homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
basic constructions:
strong axioms
further
The classical Whitehead theorem asserts that
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:
In the (∞,1)-category ∞ Grpd every weak homotopy equivalence is a homotopy equivalence.
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 for all closed subgroups (e.g. Greenlees-May 95, theorem 2.4). See at equivariant Whitehead theorem.
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.
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 $n$.
Discussion for equivariant homotopy theory includes
The $(\infty,1)$-topos version is in section 6.5 of
A formalization in homotopy type theory written in Agda is in