nLab Whitehead theorem



Homotopy theory

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Classical case

The classical Whitehead theorem asserts that

Theorem (Whitehead)

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:

Simplicial version


A simplicial map f:XYf\colon X\to Y between Kan complexes is a simplicial homotopy equivalence if and only if for any aa and bb that make the following square commute

Δ n a X ι d f Δ n b Y \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 dd that makes the upper triangle commutative and the lower triangle commutative up to a homotopy h:Δ 1×Δ nYh\colon \Delta^1\times\Delta^n\to Y that is constant on the boundary Δ 1×Δ n\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.


In the above criterion, the boundary inclusion

Δ nΔ n\partial\Delta^n\to\Delta^n

can be replaced by any weakly equivalent cofibration.


If XX or YY is not a Kan complex, one can formulate a similar criterion using barycentric subdivisions of Δ n\partial\Delta^n and Δ n\Delta^n. A simplicial map f:XYf\colon X\to Y between simplicial sets is a weak homotopy equivalence if and only if for any k0k\ge0 and for any aa and bb that make the following square commute

Sd kΔ n a X Sd kι d f Sd kΔ n b Y \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 lkl\ge k such that in the outer rectangle in the diagram

Sd lΔ n Sd kΔ n a X Sd lι Sd kι f Sd lΔ n Sd kΔ n b Y \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:Sd lΔ nXd\colon Sd^l \Delta^n \to X

that makes the upper triangle in the diagram

Sd lΔ n a X Sd lι d f Sd lΔ n b Y \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:Sd l(Δ 1×Δ n)Yh\colon Sd^l(\Delta^1\times\Delta^n)\to Y

that is constant on the boundary Sd l(Δ 1×Δ n)Sd^l(\Delta^1\times\partial\Delta^n).

Equivariant version

In GG-equivariant homotopy theory the statement is that GG-homotopy equivalences between G-CW complexes are equivalent to maps that are weak homotopy equivalences on fixed point spaces H HH^H for all closed subgroups HGH \subset G (e.g. Greenlees-May 95, theorem 2.4). See at equivariant Whitehead theorem.

In general (,1)(\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 (,1)(\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 nn; this is provable in homotopy type theory by induction on nn.


Originally proved by J. H. C. Whitehead:

  • J. H. C. Whitehead, Combinatorial homotopy. I, Bulletin of the American Mathematical Society 55:3 (1949), 213–246. doi.

The simplicial version is due to Daniel M. Kan, see Theorem 7.2 in

  • Daniel M. Kan, On c.s.s. categories, Boletín de la Sociedad Matemática Mexicana 2 (1957), 82–94. PDF.

Other sources:

Discussion for equivariant homotopy theory includes

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

The (,1)(\infty,1)-topos version is in section 6.5 of

A formalization in homotopy type theory written in Agda is in

Last revised on May 23, 2022 at 19:36:49. See the history of this page for a list of all contributions to it.