Timeline for Does the localization functor $\mathcal{C}\to S^{-1}\mathcal{C}$ preserve finite colimits when $\mathcal{C}$ is not small? (size issues in proof)
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Mar 21 at 18:41 | comment | added | Fernando Muro | @ElíasGuisadoVillalgordo yes. | |
| Mar 21 at 14:03 | comment | added | Elías Guisado Villalgordo | @FernandoMuro By "the comma category" do you mean $Y/S$? | |
| Mar 20 at 10:23 | comment | added | Fernando Muro | If the comma category is large you can't even ensure that the localisation exists. In specific cases localisations exist for a reason. Maybe, in your case, those reasons ensure that you can replace the comma category with a small cofinal one, so you would be done. Universes are tricky because you're assuming that all categories are small, but you bound sets of objects and morphisms by different inaccessible cardinals, the first one being bigger than the second one. When you localise, the cardinal of morphism sets may jump, entering the exclusion zone. | |
| Mar 20 at 9:18 | comment | added | Philippe Gaucher | @ElíasGuisadoVillalgordo Try ncatlab.org/nlab/show/Grothendieck+universe. The assertion (*) is correct in ZFC + enough Grothendieck universes. | |
| Mar 20 at 7:28 | history | edited | Elías Guisado Villalgordo | CC BY-SA 4.0 |
added 29 characters in body
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| Mar 20 at 7:17 | comment | added | Elías Guisado Villalgordo | @JonasFrey Thanks for the explanation ^^. Do you know any reference for these ideas? | |
| Mar 19 at 18:51 | comment | added | Jonas Frey | Yes, that's what I mean! It's a foundational principle promoted eg by Grothendieck: categories whose collections of objects are "classes" in the ZF-sense of the world, are very unwieldy, so instead of using them we postulate a Grothendieck universe and say that everything inside the universe is small, and things outside are not (necessarily). This way even the "large" objects are set-sized, and can be manipulated without foundational problems. | |
| Mar 19 at 17:55 | comment | added | Elías Guisado Villalgordo | @JonasFrey Do you mean "every category is small in a sufficiently large universe"? If yes, what is the meaning of these words? | |
| Mar 19 at 17:10 | comment | added | Jonas Frey | I would say every category of small in a sufficiently large universe, so it works. Size issues are only important when it comes to relative sizes. | |
| Mar 19 at 16:23 | history | asked | Elías Guisado Villalgordo | CC BY-SA 4.0 |