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.

]]>The article local model structure on simplicial presheaves states that for a site with enough points stalkwise weak equivalences of simplicial presheaves coincide with weak equivalences of simplicial presheaves in the Bousfield localization of componentwise weak equivalences with respect to all hypercovers (i.e., weak equivalences in a local model structure).

Is a proof of this statement written up somewhere? (The article cited above gives a reference to Jardine, which claims, but does not prove this statement.)

Also, is it possible to formulate an analog of this statement for sites that do not have enough points? (Presumably we would have to talk about sufficiently refined (hyper)covers instead of points.)

]]>The article [[synthetic differential ∞-groupoid]] claims that the site CartSp_synth of smooth loci of the form R^n×ℓW can also be constructed as the semidirect product CartSp⋉InfPoint. However, the cited article by Kock and Reyes has a corrigendum (http://numdam.org/item?id=CTGDC_1987__28_2_99_0) that claims that their original construction of the site CartSp_synth as a semidirect product is erroneous (the resulting category has the right objects, but too few maps). In which sense then the formula in the cited article should be interpreted? Or is it simply a reference to the original incorrect construction?

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