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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.
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?
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.
If it hasn’t been studied in the literature, I suspect that the reason is that there is not a very nice answer.
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.
Thanks, Marc! I quite like your answer:
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