Let $M$ and $N$ be topological spaces. Let $\operatorname{Sh}(M)$ denote the presentable $\infty$-category of space-valued sheaves on $M$. It seems to me that the equivalence $$\operatorname{Sh}(M) \otimes \operatorname{Sh}(N) = \operatorname{Sh}(M \times N)$$ where $\otimes$ is the standard symmetric monoidal structure of $\operatorname{Pr^L}$, the $\infty$-category of presentable categories with left adjoints, is well-known. However, I cannot find a proof in the literature. My question is if there is a standard reference for this statement? Or if there is a "simple proof" of it by assuming some well-known results? By the way, I believe that I was once told that it's in Lurie but I've failed to find it in either Higher Topos Theory or Higher Algebra.
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2$\begingroup$ This doesn’t sound right. The construction factors through locales, but the forgetful functor from topological spaces to locales doesn’t preserve products. However, the functor from locales to toposes preserves finite products, if I recall correctly. $\endgroup$– Zhen LinCommented Aug 13, 2021 at 13:26
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2$\begingroup$ @ZhenLin: This was just discussed here: mathoverflow.net/questions/401157/… $\endgroup$– Dmitri PavlovCommented Aug 13, 2021 at 15:07
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Provided at least one of $M$ and $N$ is locally compact, the $\infty$-topos $\mathrm{Sh}(M \times N)$ is the product of $\mathrm{Sh}(M)$ and $\mathrm{Sh}(N)$ in $\mathrm{RTop}$. This is HTT 7.3.1.11.
Products in $\mathrm{RTop}$ can be computed as tensor products in $\mathrm{Pr^L}$. This is HA Example 4.8.1.19.
HA doesn't include a complete proof of the latter, but the case where one of the factors is of the form $\mathrm{Sh}(M)$ is essentially HTT 7.3.3.9, up to the matter of identifying $\mathrm{Sh}(M; \mathrm{Sh}(N))$ with the tensor product $\mathrm{Sh}(M) \otimes \mathrm{Sh}(N)$, for which HA 4.8.1.17 should be useful.
answered Aug 13, 2021 at 13:58