# Do topos-valued sheaves form a topos?

Let $$\bf C$$ be a category, $$\mathcal S$$ an (elementary) topos.

If $$\mathcal S$$ is a presheaf category over $$\bf D$$, then it's easy to see $$[\mathbf C^{\rm op},\, \mathcal{S}] \cong [(\mathbf C \times \mathbf D)^{\rm op},\, \mathcal{Sets}]$$ is still a topos. In more general situations I struggle to see an easy reason for it to be true as well. Thus:

When is the category $$[\mathbf C^{\rm op},\, \mathcal S]$$ of contravariant functors from $$\bf C$$ to $$\mathcal S$$ a topos?

• If $\mathcal{S}$ is a Grothendieck topos, or more generally has large enough disjoint universal coproduct you are always fine. But If $\mathcal{S}$ is an elementary topos in general it is not going to work, but I do not know if there are nice conditions under which it works. – Simon Henry Jan 16 '20 at 23:40
• If $\mathbf C$ were an internal category in $\mathcal S$, then the category of internal $\mathcal S$-valued presheaves on $\mathbf C$ would be a topos. (This is surely in Johnstone's "Topos Theory".) I think the point of @SimonHenry's comment is that good coproducts let you regard any genuine small category as an internal category in $\mathcal S$. – Andreas Blass Jan 17 '20 at 0:09
• That was indeed where I was going. But the question seem much more interesting that this observation: For example if the category C is a groupoid I think that S-valued presheaf will be an elementary topos without needing any assumption on S, so the general question definitely do not reduce to the case I was refering too. – Simon Henry Jan 17 '20 at 0:16
• The category of $G$-sets for a large group $G$ is a cocomplete elementary topos but is not a Grothendieck topos. [1] – Zhen Lin Jul 20 '20 at 3:33
• @ZhenLin good to see you again! Yes, that's definitely relevant, and that example generalises a lot to other special large limits of toposes. – theHigherGeometer Jul 20 '20 at 6:01

Rather than a full-fledged answer, this is a sketch of a plan:

As you yourself pointed out, if $$\mathcal S=[\bf D^{\rm op},\, \mathcal \cal{Sets}]$$, the result is trivial.

Now, suppose you consider an arbitrary elementary topos $$\mathcal S$$. The steps would be:

1. Use the representation theorem of Joyal-Tierney to embed your target topos as the equivariant sheaves over a localic topos, see here.

$$E:\mathcal S \to Set^{\mathcal{L}^{op}}$$

2. Now each map $$[\bf C^{\rm op},\, \mathcal S]$$ composes with the embedding $$E$$ and thus lands into a localic topos.

3. The last step would be to see how $$[\bf C^{\rm op},\, E(\mathcal S)]$$ sits inside $$[\bf C^{\rm op},\, Set^{\mathcal{L}^{op}}]$$.

Conjecture: under some some conditions (to be determined, but see this post When is a reflective subcategory of a topos a topos? ) there is a reflection which is sufficiently exact to ensure that it is indeed a subtopos of $$[(\bf C \times \bf\mathcal{L})^{\rm op},\, \cal{Sets}]$$