nLab
weak equivalence

Context

Equality and Equivalence

Category theory

Contents

Idea

A weak equivalence is a morphism in a category CC which is supposed to be a true equivalence in a higher categorical refinement of CC.

The bare minimum of axioms to be satisfied by a weak equivalence are encoded in the concepts of category with weak equivalences and homotopical category. For such categories one can consider

Often, categories having weak equivalences also have extra structure that makes them easier to work with. A very powerful, and commonly occurring, level of such structure is called a model structure. There are also various weaker levels of structure, such as a category of fibrant objects.

Examples

Revised on September 17, 2012 23:55:00 by Urs Schreiber (82.169.65.155)