In mathematics, especially algebraic topology, a weak equivalence between simplicial sets is a map between simplicial sets that is invertible in some weak sense. Formally, it is a weak equivalence in some model structure on the category of simplicial sets (so the meaning depends on a choice of a model structure.)
An ∞-category can be (and is usually today) defined as a simplicial set satisfying the weak Kan condition. Thus, the notion is especially relevant to higher category theory.
Equivalent conditions.
(Y, V) \to \operatorname{ho}\underline{\operatorname{Hom}}(X, V)</math> is an equivalence of categories for each ∞-category "V", where ho means the homotopy category of an ∞-category,
If formula_3 are ∞-categories, then a weak equivalence between them in the sense of Joyal is exactly an equivalence of ∞-categories (a map that is invertible in the homotopy category).
Let formula_4 be a functor between ∞-categories. Then we say
Then formula_5 is an equivalence if and only if it is fully faithful and essentially surjective.