[[!redirects weak equivalence]] [[!redirects essentially surjective]] ## Idea ## ## Definition ## A [[functor]] $F: A \to B$ is **essentially surjective** if for all $b:B$, there merely exists an $a:A$ such that $F a \cong b$. We say that $F$ is a **weak equivalence** if it is [[fully faithful]] and essentially surjective. For [[categories]] there is no difference between weak equivalences and [[equivalence of precategories|equivalences]]. ## Properties ## ## See also ## [[Category theory]] [[equivalence of precategories]] [[functor]] [[fully faithful functor]] ## References ## [[HoTT book]]