Weak equivalence between simplicial sets
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
An ∞-category can be defined as a simplicial set satisfying the weak Kan condition. Thus, the notion is especially relevant to higher category theory.
Equivalent conditions
If are ∞-categories, then a weak equivalence between them in the sense of Joyal is exactly an equivalence of ∞-categories.Let be a functor between ∞-categories. Then we say
- is fully faithful if is an equivalence of ∞-groupoids for each pair of objects.
- is essentially surjective if for each object in, there exists some object such that.