Timeline for When is a finitary functor induced by Ind (co)continuous
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 14, 2020 at 1:25 | vote | accept | varkor | ||
| Sep 14, 2020 at 1:16 | comment | added | Zhen Lin | That’s right. Every locally $\kappa$-presentable category is equivalent to the category of $\kappa$-ind objects of some $\kappa$-cocomplete category, but that may be a different category from what you start with. (Not too different, in reality: it is the Cauchy completion.) | |
| Sep 14, 2020 at 1:11 | comment | added | varkor | Ah, perhaps the answer is that, if $\mathbf{Ind}_\kappa(\mathcal C)$ is cocomplete, then there is a category $\mathcal C'$ with $\kappa$-small colimits such that $\mathbf{Ind}_\kappa(\mathcal C) ≃ \mathbf{Ind}_\kappa(\mathcal C')$, but $\mathcal C$ itself may not be. | |
| Sep 14, 2020 at 1:04 | comment | added | varkor | To clarify where my confusion is coming from, I was basing my original statement of Theorem 5.5(ii) of A classification of accessible categories (which I see I must have misunderstood). That paper does not mention idempotents, so I had not appreciated their importance here. | |
| Sep 14, 2020 at 1:02 | comment | added | varkor | Sorry, I realise I omitted a critical negation in my previous comment. Thank you for your patience! Suppose $\mathbf{Ind}_\kappa(\mathcal C)$ is cocomplete. If idempotents in $\mathcal C$ split, then we know that $\mathcal C$ has $\kappa$-small colimits. However, if idempotents in $\mathcal C$ do not split, then $\mathcal C$ cannot have $\kappa$-small colimits, or otherwise its idempotents would split. Is there a specific example of a small category in which idempotents do not split, but whose $\mathbf{Ind}_\kappa$-completion is cocomplete? | |
| Sep 13, 2020 at 22:06 | comment | added | Zhen Lin | No, if we don't know that idempotents in $\mathcal{C}$ split then we cannot conclude that $\mathcal{C}$ has $\kappa$-small colimits even if $\textbf{Ind}_\kappa (\mathcal{C})$ is cocomplete. | |
| Sep 13, 2020 at 21:11 | comment | added | varkor | If I understand correctly: if $\mathcal C$ has $\kappa$-small colimits, then idempotents in $\mathcal C$ split. So is this to say that $\mathbf{Ind}_\kappa(\mathcal C)$ may be cocomplete even if $\mathcal C$ does not have $\kappa$-small colimits, but idempotents in $\mathcal C$ do split? | |
| Sep 13, 2020 at 14:03 | comment | added | Zhen Lin | Assuming idempotents in $\mathcal{C}$ split, $\textbf{Ind}_\kappa (\mathcal{C})$ has colimits if and only if $\mathcal{C}$ has $\kappa$-small colimits. | |
| Sep 13, 2020 at 12:19 | comment | added | varkor | @ZhenLin: just to clarify, what would be the corrected statement? | |
| Sep 12, 2020 at 23:27 | answer | added | Zhen Lin | timeline score: 4 | |
| Sep 12, 2020 at 22:47 | comment | added | Zhen Lin | Your "iff" is not quite correct, I think. You should assume idempotents split. | |
| Sep 12, 2020 at 17:58 | history | asked | varkor | CC BY-SA 4.0 |