Timeline for Stability of accessible $\infty$-categories under some operations
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 6, 2021 at 13:51 | vote | accept | Giulio Lo Monaco | ||
| Apr 6, 2021 at 13:51 | vote | accept | Giulio Lo Monaco | ||
| Apr 6, 2021 at 13:51 | |||||
| Apr 6, 2021 at 13:51 | vote | accept | Giulio Lo Monaco | ||
| Apr 6, 2021 at 13:51 | |||||
| Mar 11, 2021 at 12:24 | answer | added | Yonatan Harpaz | timeline score: 3 | |
| Jan 22, 2019 at 17:22 | comment | added | Tim Campion | @GiulioLoMonaco $\mathcal C$ is $\kappa$-accessible. Then $\tau$ is chosen to be strictly bigger than $\kappa$ such that $\mathcal C$ is still $\tau$-accessible. Then $\mathcal D$ is equivalent to the full subcategory of $\mathcal C$ consisting of the $\tau$-accessible objects. $\mathcal C$ has $\kappa$-filtered colimits and the $\tau$-accessible objects are closed under $\tau$-small colimits. So $\mathcal D$ is indeed closed under $\tau$-small $\kappa$-filtered colimits. It's the fact that $\tau$ is strictly bigger than $\kappa$ which ensures that $\mathcal D'$ is not arbitrary. | |
| Jan 14, 2019 at 13:50 | comment | added | Giulio Lo Monaco | $\mathcal{D}$ is actually the essential image of the Yoneda embedding $\mathcal{D}' \to \text{Ind}_{\kappa}(\mathcal{D}')$, where $\mathcal{D}'$ is a not further specified small $\infty$-category. It doesn't admit colimits in general. | |
| Jan 12, 2019 at 4:33 | comment | added | Tim Campion | Regarding (1), the categorical thinking is as follows: $D_{/d}$ has $\tau$-small $\kappa$-filtered colimits for any $d \in D$ because $D$ does, and in a slice category, colimits are computed levelwise. It is $\tau$-filtered because $C$ is $\tau$-accessible, so every object $d$ is a $\tau$-filtered colimit of objects of $D$, and a cofinality argument then shows that the canonical colimit, indexed by $D_{/d}$, is $\tau$-filtered -- the 1-categorical version of this argument is Prop 1.22 in Adamek and Rosicky, Locally Presentable and Accessible Categories. | |
| Jan 10, 2019 at 18:29 | comment | added | dhy | I believe 3.) is actually easily fixed. Simply replace the bottom arrow with $\mathcal{C}\rightarrow\operatorname{Fun}(K\times\{0\}\amalg K\times\{1\},\mathcal{C}).$ On the other hand, I got seriously stuck for a few days on your issue #2. I agree with you that the proof as written does not work, but the result still holds with a different + substantially harder proof (I think you may also need to assume some condition on the cardinalities) - let me try to remember it. I don't immediately recall any thoughts on issue 1.) | |
| Jan 10, 2019 at 18:23 | comment | added | dhy | My experience is that there are a huge number of mistakes in HTT (but remarkably, none of them seem to be essential), and proofs should be read as suggestions. Here are some comments about your specific questions: | |
| Jan 10, 2019 at 18:05 | review | First posts | |||
| Jan 10, 2019 at 18:53 | |||||
| Jan 10, 2019 at 18:03 | history | asked | Giulio Lo Monaco | CC BY-SA 4.0 |