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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 7th 2018
• (edited Jul 7th 2018)

It’s still not quite right, is it? (here) After

Moreover, up to equivalence, every Grothendieck topos arises this way:

isn’t there the clause of accessible embedding missing? I.e. instead of

the equivalence classes of left exact reflective subcategories $\mathcal{E} \hookrightarrow PSh(\mathcal{C})$ of the category of presheaves

it should have

the equivalence classes of left exact reflective and accessivley embedded subcategories $\mathcal{E} \hookrightarrow PSh(\mathcal{C})$ of the category of presheaves

Or else, by the prop that follows, it should say

the equivalence classes of left exact reflective and locally presentable subcategories $\mathcal{E} \hookrightarrow PSh(\mathcal{C})$ of the category of presheaves

No?

(This is just a question. I didn’t make an edit. Yet.)

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJul 7th 2018
• (edited Jul 7th 2018)

It looks right to me as is. There should be an essential equivalence between left exact idempotent monads on a topos and Lawvere-Tierney topologies $j: \Omega \to \Omega$, and if one has a Lawvere-Tierney topology in a presheaf topos, one ought to be recover a Grothendieck topology whose sheaves correspond to the algebras of the monad or objects orthogonal to $j$-dense monomorphisms. These statements do not bring in accessibility conditions; I’d think accessibility of the subtopos embedding should be a consequence however.

• CommentRowNumber3.
• CommentAuthorTodd_Trimble
• CommentTimeJul 7th 2018

For a small category $C$, the assignment $J \mapsto Sh(C, J)$ is a bijection from the set of Grothendieck coverages on $C$ to the class of reflective subcategories of $[C^{op}, Set]$ with cartesian reflector.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJul 7th 2018

Okay, thanks!!

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJul 7th 2018

added the additional pointer to Johnstone that Todd kindly provided. Also highlighted a bit more the remark (here) about the need to require accessibility (only) in the generality of $\infty$-toposes, lest I forget about this once again next time.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeJul 7th 2018

Yes, this is an interesting quirk. Is it known that accessibility is definitely necessary for $\infty$-toposes? I.e. is there an example of a non-accessible left exact reflective subcategory of an $\infty$-topos?