Timeline for Which Ends preserve filtered colimits?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Mar 12, 2023 at 11:25 | comment | added | D.-C. Cisinski | @MaximeRamzi You are right, $M$ can not be finitely presentable in the $\infty$-category of spaces. That means my comment about a counter example was written too quickly. | |
| Mar 11, 2023 at 17:42 | comment | added | Maxime Ramzi | @D.-C.Cisinski In your example, $M$ cannot be finitely presented as a monoid in the $\infty$-category of spaces, though, right ? At least in this case, $k[M]$ must be smooth. | |
| Mar 11, 2023 at 14:30 | comment | added | D.-C. Cisinski | In a different context: replace sets by the stable $\infty$-category of modules over a commutative ring $k$ and $C$ the $\infty$-category of perfect complexes on a derived scheme $X$. This property is then equivalent to smoothness of $X$ over $k$. If you take a finite commutative monoid $M$ with associated $k$-algebra $k[M]$ that is non smooth, this will mean that $C=$ the category with one object associated to $M$ is a non-example. E.g. the group with $2$ elements. Therefore, the property of being of finite presentation (or even finite) is not sufficient. | |
| Mar 11, 2023 at 4:27 | comment | added | Simon Henry | I should even, Cauchy completion of a finitely generated category in fact, as such a cauchy completion can have an infinite number of objects, and that doesn't change the End | |
| Mar 11, 2023 at 1:28 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Mar 11, 2023 at 1:22 | comment | added | Simon Henry | Yes, at the minimum, I'd like to know if it is equivalent to $C$ being (equivalent to a category which is) finitely generated or not. | |
| Mar 11, 2023 at 1:19 | comment | added | Zhen Lin | It is necessary and sufficient that $\textrm{Hom}_\mathcal{C} : \mathcal{C}^\textrm{op} \times \mathcal{C} \to \textbf{Set}$ be a finitely presentable object in the category $[\mathcal{C}^\textrm{op} \times \mathcal{C}, \textbf{Set}]$, but I suppose that is not a satisfactory answer for you? | |
| Mar 10, 2023 at 22:49 | history | edited | Simon Henry | CC BY-SA 4.0 |
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| Mar 10, 2023 at 22:39 | history | asked | Simon Henry | CC BY-SA 4.0 |