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Simon Wadsley
Simon Wadsley
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Recall that $\mathrm{Pro}(\mathbf{FinSet}) = *_{\text{proét}}$, the category of profinite sets, forms a site with finite jointly surjective families as covers, and that the category of sheaves on this set is the category $\mathbf{Cond}(\mathbf{Set}) = \mathbf{Sh}(\mathrm{Pro}(\mathbf{FinSet}))$ of condensed sets. One expects this to be a topos, but it is not because of size issues (the site of profinite sets is too large).

Clausen and Scholze proposed in 2023 to replace the site of profinite sets with that of light profinite sets (that of contablecountable formal cofiltered limits of finite sets), which resolves the size issues and makes the the category of light condensed sets a topos.

Now, every (Grothendieck) topos is the classifying topos of some geometric theory, and more precisely that of cover-preserving flat functors out of its site of definition. What does the topos light condensed sets classify? Equivalently, what are the cover-preserving functors out of the site of (light) profinite sets?

This question could be trivial, but I currently just can't wrap my head around it.

But, even more generally, what does the pro-étale topos of a scheme classify?

Recall that $\mathrm{Pro}(\mathbf{FinSet}) = *_{\text{proét}}$, the category of profinite sets, forms a site with finite jointly surjective families as covers, and that the category of sheaves on this set is the category $\mathbf{Cond}(\mathbf{Set}) = \mathbf{Sh}(\mathrm{Pro}(\mathbf{FinSet}))$ of condensed sets. One expects this to be a topos, but it is not because of size issues (the site of profinite sets is too large).

Clausen and Scholze proposed in 2023 to replace the site of profinite sets with that of light profinite sets (that of contable formal cofiltered limits of finite sets), which resolves the size issues and makes the the category of light condensed sets a topos.

Now, every (Grothendieck) topos is the classifying topos of some geometric theory, and more precisely that of cover-preserving flat functors out of its site of definition. What does the topos light condensed sets classify? Equivalently, what are the cover-preserving functors out of the site of (light) profinite sets?

This question could be trivial, but I currently just can't wrap my head around it.

But, even more generally, what does the pro-étale topos of a scheme classify?

Recall that $\mathrm{Pro}(\mathbf{FinSet}) = *_{\text{proét}}$, the category of profinite sets, forms a site with finite jointly surjective families as covers, and that the category of sheaves on this set is the category $\mathbf{Cond}(\mathbf{Set}) = \mathbf{Sh}(\mathrm{Pro}(\mathbf{FinSet}))$ of condensed sets. One expects this to be a topos, but it is not because of size issues (the site of profinite sets is too large).

Clausen and Scholze proposed in 2023 to replace the site of profinite sets with that of light profinite sets (that of countable formal cofiltered limits of finite sets), which resolves the size issues and makes the the category of light condensed sets a topos.

Now, every (Grothendieck) topos is the classifying topos of some geometric theory, and more precisely that of cover-preserving flat functors out of its site of definition. What does the topos light condensed sets classify? Equivalently, what are the cover-preserving functors out of the site of (light) profinite sets?

This question could be trivial, but I currently just can't wrap my head around it.

But, even more generally, what does the pro-étale topos of a scheme classify?

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xuq01
xuq01
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What does the topos of (light) condensed sets classify?

Recall that $\mathrm{Pro}(\mathbf{FinSet}) = *_{\text{proét}}$, the category of profinite sets, forms a site with finite jointly surjective families as covers, and that the category of sheaves on this set is the category $\mathbf{Cond}(\mathbf{Set}) = \mathbf{Sh}(\mathrm{Pro}(\mathbf{FinSet}))$ of condensed sets. One expects this to be a topos, but it is not because of size issues (the site of profinite sets is too large).

Clausen and Scholze proposed in 2023 to replace the site of profinite sets with that of light profinite sets (that of contable formal cofiltered limits of finite sets), which resolves the size issues and makes the the category of light condensed sets a topos.

Now, every (Grothendieck) topos is the classifying topos of some geometric theory, and more precisely that of cover-preserving flat functors out of its site of definition. What does the topos light condensed sets classify? Equivalently, what are the cover-preserving functors out of the site of (light) profinite sets?

This question could be trivial, but I currently just can't wrap my head around it.

But, even more generally, what does the pro-étale topos of a scheme classify?