Schreiber Whitehead theorem

see [[nLab:Whitehead theorem]]


Classical Whitehead theorem

The classical Whitehead theorem asserts that

Theorem (Whitehead)

Every weak homotopy equivalence? between CW-complex?es is a homotopy equivalence?.

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

Using the homotopy hypothesis?-theorem this may be reformulated:


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

Whitehead theorem in general (,1)(\infty,1)-toposes

There is a notion of homotopy group?s 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)-topos?es.

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 presnted? by a local model structure on simplicial presheaves?.


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

  • Jacob Lurie?, Higher Topos Theory?

For two fibrant simplicial sets X,YX, Y, if a map f:XYf: X \to Y is a weak equivalence, then it is a homotopy equivalence.

nLab page on Whitehead theorem

Created on October 27, 2009 at 14:00:48. See the history of this page for a list of all contributions to it.