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 $X, Y$ 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 $f : X \to Y$ be a functor between ∞-categories. Then we say
*$f$ is fully faithful if $f : \operatorname(a, b) \to \operatorname(f(a), f(b))$ is an equivalence of ∞-groupoids for each pair of objects $a, b$.
*$f$ is essentially surjective if for each object $y$ in $Y$, there exists some object $a$ such that $y \simeq f(a)$.
Then $f$ is an equivalence if and only if it is fully faithful and essentially surjective.

References

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Algebraic topology
Simplicial sets
Higher category theory