Timeline for In which categories is the union of subobjects given by the pushout over the intersection?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Jan 5, 2023 at 3:56 | vote | accept | Tim Campion | ||
| Jan 5, 2023 at 6:02 | |||||
| Jan 2, 2023 at 0:07 | answer | added | Tim Campion | timeline score: 1 | |
| Jan 1, 2023 at 10:41 | comment | added | Z. M | Do you expect any $\infty$-categorical version? Comprising $\infty$-topoi and prestable $\infty$-categories? | |
| Jan 1, 2023 at 10:16 | history | edited | Tim Campion | CC BY-SA 4.0 |
added 158 characters in body
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| Dec 29, 2022 at 9:20 | comment | added | Paul Taylor | Regularity isn't enough, eg ${\mathbf{Set}}^{op}$. There is a pushout in $\mathbf{Set}$ using sets with 1, 2, 2 and 3 elements where the pullback has 4 elements, so $3\to 4$ is not epi in $\mathbf{Set}$ (mono in ${\mathbf{Set}}^{op}$). | |
| Dec 25, 2022 at 3:29 | history | became hot network question | |||
| Dec 24, 2022 at 21:30 | comment | added | Kevin Carlson | Exactness isn’t enough: you can have pairs of subgroups with trivial intersection whose union is not the free product. | |
| Dec 24, 2022 at 21:02 | comment | added | Paul Taylor | I asked this myself, including the directed case, but got no answer. From some other question I learned that Michael Barr had written about this in Springer LNM 1348. My application was to show when extensional well founded coalgebras have "overlapping unions" like sets ($\epsilon$-structures). | |
| Dec 24, 2022 at 20:25 | answer | added | godelian | timeline score: 10 | |
| Dec 24, 2022 at 20:14 | comment | added | Martin Brandenburg | This comments seems to be an answer. | |
| Dec 24, 2022 at 19:39 | comment | added | godelian | It suffices that $\mathcal{C}$ is a coherent category. The proof is mainly due to Joyal, and it appears e.g. in "Reyes, Gonzalo: From sheaves to logic - Studies in algebraic logic, M.A.A. studies in Math., vol. 9 (1974). The Elephant has also two different proofs, one using the internal language of the category. | |
| Dec 24, 2022 at 19:26 | history | asked | Tim Campion | CC BY-SA 4.0 |