Timeline for Small objects in categories
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 4, 2014 at 13:02 | vote | accept | Marci | ||
| Jan 3, 2014 at 20:31 | answer | added | Dylan Wilson | timeline score: 5 | |
| Jan 3, 2014 at 18:03 | comment | added | Fernando Muro | Marci, I think it's clearer now. Thanks for editing. | |
| Jan 3, 2014 at 17:43 | comment | added | Marci | This might help. The first step in defining the Grothendieck group of a scheme is to restrict the objects to small ones (i.e. to coherent sheaves), otherwise the group will be trivial. Now, I would like to define the Grothendieck group of some categories. So the first step would be to define small objects in it. For example, for the category of k-schemes, I would like to obtain Dream 1. | |
| Jan 3, 2014 at 17:30 | comment | added | Marci | @Muro The only notion I know is the notion of "compact object", but the examples I know have nothing to do with actual schemes or spaces (see Yuan's comment). I would like to find some similar notion which gives me Dream 1 and Dream 2. I do not know how to state the question more clearly. | |
| Jan 3, 2014 at 17:19 | comment | added | Fernando Muro | Marci, reading your comments, I think your question is not clearly stated | |
| Jan 3, 2014 at 16:42 | answer | added | David White | timeline score: 2 | |
| Jan 3, 2014 at 14:10 | comment | added | Marci | @Lin Can I ask you why? If I think of my model category as an $(\infty, 1)$-category, then for me it seems more natural to take the hom spaces rather the hom set. | |
| Jan 3, 2014 at 1:15 | comment | added | Zhen Lin | @Marci The derived hom might change, but "smallness" is usually defined in terms of the ordinary hom. | |
| Jan 3, 2014 at 0:54 | history | edited | Marci | CC BY-SA 3.0 |
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| Jan 3, 2014 at 0:51 | comment | added | Marci | @Yuan, thank you! That's a good observation, I change the question. | |
| Jan 3, 2014 at 0:51 | comment | added | Marci | @Muro I do not see why the model structure is irrelevant. The hom spaces do not change while changing the model structure? | |
| Jan 2, 2014 at 22:56 | comment | added | Fernando Muro | Concerning 3, model structures are irrelevant for compactness, and the only compact topological spaces, if compact = $\aleph_0$-presentable, are the finite discrete spaces, if I remember well. | |
| Jan 2, 2014 at 20:13 | comment | added | Qiaochu Yuan | The compact objects in the category of $k$-algebras are exactly the finitely presented $k$-algebras, but $k$-schemes are related to these contravariantly, so I think schemes of finite type is not the right conjecture (these should be the "cocompact" objects). | |
| Jan 2, 2014 at 19:52 | comment | added | Dietrich Burde | Too many questions ? Some ideas regarding 4. are here. | |
| Jan 2, 2014 at 19:48 | history | asked | Marci | CC BY-SA 3.0 |