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James Deikun (Jan 16 2024 at 18:43):Okay, so here is a proof that the simplex category has no nontrivial Grothendieck topologies. Subtoposes of a presheaf topos correspond directly to Grothendieck topologies. So, the [[lattice of subtoposes]] of the topos of simplicial sets consists of bottom and a single atom.
Atoms of the lattice of subtoposes correspond to two-valued Boolean subtoposes. So the topos of simplicial sets is two-valued and Boolean, and in particular is Boolean. In a [[Boolean topos]], the subobject classifier in .
But the subobject classifier in simplicial sets is manifestly not isomorphic to . If it were, there would be exactly 2 sieves on each representable, while it is easy to exhibit 3 of them, for example, for the 1-simplex.
Mike Shulman (Jan 16 2024 at 18:52):The topos of simplicial sets does have plenty of nontrivial subtoposes, such as -coskeletal simplicial sets for all . So there must be something wrong with the MSE proof you linked to.
James Deikun (Jan 16 2024 at 19:36):Ah, yeah, now that I remember the Boolean presheaf toposes are exactly the ones on groupoids, aren't they? So simplicial sets even has dense subtoposes. Which presheaf toposes are two-valued again? I know they are when the category is a monoid but that's only a sufficient condition ...
Mike Shulman (Jan 16 2024 at 20:24):Hm, global subterminal objects in a presheaf topos are sub-presheaves of the terminal presheaf, which are sieves in the domain category. I think a category has no nonempty proper sieves just when it is "strongly connected" in that for any two objects and there is an arrow .