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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 9th 2012
   • (edited Mar 9th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am wondering about the following question, concerning _[[2-sheaves]]_: Let $\mathcal{B}$ be a sheaf-topos. By [Bunge-Pare, cor 2.6](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1979__20_4/CTGDC_1979__20_4_373_0/CTGDC_1979__20_4_373_0.pdf#page=19) 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 $Topos \to Cat$?

   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 $Topos \to Cat$?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMar 9th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexYes, 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 9th 2012
   • (edited Mar 9th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks! > 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...)

   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…)

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeMar 12th 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItex> 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 $X$.

   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 $X$.