Timeline for In what algebraic categories do finitely presentable objects form a dense cogenerator?
Current License: CC BY-SA 4.0
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| Nov 19, 2023 at 23:56 | comment | added | Arshak Aivazian | @MaximeRamzi Great example, thanks! | |
| Nov 19, 2023 at 23:56 | comment | added | Arshak Aivazian | @TimCampion Indeed, there was still a plausible (and fulfilled) option: the answer does not depend on ZFC. Thank you! | |
| Nov 19, 2023 at 23:54 | comment | added | Arshak Aivazian | @IvanDiLiberti Thanks a lot! | |
| Nov 19, 2023 at 23:20 | comment | added | Ivan Di Liberti | Check out Sec 2 of my paper "Codensity: Isbell duality...", most relevantly 2.17 and 2.18. | |
| Nov 19, 2023 at 15:29 | comment | added | Tim Campion | Indeed, it is rare for locally presentable categories to have cogenerators at all (although categories of sheaves of sets or abelian groups do (importantly!) have injective cogenerators). However, as long as there does not exist a proper class of measurable cardinals, $Set^{op}$ has a dense cogenerator (given by the sets of cardinality $<\lambda$, where $\lambda$ is the largest measurable cardinal (which might be $\aleph_0$)). Note that this hypothesis is an "anti-large-cardinal-hypothesis" which contradicts VP, so there's no contradiction here. | |
| Nov 19, 2023 at 13:11 | comment | added | Maxime Ramzi | For commutative rings you can easily find counterexamples: for any $\kappa$ there exists a field of cardinality $>\kappa$. Such a field cannot map to any ring of size $\leq \kappa$. | |
| Nov 19, 2023 at 12:35 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Nov 19, 2023 at 11:59 | history | edited | Arshak Aivazian | CC BY-SA 4.0 |
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| Nov 19, 2023 at 7:35 | history | asked | Arshak Aivazian | CC BY-SA 4.0 |