I have spelled out at homotopy dimension the proof that the $\infty$-presheaf topos on a site with terminal object has homotopy dimension $\leq 0$. And I have made explicit that the argument straightforwardly generalizes to imply that every local (infinity,1)-topos has homotopy dimension $\leq 0$.

This implies a whole chain of immediate pleasant results:

every local $\infty$-topos is

of cohomology dimension $\leq 0$

and hypercomplete;

hence

every cohesive (infinity,1)-topos is

of cohomology dimension $\leq 0$

and hypercomplete.

have created an entry homotopy dimension

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