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1. • CommentRowNumber1.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeMar 12th 2017
   • (edited Mar 12th 2017)
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   Author: IngoBlechschmidt
   Format: MarkdownItexIt's well known that the category of points of the presheaf topos over $Ring_{fp}^{op}$, the dual of the category of finitely presented rings, is the category of all rings (without a size or presentation restriction). In fact this holds for any algebraic theory, not only for the theory of commutative rings. One can learn about this in our entries on _[[Gabriel-Ulmer duality]]_, _[[flat functors]]_, and Moerdijk/Mac Lane. But what if we don't restrict the site to consist only of the [[compact objects]]? What are the points of the presheaf topos over the large category $Ring^{op}$, to the extent that the question is meaningful because of size-related issues? What are the points of the presheaf topos over $Ring_{\kappa}^{op}$, the dual of the category of rings admitting a presentation by $\lt \kappa$ many generators and relations, where $\kappa$ is a regular cardinal? (The category $Ring_{\kappa}^{op}$ is essentially small, so the question is definitely meaningful.) The question can be rephrased in the following way: What is an explicit description of the category of *finite* limit preserving functors $F : Ring_{\kappa}^{op} \to Set$? Any such functor gives rise to a ring by considering $F(\mathbb{Z}[X])$, but unlike in the case $\kappa = \aleph_0$ such a functor is not determined by this ring. This feels like an extremely basic question to me; it has surely been studied in the literature. I appreciate any pointers! Of course I'll record any relevant thoughts in the lab.

   It’s well known that the category of points of the presheaf topos over $Ring_{fp}^{op}$, the dual of the category of finitely presented rings, is the category of all rings (without a size or presentation restriction). In fact this holds for any algebraic theory, not only for the theory of commutative rings. One can learn about this in our entries on Gabriel-Ulmer duality, flat functors, and Moerdijk/Mac Lane.

   But what if we don’t restrict the site to consist only of the compact objects? What are the points of the presheaf topos over the large category $Ring^{op}$, to the extent that the question is meaningful because of size-related issues? What are the points of the presheaf topos over $Ring_{\kappa}^{op}$, the dual of the category of rings admitting a presentation by $\lt \kappa$ many generators and relations, where $\kappa$ is a regular cardinal? (The category $Ring_{\kappa}^{op}$ is essentially small, so the question is definitely meaningful.)

   The question can be rephrased in the following way: What is an explicit description of the category of finite limit preserving functors $F : Ring_{\kappa}^{op} \to Set$? Any such functor gives rise to a ring by considering $F(\mathbb{Z}[X])$, but unlike in the case $\kappa = \aleph_0$ such a functor is not determined by this ring.

   This feels like an extremely basic question to me; it has surely been studied in the literature. I appreciate any pointers! Of course I’ll record any relevant thoughts in the lab.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 13th 2017
   • (edited Mar 13th 2017)
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   Author: Urs
   Format: MarkdownItexI don't know the answer to your question. But since already the well known statement you refer to is not really reflected on the $n$Lab, might you have a minute to state it with decent references [here](https://ncatlab.org/nlab/show/point+of+a+topos#Rings)?

   I don’t know the answer to your question. But since already the well known statement you refer to is not really reflected on the $n$Lab, might you have a minute to state it with decent references here?

3. • CommentRowNumber3.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeMar 13th 2017
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   Author: IngoBlechschmidt
   Format: MarkdownItexYes, I will! I just need a second to gather suitable references. The result is so ingrained to me that I forgot where I actually learned this from.

   Yes, I will! I just need a second to gather suitable references. The result is so ingrained to me that I forgot where I actually learned this from.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeMar 14th 2017
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   Author: Mike Shulman
   Format: MarkdownItexIf it hasn't been studied in the literature, I suspect that the reason is that there is not a very nice answer.

   If it hasn’t been studied in the literature, I suspect that the reason is that there is not a very nice answer.

5. • CommentRowNumber5.
   • CommentAuthorMarc Hoyois
   • CommentTimeMar 14th 2017
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   Author: Marc Hoyois
   Format: MarkdownItexThis may be obvious but the points of a presheaf topos PSh(C), aka left exact functors C → Set, are the same thing as ind-objects in C^op.

   This may be obvious but the points of a presheaf topos PSh(C), aka left exact functors C → Set, are the same thing as ind-objects in C^op.

6. • CommentRowNumber6.
   • CommentAuthorIngoBlechschmidt
   • CommentTimeMay 27th 2017
   • (edited May 27th 2017)
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   Author: IngoBlechschmidt
   Format: MarkdownItexThanks, Marc! I quite like your answer: * It gives a simple reason why $Pt(PSh(Ring_{fp}^{op})) \simeq Ring$: It's known that $Ind(Ring_{fp}) \simeq Ring$. * It also explains why one shouldn't expect a simple description of $Pt(PSh(Ring_\kappa^{op}))$ for $\kappa \gt \aleph_0$. * It gives intuition for the general case. I believe we can even pretend that $PSh(C)$ is the classifying topos of "ind-objects of $C^{op}$".

   Thanks, Marc! I quite like your answer:

   • It gives a simple reason why $Pt(PSh(Ring_{fp}^{op})) \simeq Ring$: It’s known that $Ind(Ring_{fp}) \simeq Ring$.
   • It also explains why one shouldn’t expect a simple description of $Pt(PSh(Ring_\kappa^{op}))$ for $\kappa \gt \aleph_0$.
   • It gives intuition for the general case. I believe we can even pretend that $PSh(C)$ is the classifying topos of “ind-objects of $C^{op}$”.