Timeline for Proof that objects are colimits of generators
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Nov 21, 2015 at 4:17 | history | edited | Omar Antolín-Camarena | CC BY-SA 3.0 |
fixed LaTeX
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| Nov 21, 2010 at 16:53 | answer | added | Buschi Sergio | timeline score: 1 | |
| Jun 7, 2010 at 14:08 | vote | accept | Joey Hirsh | ||
| Jun 6, 2010 at 23:19 | answer | added | Mike Shulman | timeline score: 19 | |
| Jun 6, 2010 at 23:13 | comment | added | Todd Trimble | Ditto Mike's last sentence. The last example I tried looking at was where $G$ is the 2-dimensional oriental in the category 2-Cat, which I'm pretty sure is not dense (because Street's nerve functor is not full on 2-Cat). | |
| Jun 6, 2010 at 19:44 | comment | added | Todd Trimble | Sorry, Joey -- compact Hausdorff spaces are a cocomplete category. This is well-known. (You are right however that colimits there are not computed as they would be in $Top$.) I don't think however the question has been totally settled by Mike's example, because one can still present a compact Hausdorff space as a colimit of a diagram consisting totally of 1's (essentially because compact Hausdorff spaces are monadic over sets). | |
| Jun 6, 2010 at 18:54 | comment | added | Joey Hirsh | Compact Hausdorff isn't closed under colimits... | |
| Jun 6, 2010 at 18:34 | comment | added | Mike Shulman | If that's what is meant, then the statement is false. In the category of compact Hausdorff spaces, the terminal object is a strong generator. But since it has no endomorphisms, that slice category is discrete, so the colimit is just the coproduct of 1 over the set Hom(1,X), and this is the Stone-Cech compactification of the underlying set of X -- generally not the same as X. | |
| Jun 6, 2010 at 16:09 | comment | added | Harry Gindi | The colimit is taken over the slice category as $$\varinjlim_{f\in(G\downarrow X)}s(f)$$ where $s$ is the canonical functor $(G\downarrow X)\to C$ sending every object to its source (in this case G) and every commutative triangle to the corresponding morphism in $C$. Joey's notation is a bit too abusive for my tastes, but I understood what he meant. | |
| Jun 6, 2010 at 15:47 | comment | added | Omar Antolín-Camarena | I don't understand what the colimit in the last paragraph is indexed over. Is it the set Hom(G,X)? In that case it seems to me you could only mean the coproduct, which is probably not what you want. | |
| Jun 6, 2010 at 15:45 | comment | added | Omar Antolín-Camarena | For what is commonly called a generator (i.e., what Martin mentions in his comment), the result fails: In Top a point is a generator, but colimits of points are just discrete spaces. | |
| Jun 6, 2010 at 15:32 | history | edited | Joey Hirsh | CC BY-SA 2.5 |
changed "generator" to "strong generator"
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| Jun 6, 2010 at 14:18 | comment | added | Mike Shulman | What you called a "generator" is usually called a strong generator. Although a few people do confusingly call it merely a "generator," and instead say "separator" for the notion mentioned by Martin (which is what is more commonly called a "generator"). | |
| Jun 6, 2010 at 8:34 | comment | added | Martin Brandenburg | I know the following definition of a generator (from Mac Lane's book): For every distinct $f,g : X \to Y$, there is some $h : G \to X$ such that $fh, gh$ are also distinct. I doubt that this is equivalent to your one. | |
| Jun 6, 2010 at 2:57 | history | asked | Joey Hirsh | CC BY-SA 2.5 |