Timeline for Do cocontinuous SET-valued functors separate points?
Current License: CC BY-SA 3.0
6 events
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| Jul 1, 2013 at 20:41 | comment | added | Theo Johnson-Freyd | @Dylan ...cosheaves are determined by their values on any generating set, so somehow the data being "forgotten" by cosheafification is the values of the functor on all the other objects.) | |
| Jul 1, 2013 at 20:39 | comment | added | Theo Johnson-Freyd | @Dylan: If $C$ is small, the category of cosheaves on $C$ is presentable. Since colimits are computed pointwise and "colimits commute", the inclusion of cosheaves into all functors is cocontinuous. It follows from the special adjoint functor theorem for presentable categories that this inclusion has a right adjoint, which you could call "cosheafification". So in that sense the answer to your question is "yes". It is a little funny: usually, "forget" is the right adjoint, and "free" is the left adjoint. So "cosheafification" is "forgetting" data from a functor to a cosheaf. (Namely, ... | |
| Jul 1, 2013 at 15:56 | vote | accept | Theo Johnson-Freyd | ||
| Jul 1, 2013 at 7:28 | answer | added | Qiaochu Yuan | timeline score: 7 | |
| Jul 1, 2013 at 5:47 | comment | added | Dylan Wilson | Ok so it's weird to me that corepresentable functors are not cosheaves... Is there a "cosheafification"? Does the cosheafification of $\text{hom}(Y, -)$ suffice? | |
| Jul 1, 2013 at 5:40 | history | asked | Theo Johnson-Freyd | CC BY-SA 3.0 |