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 the standard model structure on the category of simplicial sets.

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.

References

• Cisinski, Denis-Charles (2023). Higher Categories and Homotopical Algebra (PDF) (in en%). Cambridge University Press. ISBN 978-1108473200.{{cite book}}: CS1 maint: unrecognized language (link)
• 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