Timeline for Grothendieck toposes in (very) weak foundation
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Aug 26, 2015 at 13:50 | comment | added | Jonas Frey | Yet another comment. C2.2.8(vii) in the Elephant characterizes Grothendieck toposes as infty-pretoposes with a separating set of objects. But infty-pretoposes are by definition well-powered for Johnstone, and any well powered regular category is also locally small since morphisms from A to B can be embedded into subobjects of AxB as graphs. | |
| Aug 26, 2015 at 13:35 | comment | added | Jonas Frey | I think I formulated my comment badly since I made it sound like the conditions of Giraud's theorem are complete without local smallness. The conclusion should rather be that local smallness has to be included explicitely. | |
| Aug 26, 2015 at 13:21 | comment | added | Jonas Frey | On the question whether local smallness is implied by the axioms in classical strong foundations, what is wrong with the following counterexample? Assume that U and V are two Grothendieck universes, with U contained in V. Then V is exact and extensive and has a U-small coproducts and a U-small separating set (in fact 1 alone separates). Thus V satisfies the hypotheses of Giraud's theorem relative to U. However, V is not locally small relative to U. This shows that local smallness is not implied by Giraud's axioms. Did I overlook something? | |
| Aug 25, 2015 at 21:20 | comment | added | Toby Bartels | For now, I've added some hedging and a pointer here to the nLab page. | |
| Aug 25, 2015 at 21:08 | comment | added | Toby Bartels | I've gotten back into M.O, but I'm not ready to answer yet, since I'm waiting on getting ahold of the Elephant again. In the meantime, Simon's last comment seems very plausible. Then my last paragraph would not be a theorem, but would have to be an additional hypothesis. If so, I'll edit the nLab article accordingly. (Or if anybody else feels confident about this, then you also have the right to edit the nLab article.) | |
| Aug 14, 2015 at 20:31 | comment | added | Simon Henry | Ok, reading the elephant it seems that the trick is that one ask for the object of the separating sets to have have small set of maps out of it... | |
| Aug 14, 2015 at 10:05 | comment | added | Simon Henry | Thank you ! If the last paragraph is true that would be indeed very interesting... But I don't really see how one can get something like that. Or maybe we take it as an additional assumption on the sets of generators ? | |
| Aug 14, 2015 at 9:44 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Aug 14, 2015 at 9:20 | history | answered | Todd Trimble | CC BY-SA 3.0 |