nLab Whitehead theorem

Classical case

The classical Whitehead theorem asserts that:

Theorem

(Whitehead 1949) Every weak homotopy equivalence between CW-complexes is a homotopy equivalence.

(See also the discussion at m-cofibrant space).

Using the homotopy hypothesis-theorem this may be reformulated:

Corollary

In the (∞,1)-category ∞Grpd every weak homotopy equivalence is a homotopy equivalence.

Simplicial version

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 $\Delta^1\times\partial\Delta^n$.

Of course, this statement can be reformulated using homotopy groups like the version for topological spaces, but the above statement is more practical.

Remark

In the above criterion, the boundary inclusion

$$\partial\Delta^n\to\Delta^n$$

can be replaced by any weakly equivalent cofibration.

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 $Sd^l(\Delta^1\times\partial\Delta^n)$.

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 subgroups $H \subset G$ (e.g. Greenlees-May 95, theorem 2.4). 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 objects.

The ∞-stack (∞,1)-toposes in which the Whitehead theorem does hold are the hypercomplete (∞,1)-toposes. These are precisely the ones that are 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

• CW-approximation theorem

• cellular approximation theorem

References

The original theorem for maps between CW complexes:

• 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, 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 [pdf, pdf]

Review:

• Glen Bredon, Cor. 11.14 in: Topology and Geometry, Graduate texts in mathematics 139, Springer 1993 (doi:10.1007/978-1-4757-6848-0, pdf)

• 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 Amsterdam: North-Holland, pp. 213–253, (pdf)

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Thm. 6.3.31 in: Algebraic topology from a homotopical viewpoint, Springer (2008) (doi:10.1007/b97586)

Discussion of the equivariant Whitehead theorem:

• John Greenlees, Peter May, Equivariant stable homotopy theory, in I.M. James (ed.), Handbook of Algebraic Topology , pp. 279-325. 1995. (pdf)

The $(\infty,1)$-topos theoretic version:

• Jacob Lurie, section 6.5 of Higher Topos Theory

The analogous formulation in homotopy type theory:

• Univalent Foundations Project, §8.8 in: Homotopy Type Theory – Univalent Foundations of Mathematics (2013) [web, pdf]

Corresponding formalization in Agda:

• Dan Licata, Whitehead.agda

category: foundational axiom