nLab
equivalence of quasi-categories

Definition

… quasi-category … Joyal’s model structure for quasi-categories …

Properties

Lemma

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

• On the underlying simplicial sets it is a weak categorical equivalence in Joyal’s model structure for quasi-categories.

• 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 equivalence of Kan complexes (a homotopy equivalence).

Proof

This is HTT, lemma 3.1.3.2.

Related concepts

• equality

• isomorphism

• equivalence

• weak equivalence

• homotopy equivalence, weak homotopy equivalence

• homotopy equivalence of toposes

• equivalence in an (∞,1)-category

• equivalence of categories, adjoint functor

• equivalence of 2-categories, 2-adjunction

• equivalence of (∞,1)-categories, adjoint (∞,1)-functor