Paths and cylinders
(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
The classical Whitehead theorem asserts that
(See also the discussion at m-cofibrant space).
Using the homotopy hypothesis-theorem this may be reformulated:
In general -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 -topos Top is presented by the model structure on simplicial presheaves on the point, namely the model structure on simplicial sets.
The -topos version is in section 6.5 of
A formalization in homotopy type theory written in Agda is in
Revised on March 3, 2013 04:22:49
by Urs Schreiber