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A Grothendieck $0$-topos is the same as a frame (see here). Another relationship between Grothendieck toposes and frames is the following: whenever $\mathcal O$ is a frame, then $\mathrm{Sh}(\mathcal O)$, the category of sheaves on $\mathcal O$, is a Grothendieck topos.

A elementary $0$-topos is the same as a Heyting algebra (see here). Another relationship between elementary toposes and Heyting algebras is the following: if $\mathcal E$ is an elementary topos and $X\in \mathcal E$ is an object, then $\mathrm{Sub}_\mathcal E(X)$ is a Heyting algebra.

These facts suggest to me these question:

  1. If $H$ is a Heyting algebra, does $H$ induce an elementary topos (in the same way a frame $\mathcal O$ induces the Grothendieck topos $\mathrm{Sh}(\mathcal O)$)?

  2. If $\mathcal E$ is a Grothendieck topos and $X\in\mathcal E$ an object, is $\mathrm{Sub}_\mathcal E(X)$ a frame?

  3. Do these connections hold in general for (Grothendieck or elementary) $n$-toposes? That is, whenever $\mathcal E$ is a (Grothendieck or elementary) $n$-topos and $X\in\mathcal E$, then $\mathrm{Sub}_\mathcal E(X)$ is a (Grothendieck or elementary) $n-1$-topos, and whenever $\mathcal E$ is a (Grothendieck or elementary) $n$-topos, then there's an induced $n+1$-topos $\mathrm{Sh}(\mathcal E)$?

Shaun
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user978360
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    For 2: in a Grothendieck topos every subobject lattice is a complete Heyting algebra, so in particular it is a frame. – Mark Kamsma Oct 10 '21 at 16:06
  • Thanks, Mark Kamsma. – user978360 Oct 10 '21 at 19:10
  • By the way, a complete Heyting algebra is the same thing as a frame. :P – user978360 Oct 10 '21 at 19:37
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    Yes it is as a lattice, but the category of complete Heyting algebras is different from the category of frames. The implication operation should really not be considered a part of the structure of a frame (even if we can 'build' it). – Mark Kamsma Oct 10 '21 at 20:38
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    I know that. Morphisms don't matter in my question at all, so I'm wondering why you are pointing that out. – user978360 Oct 11 '21 at 12:50
  • Just because it is good to realise, and I did not know if you did. Also, it is relevant for other people that might find this question (as you can see by the upvotes, other people agree about the relevance). I could also imagine an answer being phrased in terms of functors between the category of Grothendieck/elementary toposes and the category of frames/complete Heyting algebras, in which case it could be relevant. – Mark Kamsma Oct 11 '21 at 14:57
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    For any Heyting algebra, finite sheaves on it form an elementary topos (w/o NNO). – Jonas Frey Oct 11 '21 at 19:14
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    Asked on mathoverflow: https://mathoverflow.net/questions/406060/connections-between-0-toposes-and-1-toposes-grothendieck-and-elementary – user978360 Oct 12 '21 at 18:08

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