It’s well known that the category of points of the presheaf topos over , 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 , to the extent that the question is meaningful because of size-related issues? What are the points of the presheaf topos over , the dual of the category of rings admitting a presentation by many generators and relations, where is a regular cardinal? (The category 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 ? Any such functor gives rise to a ring by considering , but unlike in the case 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.
]]>