Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
1 to 6 of 6
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.)
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.
See also Johnstone’s C.2.1.11:
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.
Okay, thanks!!
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?
1 to 6 of 6