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I am wondering about the following question, concerning 2-sheaves:
Let ℬ be a sheaf-topos. By Bunge-Pare, cor 2.6 every geometric morphism
f:ℰ→ℬinto ℬ is a 2-sheaf on ℬ equipped with the canonical topology, in the sense that the corresponding indexed category
𝒞/f*(−):ℬop→Catis 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 ℬ.
So we have this sequence of 2-functors
ℬ≃Sh1(ℬ)≃(Topos/ℬ)et↪Topos/ℬ↪Sh2(ℬ).Unless I am missing something.
Is the last inclusion that of 2-sheaves which factor through the forgetful functor Topos→Cat?
Yes, that’s right. Except there is a size switch: 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 Sh1(ℬ)→Sh2(ℬ) is not the obvious inclusion of 1-sheaves in 2-sheaves.
I don’t understand your last question.
Thanks!
In particular, the composite Sh1(ℬ)→Sh2(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∈ℬ to the functor A↦ℬ/X×A. So how should we think of this inclusion?
I don’t understand your last question.
For a 2-sheaf ℬop→Cat we can ask if it is equivalent to a composite ℬop→Topos→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, Sh1(B) is a “category of sets” and Sh2(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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