Timeline for Checking that (hyper) sheafification is fibrant in local projective model structure on simplicial presheaves
Current License: CC BY-SA 4.0
9 events
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| Mar 17, 2021 at 19:56 | vote | accept | cheyne | ||
| Mar 17, 2021 at 19:56 | comment | added | cheyne | Thank you very much, Dmitri. These comments and our separate conversation helped immensely. | |
| Mar 17, 2021 at 18:41 | comment | added | Dmitri Pavlov | @cheyne: To rephrase: any attempt to write down a sheafification functor as a single-step construction using only subobjects of representable functors (i.e., sieves) cannot work because of the many counterexamples (for example, see Section III.5, page 130, in Mac Lane and Moerdijk's Sheaves in Geometry and Logic). In order to get a correct sheafification functor, you need to “access” higher homotopy groups, which can be either done one step at a time using transfinite constructions, or by generalizing from subobjects of representable presheaves (i.e., sieves) to hypercovers. | |
| Mar 17, 2021 at 18:26 | comment | added | Dmitri Pavlov | @cheyne: Yes, Lurie's F^† is the same functor. If you look at the proof of Proposition 6.2.2.7, Lurie has to construct a transfinite sequence T_β in the manner I described. So Lurie's approach corresponds to the first solution in my answer. At the beginning of Section 6.5.3 you can find a discussion similar to my answer, phrased in terms of ∞-categories. In particular, in the 4th paragraph of Section 6.5.3 he discusses how to compute sheafifications using hypercovers, in a single step. | |
| Mar 17, 2021 at 17:36 | comment | added | cheyne | I am aware of these issues you are referring to but I would like to focus on your "hypercovers(which are not subfunctors of representable functors)" comment. My understanding was that Lurie is actually taking a hocolim over all covering sieves (Remark 6.2.2.12) and these are in bijective correspondence with subfunctors of the representable (Prop 6.2.2.5). So in this sense, according to Lurie, it seems we should be able to have the conversation without saying "hypercover" but I don't see how to resolve my question. Thank you! | |
| Mar 17, 2021 at 17:05 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
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| Mar 17, 2021 at 17:04 | comment | added | Dmitri Pavlov | @cheyne: F^† does not compute the sheafification even for ordinary sheaves of sets, there are counterexamples that show you must apply it twice. So there is no way to make it work even in the elementary cases. That's why one must either expand covers to hypercovers (which are not subfunctors of representable functors), or iterate the construction. Indeed, it is a classical theorem in sheaf theory that F^†† computes the sheafification of a presheaf of sets, and Lurie proves in higher topos theory that iterating † n+2 times computes the sheafification of a presheaf of n-groupoids. | |
| Mar 17, 2021 at 17:00 | comment | added | cheyne | Thank you for this response, Dmitri. I am trying to do this S-localization whereby I replace hypercovers with subfunctors of Yoneda. Perhaps you are saying the answer to my question lies within Verdier's theorem? Is there no answer to my question without passing to a discussion about covers or hypercovers? | |
| Mar 17, 2021 at 16:53 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |