Timeline for "Sums-compact" objects = f.g. objects in categories of modules?
Current License: CC BY-SA 2.5
28 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Jun 6, 2021 at 10:42 | vote | accept | Sasha | ||
| Jan 30, 2014 at 11:20 | vote | accept | Sasha | ||
| S Jun 6, 2021 at 10:42 | |||||
| Apr 18, 2012 at 18:34 | answer | added | Jeremy Rickard | timeline score: 14 | |
| Feb 5, 2012 at 2:45 | answer | added | Todd Trimble | timeline score: 7 | |
| Nov 26, 2011 at 12:26 | comment | added | Todd Trimble | Tom, I've been thinking about this recently, and it seems to me that "connectedness" is a great word for the concept if the category is extensive (like topological spaces or a topos). But in other cases like modules, not so much. There's another difference in that the connectedness usually refers to Set-based homs, whereas that would hardly make good sense here. | |
| Nov 22, 2011 at 18:41 | comment | added | Tom Leinster | The condition that Hom(X, -) preserves coproducts is usually called connectedness of X. (Think about the category of topological spaces.) There's some flexibility about whether you ask for preservation of all small coproducts or just the finite ones. See for instance ncatlab.org/nlab/show/connected+object | |
| Nov 22, 2011 at 16:30 | answer | added | Martin Brandenburg | timeline score: 8 | |
| Nov 22, 2011 at 13:52 | comment | added | David White | @Sasha: I assume you've read the nLab article on compact objects. It seems to me that their definition of compact for an additive category is what you're calling sumpact, so the references there might help you (unless I'm misunderstanding the definitions here). Anyway, they also give a theorem about when an abelian category $\mathcal{C} \cong R$-mod for some $R$, which may interest you since this is the situation you seem to want to work in above. This link hits the relevant section: ncatlab.org/nlab/show/… | |
| Nov 20, 2011 at 19:09 | comment | added | user2035 | @G. Rodrigues: I hat not noticed that Theo's little only requires that $\mathrm{Hom}(X,{-})$ preserves finite coproducts. I think this is a typo, since in an additive category, finite coproducts are "the same" as finite products, so $\mathrm{Hom}(X,{-})$ commutes with finite coproducts, for any $X$. However, this also shows that if $\mathrm{Hom}(X,{-})$ commutes with filtered colimits, it commutes with arbitrary coproducts (by the argument I gave above). | |
| Nov 20, 2011 at 12:05 | comment | added | G. Rodrigues | @a-fortiori: right, but a finite coproduct is not a filtered colimit since the indexing category is a finite set which is not filtered. Taking the poset of finite subsets does not buy you anything because this poset has a top element so the colimit is trivially computed. So little-ness is still not weaker (or stronger) than compactness, right? (I have the strange feeling, that I am just about to have one of those doh moments...) | |
| Nov 20, 2011 at 11:18 | comment | added | user2035 | @G.Rodrigues: An arbitrary coproduct is the filtered colimit of its finite sub-coproducts. | |
| Nov 20, 2011 at 10:25 | answer | added | Buschi Sergio | timeline score: 3 | |
| Nov 19, 2011 at 22:38 | answer | added | Anonymous | timeline score: 0 | |
| Nov 19, 2011 at 14:00 | comment | added | G. Rodrigues | Maybe I am missing something obvious, but since when is a coproduct a filtered colimit? In other words, why is <i>little-ness</i> weaker than compactness? | |
| Nov 19, 2011 at 11:32 | answer | added | Pierre-Yves Gaillard | timeline score: 17 | |
| Nov 19, 2011 at 10:24 | history | edited | Martin Brandenburg |
edited tags
|
|
| Mar 23, 2011 at 21:29 | comment | added | Theo Johnson-Freyd | Ah, I'm much happier now. +1, by the way. | |
| Mar 23, 2011 at 17:22 | comment | added | Martin Brandenburg | 1+, good question. By the way, I have seen many texts on homological algebra which define a compact object to be an object $X$ such that $Hom(X,-)$ preserves coproducts. | |
| Mar 23, 2011 at 16:50 | answer | added | Fernando Muro | timeline score: 13 | |
| Mar 23, 2011 at 15:48 | comment | added | Sasha | OK, thanks. I edited my question accordingly. | |
| Mar 23, 2011 at 15:47 | history | edited | Sasha | CC BY-SA 2.5 |
added 283 characters in body; edited title
|
| Mar 23, 2011 at 15:28 | comment | added | Theo Johnson-Freyd | With @Fernando, I object to your word "compact", because it already has some precise technical meaning in the category theory literature. I would prefer you to pick some other word --- "A module $M$ in $R\text{-mod}$ is little if $\hom(M,-) : R\text{-mod} \to \text{Ab}$ preserves finite coproducts. It is clear that finitely-generated implies little. Is the converse true? Note: little is weaker than compact, which is that $\hom(M,-)$ preserves filtered colimits." | |
| Mar 23, 2011 at 12:49 | comment | added | Sasha | @Tilman: If I am not missing something, in that case we do not have a morphism from our module $M$ to the direct sum $\osum M / N_i$, since for some $x \in M$ it might be that there are infinitely many $N_i$ such that $x$ is not in $N_i$. | |
| Mar 23, 2011 at 12:43 | comment | added | Tilman | Your non-finitely generated module might not be the colimit of a countable sequence of finitely generated submodules, but the finitely generated submodules do form a directed set whose colimit is your module. And then your argument should still work, or am I missing something here? | |
| Mar 23, 2011 at 11:54 | comment | added | Sasha | And I agree that the version you say is (as far as I understand) much more meaningful, but still, I wondered. | |
| Mar 23, 2011 at 11:53 | comment | added | Sasha | Yes, thank you. That version I know. I wondered what happens when the definition which I use in the text is used. | |
| Mar 23, 2011 at 11:34 | comment | added | Fernando Muro | The usual definition of compact object in an abelian category, such as a module category, is that the covariant representable functor preserves filtered colimits, not just coproducts. In this case, compact is equivalent to finitely presented. | |
| Mar 23, 2011 at 10:39 | history | asked | Sasha | CC BY-SA 2.5 |