Weak equivalence between simplicial sets

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

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Theorem—^[1] Let $f:X\to Y$ {\displaystyle f:X\to Y} be a map between simplicial sets. Then the following are equivalent:

$f$ {\displaystyle f} is a weak equivalence in the sense of Joyal (Joyal model category structure).
$f^{*}:\operatorname {ho} {\underline {\operatorname {Hom} }}(Y,V)\to \operatorname {ho} {\underline {\operatorname {Hom} }}(X,V)$ {\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,
$f^{*}:{\underline {\operatorname {Hom} }}(Y,V)^{\simeq }\to {\underline {\operatorname {Hom} }}(X,V)^{\simeq }$ {\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 $\simeq$ {\displaystyle \simeq } means the core.

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).^[2]

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

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

Then $f$ {\displaystyle f} is an equivalence if and only if it is fully faithful and essentially surjective.^[3]^[4]^[5]^{[clarification needed ]}

Notes

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

References

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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.

Equivalent conditions

Notes

References

Further reading