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Let $C, D$ two Grothendieck sites. Since the corresponding toposes $E, F$ are locally presentable categories, then (by Gabriel-Ulmer duality) they correspond to limit theories $X, Y$ (that is, small categories having all $\lambda$-small limits; here $\lambda$ is chosen larger of two). Now we can consider $X$-objects in $F$ or $Y$-objects in $E$. This gives the same locally-representable category $H$ (this is known as: the tensor product of limit theories is commutative).

  1. Is it true that this category $H$ is a Grothendieck topos? (I have almost no doubt about this)
  2. Is there a natural construction (in terms of sites) defining a site $C \otimes D$, the topos of sheaves over which is equivalent to $H$? It is natural to expect that as a category it would be $C \times D$ (then, in view of the currying, this could exactly be treated as $F$-valued sheaves on $C$ or $E$-valued sheaves on $D$). Is it possible to take just the direct product of the covering families in $C$ and $D$ as covering families?

Jonstone (in the $C$ part) defines a semidirect product of sites, but it talks about sites internal to the toposes. I'm not sure if this is related to my question yet.

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The category $H$ can be described as the category of $E$-valued sheaves on $D$, or $F$-valued sheaves on $C$.

You get a site by taking the category $C \times D$ and taking the topology generated by the $(c_i,d) \to (c,d)$ for $c_i \to c$ a covering family in $C$, and the $(c,d_i) \to (c,d)$ for $d_i \to d$ a covering family in $D$.

(or equivalently, the generators are the $\{(c_i,d_j) \to (c,d)\}_{i\in I,j \in J}$ for $c_i \to c$ and $d_j \to d$ covering families in $C$ and $D$ respectively - this is probably easier to check the claim above with this definition).

In particular, $H$ is a Grothendieck topos.

In terms of the construction of a semi-direct product site in the Elephant, this should corresponds to the case of a "constant" internal site.

Finally, this construction makes $H$ the product of $E$ and $F$ in the category of Grothendieck topos and geometric morphisms (in the geometric direction, so coproduct in the algebraic direction).

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  • $\begingroup$ Wow, this is the categorical product! Thank you so much! $\endgroup$ May 25, 2023 at 16:13
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    $\begingroup$ Yes! it is not to surprising if you are familiar with the analogies between commutative rings and Grothendieck toposes. This is directly analogous to the fact that coproduct in the category of commutative ring is given by the tensor product. $\endgroup$ May 25, 2023 at 19:04
  • $\begingroup$ Indeed, I've heard a bit about it, but don't know the details yet (I'll watch this lecture in the summer). So it sounds even more magical to me now... $\endgroup$ May 25, 2023 at 20:51

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