Timeline for Drect limit of sequences
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| May 19, 2013 at 12:18 | comment | added | Paul Taylor | It seems to me that clarifying general issues about how categorical properties are limits or colimits and when these might be expected to commute with one another is more valuable than quoting specific definitions in particular subjects. | |
| May 19, 2013 at 9:34 | comment | added | hamid | Roughly speaking, a Grothendieck category is an abelian category with direct limits, pull back, push out. | |
| May 18, 2013 at 15:28 | history | edited | Paul Taylor | CC BY-SA 3.0 |
added 285 characters in body
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| May 18, 2013 at 14:29 | comment | added | Paul Taylor | I thought that might be the case, but maybe for completeness you could either say explicitly what a Grothendieck category is or provide a link to a definition. | |
| May 18, 2013 at 13:34 | comment | added | Zhen Lin | To be clear, filtered colimits do commute with finite limits in a Grothendieck category – but this is basically part of the definition! For an example of a Grothendieck category that is not locally finitely presentable, one just has to look for a topos that is not locally finitely presentable. | |
| May 18, 2013 at 12:51 | history | edited | Paul Taylor | CC BY-SA 3.0 |
expanded further
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| May 18, 2013 at 12:36 | comment | added | Zhen Lin | @DavidWhite Filtered colimits commute with finite limits in any finitely accessible category (hence Paul's emphasis on finitary algebraic theory). In particular it will be true in a locally finitely presentable abelian category, but Grothendieck categories need not be locally finitely presentable. Moreover it is not true in general that filtered colimits commmute with finite limits: take $\mathbf{Set}^\mathrm{op}$, for example. | |
| May 18, 2013 at 11:57 | history | edited | Paul Taylor | CC BY-SA 3.0 |
incorporated answer to David White
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| May 18, 2013 at 11:17 | comment | added | David White | A Grothendieck category has a lot of smallness (I think it's locally presentable for example) so that means Hom commutes with colimits, maybe with some hypothesis on the kind of colimit or its length. It seems your argument might hold more generally and not really need this smallness hypothesis | |
| May 18, 2013 at 11:15 | comment | added | David White | I thought filtered colimits always commute with finite limits. Isn't that basically what filtered colimits are good for? There's an old mathoverflow question on this subject | |
| May 18, 2013 at 11:10 | history | answered | Paul Taylor | CC BY-SA 3.0 |