+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Homotopy theory +-- {: .hide} [[!include homotopy - contents]] =-- #### (∞,1)-Topos theory +-- {: .hide} [[!include (infinity,1)-topos - contents]] =-- #### Foundations +-- {: .hide} [[!include foundations - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Classical case The classical **Whitehead theorem** asserts that: \begin{theorem} ([Whitehead 1949](#Whitehead49)) Every [[weak homotopy equivalence]] between [[CW-complexes]] is a [[homotopy equivalence]]. \end{theorem} (See also the discussion at *[[m-cofibrant space]]*). Using the [[homotopy hypothesis]]-theorem this may be reformulated: +-- {: .num_cor } ###### Corollary In the [[(∞,1)-category]] [[∞Grpd]] every [[weak homotopy equivalence]] is a [[homotopy equivalence]]. =-- ## Simplicial version \begin{theorem} A [[simplicial map]] $f\colon X\to Y$ between [[Kan complexes]] is a [[simplicial homotopy equivalence]] if and only if for any $a$ and $b$ that make the following square commute $$\array{ \partial\Delta^n&\stackrel{a}{\to}&X\\ \downarrow^\iota&{}^{\exists d}\nearrow&\downarrow^f\\ \Delta^n &\stackrel{b}{\to}&Y\\ } $$ there is a diagonal arrow $d$ that makes the upper triangle commutative and the lower triangle commutative up to a homotopy $h\colon \Delta^1\times\Delta^n\to Y$ that is constant on the boundary $\Delta^1\times\partial\Delta^n$. \end{theorem} Of course, this statement can be reformulated using homotopy groups like the version for topological spaces, but the above statement is more practical. \begin{remark} In the above criterion, the boundary inclusion $$\partial\Delta^n\to\Delta^n$$ can be replaced by any weakly equivalent [[cofibration]]. \end{remark} \begin{remark} If $X$ or $Y$ is not a [[Kan complex]], one can formulate a similar criterion using [[barycentric subdivisions]] of $\partial\Delta^n$ and $\Delta^n$. A [[simplicial map]] $f\colon X\to Y$ between [[simplicial sets]] is a [[weak homotopy equivalence]] if and only if for any $k\ge0$ and for any $a$ and $b$ that make the following square commute $$\array{ Sd^k \partial\Delta^n&\stackrel{a}{\to}&X\\ \downarrow^{Sd^k \iota}&{}^{\exists d}\nearrow&\downarrow^f\\ Sd^k \Delta^n &\stackrel{b}{\to}&Y\\ } $$ there is $l\ge k$ such that in the outer rectangle in the diagram $$\array{ Sd^l \partial\Delta^n&\to&Sd^k \partial\Delta^n&\stackrel{a}{\to}&X\\ \downarrow^{Sd^l \iota}&&\downarrow^{Sd^k \iota}&&\downarrow^f\\ Sd^l \Delta^n &\to&Sd^k \Delta^n &\stackrel{b}{\to}&Y\\ } $$ we can find a diagonal arrow $$d\colon Sd^l \Delta^n \to X$$ that makes the upper triangle in the diagram $$\array{ Sd^l \partial\Delta^n&\stackrel{a}{\to}&X\\ \downarrow^{Sd^l \iota}&{}^{\exists d}\nearrow&\downarrow^f\\ Sd^l \Delta^n &\stackrel{b}{\to}&Y\\ } $$ commutative and the lower triangle commutative up to a homotopy $$h\colon Sd^l(\Delta^1\times\Delta^n)\to Y$$ that is constant on the boundary $Sd^l(\Delta^1\times\partial\Delta^n)$. \end{remark} ## Equivariant version In $G$-[[equivariant homotopy theory]] the statement is that $G$-homotopy equivalences between [[G-CW complexes]] are equivalent to maps that are weak homotopy equivalences on [[fixed point]] spaces $H^H$ for all [[closed subset|closed]] [[subgroups]] $H \subset G$ (e.g. [Greenlees-May 95, theorem 2.4](#GreenleesMay95)). See at _[[equivariant Whitehead theorem]]_. ## In general $(\infty,1)$-toposes There is a notion of [[homotopy groups]] for objects in every [[∞-stack]] [[(∞,1)-topos]], as described at [[homotopy group (of an ∞-stack)]]. Accordingly, there is a notion of [[weak homotopy equivalence]] in every [[∞-stack]] [[(∞,1)-topos]] and hence an analog of the statement of Whiteheads theorem. One finds that **Warning** Whitehead's theorem **fails** for general [[(∞,1)-toposes]] and non-[[truncated object in an (infinity,1)-category|truncated]] objects. The [[∞-stack]] [[(∞,1)-topos]]es in which the Whitehead theorem does hold are the [[hypercomplete (∞,1)-topos]]es. These are precisely