nLab weak equivalence

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.

  • The Dwyer-Kan localization uses simplicially enriched categories to model $(\infty,1)$-categories.

  • If we use complete Segal spaces or quasicategories to model $(\infty,1)$-categories, then the construction is a version of fibrant replacement.

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

• A weak homotopy equivalence is a weak equivalence in the classical model structure on topological spaces.

• A simplicial weak equivalence is a weak equivalence in the classical model structure on simplicial sets.

• An equivariant weak homotopy equivalence is a weak equivalence in the fine model structure on topological G-spaces.

• A weak equivalence in the folk model structure on Cat is a functor that is fully faithful and essentially surjective.

• In homotopy type theory, a functor $F: A \to B$ between internal categories in HoTT $A$ and $B$ is called a “weak equivalence” if it is fully faithful and essentially surjective.

  For univalent categories there is no difference between weak equivalences and equivalences.

Related concepts

• equality

• isomorphism

• equivalence

• weak equivalence

• homotopy equivalence, weak homotopy equivalence

• equivalence in an (∞,1)-category

• equivalence of (∞,1)-categories