category theory

# Contents

## Idea

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

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

• the corresponding homotopy category, which is the universal solution to turning all weak equivalences into isomorphisms;

• the corresponding $(\infty,1)$-category, which is, roughly, the universal solution to turning all weak equivalences into higher categorical equivalences. There are various versions of this construction depending on what model for $(\infty,1)$-categories is chosen.

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)