Timeline for Is there a good general definition of "sheaves with values in a category"?
Current License: CC BY-SA 4.0
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| Apr 6, 2021 at 10:20 | comment | added | Ivan Di Liberti | Thanks for this clarification, it's an interesting insight. I will think about it in the next days and hopefully come back to the question. | |
| Apr 6, 2021 at 10:17 | comment | added | Zhen Lin | To answer your question... I think categories of sheaves of <whatever> should form a stack on the category of Grothendieck toposes. For structures that are axiomatised by geometric theories, we get representable stacks, and furthermore the pullback functors have good properties in regards to (co)limits. My desiderata mostly stem from this observation. | |
| Apr 6, 2021 at 9:04 | history | edited | Ivan Di Liberti | CC BY-SA 4.0 |
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| Apr 6, 2021 at 8:57 | history | edited | Ivan Di Liberti | CC BY-SA 4.0 |
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| Apr 5, 2021 at 14:24 | history | edited | Ivan Di Liberti | CC BY-SA 4.0 |
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| Apr 5, 2021 at 14:15 | history | edited | Ivan Di Liberti | CC BY-SA 4.0 |
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| Apr 5, 2021 at 14:08 | history | edited | Ivan Di Liberti | CC BY-SA 4.0 |
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| Apr 5, 2021 at 14:07 | comment | added | Zhen Lin | Sure, for the purposes of actually doing homotopy theory, there are better models. But it illustrates that there is (1) a good definition of sheaves of objects that is not just the naïve one and (2) works well for categories that are not necessarily complete or cocomplete. | |
| Apr 5, 2021 at 14:05 | history | edited | Ivan Di Liberti | CC BY-SA 4.0 |
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| Apr 5, 2021 at 14:04 | comment | added | Ivan Di Liberti | Yes, but isn't it the same of the theory of model topoi in the sense of Rezk? | |
| Apr 5, 2021 at 14:02 | comment | added | Zhen Lin | Sheaves of Kan complexes were considered in Ken Brown's paper introducing categories of fibrant objects. He gives (what I consider to be equivalent to) the correct definition: a sheaf of Kan complexes on a topological space is a simplicial sheaf whose stalks are Kan complexes. This generalises to internal Kan complexes in a topos. | |
| Apr 5, 2021 at 13:58 | history | edited | Ivan Di Liberti | CC BY-SA 4.0 |
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| Apr 5, 2021 at 13:52 | history | answered | Ivan Di Liberti | CC BY-SA 4.0 |