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:

Corollary

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

Whitehead theorem in general $(\infty,1)$-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?.

References

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

• Jacob Lurie?, Higher Topos Theory?

---

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

nLab page on Whitehead theorem