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

Theorem—^[1] Let $f:X\to Y$ be a map between simplicial sets. Then the following are equivalent:

$f$ is a weak equivalence in the sense of Joyal (Joyal model category structure).
$f^{*}:\operatorname {ho} {\underline {\operatorname {Hom} }}(Y,V)\to \operatorname {ho} {\underline {\operatorname {Hom} }}(X,V)$ is an equivalence of categories for each ∞-category V, where ho means the homotopy category of an ∞-category,
$f^{*}:{\underline {\operatorname {Hom} }}(Y,V)^{\simeq }\to {\underline {\operatorname {Hom} }}(X,V)^{\simeq }$ is a weak homotopy equivalence for each ∞-category V, where the superscript $\simeq$ means the core.

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).^[2]

Let $f:X\to Y$ be a functor between ∞-categories. Then we say

$f$ is fully faithful if $f:\operatorname {Map} (a,b)\to \operatorname {Map} (f(a),f(b))$ 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 $y\simeq f(a)$ .