Weak equivalence between simplicial sets

Short description: Concept in algebraic topology

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 —  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).
• $f^{*}:\mathrm{ho}\underset{\_}{\mathrm{Hom}}(Y,V)\to \mathrm{ho}\underset{\_}{\mathrm{Hom}}(X,V)$ is an equivalence of categories for each ∞-category V, where ho means the homotopy category of an ∞-category,
• $f^{*}:\underset{\_}{\mathrm{Hom}}(Y,V{)}^{\simeq }\to \underset{\_}{\mathrm{Hom}}(X,V{)}^{\simeq }$ 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).

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

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

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

References

• Cisinski, Denis-Charles (2023) (in en). Higher Categories and Homotopical Algebra. Cambridge University Press. ISBN 978-1108473200. https://cisinski.app.uni-regensburg.de/CatLR.pdf. 
• Quillen, Daniel G. (1967), Homotopical algebra, Lecture Notes in Mathematics, No. 43, 43, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0097438, ISBN 978-3-540-03914-3 
• Rezk, Charles (2022). "Introduction to quasicategories". https://ncatlab.org/nlab/files/Rezk-IntroToQuasicategories.pdf. 

Further reading

• equivalence of (infinity,1)-categories in nLab
• Kelly, Shane. "Derived Algebraic Geometry (Lecture 3: Categories), UTokyo Spring Semester 2025". https://www.ms.u-tokyo.ac.jp/~kelly/Course2025DAG/2025DAG03.pdf. 
• Casacuberta, Carles. "Quasicategories, 12 November 2018". https://www.ub.edu/topologia/seminars/Quasicategories.pdf. 
• "T4.6.2 Fully Faithful and Essentially Surjective Functors". https://kerodon.net/tag/01JG. 
• Haugseng, Rune (2017). "Introduction to ∞-Categories". https://runegha.folk.ntnu.no/notes.html.