nLab hypercompletion

Idea

The hypercompletion (Lurie) or $t$-completion (Rezk, ToënVezzosi) of an (∞,1)-topos of (∞,1)-sheaves is a further localization/(∞,1)-sheafification which corresponds to retaining only those (∞,1)-sheaves which satisfy descent with respect to all hypercovers.

Definition

Definition

An (∞,1)-topos of (∞,1)-sheaves is a hypercomplete (∞,1)-topos if every $\infty$-connective morphism is an equivalence.

Remark

This may be read as saying that the Whitehead theorem is valid in the (∞,1)-topos.

Examples

• The $(\infty,1)$-category defined by the Joyal-Jardine local model structure on simplicial presheaves with weak equivalences those morphisms that induce isomorphisms on homotopy sheaves is the hypercompletion of the localization of simplicial presheaves at ordinary covers. For more details see descent for simplicial presheaves.

Related concepts

• topological localization
• cotopological localization

References

Section 10 of

• Charles Rezk, Toposes and homotopy toposes (pdf)

Section 6.5 of

• Jacob Lurie, Higher Topos Theory

• Bertrand Toën, Gabriele Vezzosi, (pdf)