## Theorems

#### (∞,1)-Topos theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Classical case

The classical Whitehead theorem asserts that

Every weak homotopy equivalence between CW-complexes is a homotopy equivalence.

Using the homotopy hypothesis-theorem this may be reformulated:

## In general $\left(\infty ,1\right)$-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 $\left(\infty ,1\right)$-topos Top is presented by the model structure on simplicial presheaves on the point, namely the model structure on simplicial sets.

## References

The $\left(\infty ,1\right)$-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 (89.204.137.139)