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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?

David Roberts
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seldon
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    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
  • Can you elaborate/give references for these claims? – seldon Jan 16 '20 at 23:41
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    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 00:09
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    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 00:16
  • @SimonHenry presumably there's something happening with colimits indexed by each connected component of $\mathbf{C}$? – David Roberts Jul 19 '20 at 23:38
  • @DavidRoberts : Maybe, but that shouldn't be enough: $C$ conected isn't enough I think. – Simon Henry Jul 19 '20 at 23:47
  • @SimonHenry I mean that even if the category is connected and even skeletal, then colimits indexed by it require (if one is being naive) a coproduct indexed by the objects. Though I'm slightly worried about getting the full coqualiser diagram, so maybe needing the set of arrows of each connected component to be small enough to get coproducts (or maybe colimits) of that size in the topos ...? – David Roberts Jul 19 '20 at 23:55
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    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 03:33
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    @ZhenLin good to see you again! Yes, that's definitely relevant, and that example generalises a lot to other special large limits of toposes. – David Roberts Jul 20 '20 at 06:01
  • Right. I think it settles the groupoid case, anyway. For the category case, we probably want the category to be "locallly essentially small" in the sense that each slice is equivalent to a "small" category (relative to $\mathcal{S}$). Just setting up a context where we can compare "large" external categories and internal categories in $\mathcal{S}$ seems to be tedious but I think it can be done if $\mathcal{S}$ has pullback-stable unions of "large" disjoint families of subobjects – we use extensivity to replace morphisms $A \to \coprod_X 1$ with decompositions of $A$ indexed by $X$, etc. – Zhen Lin Jul 20 '20 at 12:00
  • (cont.) An example of a "locally essentially small" category in this sense would be the poclass of ordinal numbers $\mathbf{ON}$, and $[\mathbf{ON}^\mathrm{op}, \mathbf{Set}]$ is another example of a non-Grothendieck topos. – Zhen Lin Jul 20 '20 at 12:06

1 Answers1

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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 $[(\bf C \times \bf\mathcal{L})^{\rm op},\, \cal{Sets}]$

Mirco A. Mannucci
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