nLab equivalence in an (infinity,1)-category

Context

$(\infty,1)$-Category theory

(∞,1)-category theory

Contents

Definition

For $C$ a quasi-category, a morphism $f : x \to y$ in $C$ (an edge in the underlying simplicial set) is an equivalence if its image in the homotopy category $Ho(C)$ is an isomorphism.

Equivalently, $f$ is an equivalence if it is the image of a functor of quasi-categories (i.e. a map of simplicial sets) out of the nerve $N(J)$, where $J$ is the interval groupoid. This is a quasi-categorical version of the general theorem-schema in higher category theory that any equivalence can be improved to an adjoint equivalence.

Revised on January 14, 2014 03:57:24 by Urs Schreiber (89.204.138.148)