.

• For every simplicial set $K$ the induced morphism $f_{*}:\mathrm{Core}(\mathrm{SSet}(K,C))\to \mathrm{Core}(\mathrm{SSet}(K,D))$ on the maximal Kan complexes is a weak categorical equivalence.

Proof

This is lemma 3.1.3.2 in HTT.