nLab
equivalence in an (infinity,1)-category

Context

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

Equality and Equivalence

Contents

Definition

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

Equivalently, ff 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)N(J), where JJ 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)