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 (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

If 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 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 .

Then is an equivalence if and only if it is fully faithful and essentially surjective.

Notes

References

Further reading