the ones that are [[presentable (∞,1)-category|presented]] by a local [[model structure on simplicial presheaves]]. For instance the hypercomplete $(\infty,1)$-topos [[Top]] is presented by the model structure on simplicial presheaves on the point, namely the [[model structure on simplicial sets]]. ## In homotopy type theory Since [[homotopy type theory]] admits models in [[(∞,1)-toposes]] (and in particular in non-hypercomplete ones), Whitehead's theorem is not provable when regarded as a statement about types in homotopy type theory. From this perspective, the truth of Whitehead's theorem is a [[foundational axiom]] that may be regarded as a "classicality" property, akin to [[excluded middle]] or the [[axiom of choice]] --- we call it **Whitehead's principle** (not to be confused with [[Whitehead's problem]], another statement that is independent of the usual axioms of set theory). Whitehead's principle does hold, however, for maps between [[homotopy n-types]] for any finite $n$; this is provable in homotopy type theory by [[induction]] on $n$. ## Related concepts * [[mod p Whitehead theorem]] * [[equivariant Whitehead theorem]] * [equivariant stable Whitehead theorem](G-spectrum#EquivariantWhitehead) * [[CW-approximation theorem]] * [[cellular approximation theorem]] ## References The original theorem for maps between [[CW complexes]]: * {#Whitehead49} [[J. H. C. Whitehead]], *Combinatorial homotopy. I*, Bulletin of the American Mathematical Society **55** 3 (1949) 213–245 [[doi:10.1090/s0002-9904-1949-09175-9](http://dx.doi.org/10.1090/s0002-9904-1949-09175-9), [pdf](https://www.ams.org/journals/bull/1949-55-03/S0002-9904-1949-09175-9/S0002-9904-1949-09175-9.pdf)] The version in [[simplicial homotopy theory]] is due to: * [[Daniel M. Kan]], Theorem 7.2 in: *On c.s.s. categories*, Boletín de la Sociedad Matemática Mexicana **2** (1957) 82–94 [<a href="https://www.boletin.math.org.mx/pdf/2/2/BSMM(2).2.82-94.pdf">pdf</a>, [pdf](https://dmitripavlov.org/scans/kan-on-css-categories.pdf)] Review: * {#Bredon93} [[Glen Bredon]], Cor. 11.14 in: _Topology and Geometry_, Graduate Texts in Mathematics **139**, Springer (1993) [[doi:10.1007/978-1-4757-6848-0](https://link.springer.com/book/10.1007/978-1-4757-6848-0), [pdf](http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf)] * {#ElmendorfKrizMay95} [[Anthony Elmendorf]], [[Igor Kriz]], [[Peter May]], section 1 of: _[[Modern foundations for stable homotopy theory]]_, in [[Ioan Mackenzie James]] (ed.), _[[Handbook of Algebraic Topology]]_ (1995) 213-253 [[pdf](http://hopf.math.purdue.edu/Elmendorf-Kriz-May/modern_foundations.pdf)] * {#Hatcher02} [[Allen Hatcher]], pp. 346 in: *Algebraic Topology*, Cambridge University Press (2002) \[<a href="https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401">ISBN:9780521795401</a>, [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)\] * [[Marcelo Aguilar]], [[Samuel Gitler]], [[Carlos Prieto]], Thm. 6.3.31 in: *Algebraic topology from a homotopical viewpoint*, Springer (2008) [[doi:10.1007/b97586](https://link.springer.com/book/10.1007/b97586)] Discussion of the [[equivariant Whitehead theorem]]: * {#GreenleesMay95} [[John Greenlees]], [[Peter May]]: _Equivariant stable homotopy theory_, in I.M. James (ed.), _Handbook of Algebraic Topology_ (1995) 279-325 [[pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf)] The [[(infinity,1)-topos|$(\infty,1)$-topos]] theoretic version: * [[Jacob Lurie]], section 6.5 of [[Higher Topos Theory]] The analogous formulation in [[homotopy type theory]]: * {#UFP13} [[Univalent Foundations Project]], §8.8 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)] Corresponding formalization in [[Agda]]: * [[Dan Licata]], _[Whitehead.agda](https://github.com/dlicata335/hott-agda/blob/master/homotopy/Whitehead.agda)_ category: foundational axiom [[!redirects Whitehead's theorem]] [[!redirects Whitehead's principle]] [[!redirects Whitehead principle]] [[!redirects simplicial Whitehead theorem]]