A geometric embedding into a Grothendieck topos can be characterised by giving the Lawvere-Tierney topology that induces it. This lets us reduce questions about subtoposes to more elementary order-theoretic questions about sublocales (of the subobject classifier).
What prevents us from applying the same analogy to geometric surjections? Is there a more elementary description of the collection of geometric surjections into out of a Grothendieck topos?
Starting from a discrete topos, i.e. a topos $\mathcal{E}$ of the form $Set^X$ for some set $X$, one can identify surjections out of $\mathcal{E}$ with exact comonads on $\mathcal{E}$ as one does in Richard Garner's ionads. But even ignoring the requirement of enough points for a moment, still the question remains, does the collection of such comonads have a more elementary order-theoretic description?
