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This question on MO made me realize that I don’t even know the right definition of surjective geometric morphism for (∞,1)-toposes. For 1-toposes, it is equivalent for f* to be (1) comonadic, (2) conservative, and (3) faithful. Of course (2)⇔(3) doesn’t categorify (what does faithfulness even mean for (∞,1)-categories?). But I don’t see that (1)⇔(2) categorifies either, because the (∞,1)-comonadicity theorem (unlike the 1-categorical one) is not about finite limits, but f* is still only assumed to preserve finite limits.
Of course we want a (surjection, inclusion) factorization. For that, it seems almost certain that f* being conservative is the right definition, at least if we take the obvious definition of “inclusion” as f* being fully faithful. We can then produce the desired factorization of f:ℱ→ℰ by localizing ℰ at the class of morphisms inverted by f*.
But what happens if instead we construct the category of coalgebras of the comonad f*f*? Is the category of coalgebras of an (accessible) lex comonad on an (∞,1)-topos again an (∞,1)-topos? If so, what sort of factorization does this produce?
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