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I am wondering about the following question, concerning 2-sheaves:
Let $\mathcal{B}$ be a sheaf-topos. By Bunge-Pare, cor 2.6 every geometric morphism
$f : \mathcal{E} \to \mathcal{B}$into $\mathcal{B}$ is a 2-sheaf on $\mathcal{B}$ equipped with the canonical topology, in the sense that the corresponding indexed category
$\mathcal{C}_{/f^*(-)} : \mathcal{B}^{op} \to Cat$is a 2-sheaf.
Moreover, by inspection and by a standard fact, if $f$ is an etale geometric morphism, then this 2-sheaf is actually a 1-sheaf, equivalently: is represented by an object of $\mathcal{B}$.
So we have this sequence of 2-functors
$\mathcal{B} \simeq Sh_{1}(\mathcal{B}) \simeq (Topos_{/\mathcal{B}})_{et} \hookrightarrow Topos_{/\mathcal{B}} \hookrightarrow Sh_2(\mathcal{B}) \,.$Unless I am missing something.
Is the last inclusion that of 2-sheaves which factor through the forgetful functor $Topos \to Cat$?
Yes, that’s right. Except there is a size switch: $\mathcal{B}$ is equivalent to the category of small 1-sheaves on itself, whereas the indexed category corresponding to a geometric morphism is a large 2-sheaf. In particular, the composite $Sh_1(\mathcal{B}) \to Sh_2(\mathcal{B})$ is not the obvious inclusion of 1-sheaves in 2-sheaves.
I don’t understand your last question.
Thanks!
In particular, the composite $Sh_1(\mathcal{B}) \to Sh_2(\mathcal{B})$ is not the obvious inclusion of 1-sheaves in 2-sheaves.
I noticed this while running to the train. It sends an object $X \in \mathcal{B}$ to the functor $A \mapsto \mathcal{B}_{/ X \times A}$. So how should we think of this inclusion?
I don’t understand your last question.
For a 2-sheaf $\mathcal{B}^{op} \to Cat$ we can ask if it is equivalent to a composite $\mathcal{B}^{op} \to Topos \to Cat$.
I was wondering if those that have this factorization are those in the image of the last inclusion.
(I should be able to figure that out now. Currently I am throwing out these questions while travelling…)
So how should we think of this inclusion?
I would say, $Sh_1(B)$ is a “category of sets” and $Sh_2(B)$ is a “2-category of categories”, and this functor sends each “set” $X$ to the “overcategory” of “sets” over $X$.
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