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.

## Contents

* 1 Equivalent conditions
* 2 Notes
* 3 References
* 4 Further reading

## Equivalent conditions

Theorem -- [1] Let f:X->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*:ho⁡Hom_(Y,V)->ho⁡Hom_(X,V) is an equivalence of categories for each ∞-category V, where ho means the homotopy category of an ∞-category,
* f*:Hom_(Y,V)≃->Hom_(X,V)≃ is a weak homotopy equivalence for each ∞-category V, where the superscript ≃ 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->Y be a functor between ∞-categories. Then we say

* f is fully faithful if f:Map⁡(a,b)->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≃f(a).

Then f is an equivalence if and only if it is fully faithful and essentially surjective.[3][4][5][clarification needed]

## Notes

1. ↑ Cisinski 2023, Theorem 3.6.8.
2. ↑ Cisinski 2023, Corollary 3.6.6.
3. ↑ Cisinski 2023, Theorem 3.9.7.
4. ↑ Rezk 2022, 48.2. Theorem (Fundamental theorem of quasicategories).
5. ↑ 4.6.2 Fully Faithful and Essentially Surjective Functors in Kerodon, Theorem 4.6.2.21.

## 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.

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