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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?

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    $\begingroup$ Do you really mean surjections into $\mathcal E$? Or surjections out of $\mathcal E$? $\endgroup$ Nov 9, 2014 at 6:00
  • $\begingroup$ You're right. I should be asking about surjections out of $\mathcal{E}$. Even an ionad is a surjection out of $Set^X$. $\endgroup$ Nov 9, 2014 at 7:41
  • $\begingroup$ I see. Well, by analogy with the "toy case" this must be essentially more difficult. Subsets of a set form a Boolean algebra, while the lattice of quotient sets is much more nasty. It is also very special in a sense, but not even modular (in most cases). $\endgroup$ Nov 9, 2014 at 7:52
  • $\begingroup$ Another complication stems from the fact that while subtoposes of $\mathcal E$ are defined over $\mathcal E$, quotient toposes of $\mathcal E$ are defined "under $\mathcal E$", in a sense $\mathcal E$ "does not see them". Another related thing is that surjections are not necessarily quotients, so passing to the surjective image you not only "make $\mathcal E$ smaller" but also endow it with additional structure. $\endgroup$ Nov 9, 2014 at 7:56
  • $\begingroup$ So, you are saying, since there is no device like a subobject classfier that classifies quotients, we cannot hope for an analogous result. $\endgroup$ Nov 9, 2014 at 8:19

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