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Let $\mathcal E=\mathsf{Sh}(\mathsf C,J)$ be a Grothendieck topos. Suppose, for some representable sheaf $\mathbf{ay}C$, there are $H,G\in \Omega ^{\mathbf{ay}C}$ such that $H\cup G=\mathbf{ay}C$. I am having some trouble interpreting this statement with sheaf semantics. Below is a quick interpretation which I am unable to follow. I tried to add the details of what I can understand, and would appreciate some help filling in the rest.

enter image description here

First of all, since the Yoneda embedding is dense and sheafification commutes with colimits, I think we can may always assume generalized elements are representables. Hence $H,G\in \Omega ^{\mathbf{ay}C}$ correspond to a pair of arrows $\mathbf{ay}D\rightrightarrows \Omega ^{\mathbf{ay}C}$ in $\mathcal E$. Then, by the universal property of $\Omega$ and Cartesian closedness this amounts to two subobjects $$H\rightarrowtail \mathbf{ay}C\times \mathbf{ay}D,\; G\rightarrowtail \mathbf{ay}C\times \mathbf{ay}D,$$ whose union is $\mathbf{ay}C\times \mathbf{ay}D$. The union of these subobjects is $\mathbf{ay}C\times \mathbf{ay}D$.

That's the first three sentences, but the fourth one I'm confused about. This is probably because I don't understand the internal logic, what how is "a cover $E_\alpha\to C\times D$ possible? These should be arrows in the site $\mathsf C$, but the Yoneda embedding does not commute with products, so $\mathbf{ay}C\times \mathbf{ay}D\ncong \mathbf{ay}(C\times D)$ (I don't know whether this is even relevant)... I am confused here.

I know from Maclane and Moerdijk that $C\models \phi(\alpha)\vee \psi(\alpha)\iff$ there's a covering $(f_i:C_i\to C)$ such that for each index, $C_i\models \phi(\alpha f_i)\vee C_i\models \phi(\alpha f_i)$, but am confused about applying this.

How to understand the excerpt in the image?

Added. This comes from page 149 of the paper Local Concepts in Synthetic Differential Geometry and Germ Representability by Bunge and Dubuc.

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  • $\begingroup$ $C$ should be kept general here, shouldn't it? Why are you considering only (the sheafification of) a representable $a(y(C))$? $\endgroup$
    – Andrej Bauer
    Commented Oct 21, 2016 at 11:04
  • $\begingroup$ @AndrejBauer it just happens that I'm only interested in internal binary covers of representables :) $\endgroup$
    – Arrow
    Commented Oct 21, 2016 at 11:09
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    $\begingroup$ Last I checked, the Yoneda embedding does commute with products. $\endgroup$
    – Mike Shulman
    Commented Oct 21, 2016 at 17:47
  • $\begingroup$ @MikeShulman sorry about that. Still though, I'm not really sure how that resolves my problem. Could you help me with the rest? $\endgroup$
    – Arrow
    Commented Oct 21, 2016 at 18:18
  • $\begingroup$ Are you sure about "These should be arrows in the site"? Since the passage you quoted seems to be about general C, not necessarily the associated sheaf of a representable presheaf, I'd be inclined to think all of C, D, H, G are objects in the sheaf topos, not in the underlying site. At some point, the author of that passage seems to retreat from the sheaf topos to the presheaf topos, and the stuff about covering should probably be understood in terms of presheaves. $H$ and $G$ cover $C$ as sheaves iff they locally cover it as presheaves. $\endgroup$
    – Andreas Blass
    Commented Oct 21, 2016 at 18:33

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