Hello!

Is there a simple 'topological' condition to detect whenever a morphism of locales $f : X \rightarrow Y$ induces a surjection of infinity-toposes $f : \mathrm{Sh}_{\infty}(X) \rightarrow \mathrm{Sh}_{\infty}(Y)$ (i.e. such that $f^*$ is conservative)?

It's not enough to assume that f is a surjection of locales: indeed, if we take a topological space $X$ such that $\mathrm{Sh}_{\infty}(X)$ is not hypercomplete, and $X^{\mathrm{disc}}$ is its space of points endowed with the discrete topology, then $\mathrm{Sh}_{\infty}(X^{\mathrm{disc}}) \rightarrow \mathrm{Sh}_\infty (X)$ can't be a surjection, because the pullback of an $\infty$-connected map in $\mathrm{Sh}_\infty (X)$ is a weak equivalence in $\mathrm{Sh}_{\infty}(X^{\mathrm{disc}})$...

Thank you!

finitelimits, but $f^*$ is still only assumed to preserve finite limits. $\endgroup$