nLab
hypercompletion

Idea

The hypercompletion (Lu) or $t$-completion (ToVe) 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 p. 530 of HTT

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

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

Remarks

Notice that in section 6.5.4 of Higher Topos Theory it is argued that it may be more natural not to pass to the hypercompletion of a given $(\mathrm{\infty }\mathrm{,1})$-topos.

Examples

• The $(\mathrm{\infty }\mathrm{,1})$-category defined by the model structure on simplicial presheaves is the hypercompletion of the localization of simplicial presheaves at ordinary covers. For more details see descent for simplicial presheaves.