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 {\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). Let f : X → Y {\displaystyle f:X\to Y} be a functor between ∞-categories. Then we say
f {\displaystyle f} is fully faithful if f : Map ( a , b ) → Map ( f ( a ) , f ( b ) ) {\displaystyle f:\operatorname {Map} (a,b)\to \operatorname {Map} (f(a),f(b))} is an equivalence of ∞-groupoids for each pair of objects a , b {\displaystyle a,b} .
f {\displaystyle f} is essentially surjective if for each object y {\displaystyle y} in Y {\displaystyle Y} , there exists some object a {\displaystyle a} such that y ≃ f ( a ) {\displaystyle y\simeq f(a)} . Then f {\displaystyle f} is an equivalence if and only if it is fully faithful and essentially surjective.
Notes
References Cisinski, Denis-Charles (2023). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1108473200. Quillen, Daniel G. (1967), Homotopical algebra, Lecture Notes in Mathematics, No. 43, vol. 43, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0097438, ISBN 978-3-540-03914-3, MR 0223432 Rezk, Charles (2022). "Introduction to quasicategories" (PDF) – via ncatlab.org.
Further reading equivalence of (infinity,1)-categories at the nLab Kelly, Shane. "Derived Algebraic Geometry (Lecture 3: Categories), UTokyo Spring Semester 2025" (PDF). Casacuberta, Carles. "Quasicategories, 12 November 2018" (PDF). "T4.6.2 Fully Faithful and Essentially Surjective Functors". Kerodon. Haugseng, Rune (2017). "Introduction to ∞-Categories".