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    • CommentRowNumber1.
    • CommentAuthorIngoBlechschmidt
    • CommentTimeMar 12th 2017
    • (edited Mar 12th 2017)

    It’s well known that the category of points of the presheaf topos over Ring fp opRing_{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 opRing^{op}, to the extent that the question is meaningful because of size-related issues? What are the points of the presheaf topos over Ring κ opRing_{\kappa}^{op}, the dual of the category of rings admitting a presentation by <κ\lt \kappa many generators and relations, where κ\kappa is a regular cardinal? (The category Ring κ opRing_{\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 κ opSetF : Ring_{\kappa}^{op} \to Set? Any such functor gives rise to a ring by considering F([X])F(\mathbb{Z}[X]), but unlike in the case κ= 0\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.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 13th 2017
    • (edited Mar 13th 2017)

    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 nnLab, might you have a minute to state it with decent references here?

  1. 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.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeMar 14th 2017

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

    • CommentRowNumber5.
    • CommentAuthorMarc Hoyois
    • CommentTimeMar 14th 2017

    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.

    • CommentRowNumber6.
    • CommentAuthorIngoBlechschmidt
    • CommentTimeMay 27th 2017
    • (edited May 27th 2017)

    Thanks, Marc! I quite like your answer:

    • It gives a simple reason why Pt(PSh(Ring fp op))RingPt(PSh(Ring_{fp}^{op})) \simeq Ring: It’s known that Ind(Ring fp)RingInd(Ring_{fp}) \simeq Ring.
    • It also explains why one shouldn’t expect a simple description of Pt(PSh(Ring κ op))Pt(PSh(Ring_\kappa^{op})) for κ> 0\kappa \gt \aleph_0.
    • It gives intuition for the general case. I believe we can even pretend that PSh(C)PSh(C) is the classifying topos of “ind-objects of C opC^{op}”.