nLab
equivalence of (infinity,1)-categories

Context

(,1)(\infty,1)-Category theory

Equality and Equivalence

Contents

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:CDf : C \to D is an equivalence in (∞,1)Cat if the following equivalent conditions hold

Proof

This is HTT, lemma 3.1.3.2.

Revised on October 3, 2012 10:39:39 by Urs Schreiber (82.169.65.155)