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 ${\displaystyle f:X\to Y}$ be a map between simplicial sets. Then the following are equivalent:

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

If ${\displaystyle X,Y}$ 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 ${\displaystyle f:X\to Y}$ be a functor between ∞-categories. Then we say

• ${\displaystyle f}$ is fully faithful if ${\displaystyle f:\operatorname {Map} (a,b)\to \operatorname {Map} (f(a),f(b))}$ is an equivalence of ∞-groupoids for each pair of objects ${\displaystyle a,b}$.
• ${\displaystyle f}$ is essentially surjective if for each object ${\displaystyle y}$ in ${\displaystyle Y}$, there exists some object ${\displaystyle a}$ such that ${\displaystyle y\simeq f(a)}$.

Then ${\displaystyle f}$ is an equivalence if and only if it is fully faithful and essentially surjective.^{[3][clarification needed]}