Timeline for If-and-only-if Linton monadicity theorem over presheaves
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 21 at 23:40 | comment | added | Kevin Carlson | I checked with Mike Shulman, who wrote that line in 2010, and he can’t immediately remember what he meant. | |
| Jun 21 at 22:21 | comment | added | Kevin Carlson | I've been digging around but I'm having trouble understanding what that sentence could be referring to other than Duskin's theorem (which is both necessary and sufficient for presheaf categories, I've just edited that in), which doesn't really feel much like a form of Linton's theorem to me since it does involve $U$-split congruences. | |
| Jun 21 at 21:07 | comment | added | Kevin Carlson | Perhaps even simpler than thinking about the proof and the axiom of choice, it looks to me like Linton's theorem implies very directly that $D$ is Barr exact, since all the properties needed for Barr exactness are created by $U$. But since monadic categories over presheaves are not always Barr exact or even regular, you'd have to dramatically modify the statement to have any chance here. In particular, you can't ask for anything like the creation of regular epis. Seems like you're going to get pushed back to $U$-split coequalizers and away from Linton's formulation. | |
| Jun 21 at 9:35 | comment | added | varkor | As Paul Taylor suggests, the proof in the case of $\mathrm{Set}$ seems to rely heavily on epimorphisms splitting. For instance, Linton proves a generalisation of his monadicity theorem for $\mathrm{Set}$-like categories in Theorem 3 of Applied functorial semantics, II, but requires epimorphisms in the base category to split. So I suspect there will not be a similar-looking theorem for presheaf categories. | |
| Jun 20 at 18:22 | comment | added | Paul Taylor | I suggest checking whether this statement relies on Choice in $\mathbf{Set}$, ie that epis split. | |
| Jun 20 at 17:32 | history | edited | Ilk | CC BY-SA 4.0 |
added 55 characters in body
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| Jun 20 at 17:26 | history | asked | Ilk | CC BY-SA 4.0 |