nLab
equivalence of (infinity,1)-categories

Definition

An (∞,1)-functor between (∞,1)-categories is an equivalence in (∞,1)Cat precisely if it is an essentially surjective (∞,1)-functor and a full and faithful (∞,1)-functor.

When (∞,1)-categories are presented by quasi-categories, an equivalence between them is presented by a weak equivalence in the 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