(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
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.
An (∞,1)-topos of (∞,1)-sheaves is a hypercomplete (∞,1)-topos if every $\infty$-connective morphism is an equivalence.
This may be read as saying that the Whitehead theorem is valid in the (∞,1)-topos.
Section 10 of
Section 6.5 of