Timeline for Is there an "Abelian envelope" 2-functor?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Feb 12, 2012 at 18:50 | vote | accept | Vanessa | ||
| Feb 12, 2012 at 18:28 | comment | added | Vanessa | @Will, the functor corresponding to the group G is tensor product with G. In particular you can take Z/2 for G | |
| Feb 12, 2012 at 16:46 | comment | added | Todd Trimble | Gentlemen, the answer I gave below is for the forgetful 2-functor to $Cat$ from the 2-category of abelian categories, exact functors, and natural transformations. | |
| Feb 12, 2012 at 15:56 | comment | added | Will Sawin | What's the kernel of the map $\mathbb Z \to \mathbb Z$ given by $2$? I believe it's a functor that measures two-torsion. If that's not true, how are you defining kernels of natural transformation? | |
| Feb 12, 2012 at 15:39 | history | edited | Vanessa | CC BY-SA 3.0 |
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| Feb 12, 2012 at 15:36 | comment | added | Vanessa | @Will the result is either the category of finitely generated Abelian groups or the category of all Abelian groups, depending whether you take Ab3 in your definition of an "Abelian category". It's realization as functors is via tensor product. See also my edit of the question | |
| Feb 12, 2012 at 15:31 | history | edited | Vanessa | CC BY-SA 3.0 |
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| Feb 12, 2012 at 8:34 | comment | added | Vanessa | @Fernando, you are right I need to specify what counts as morphisms. I think 1-morphisms are exact functors and 2-morphisms are arbitrary natural transformations. When you say "it depends" you mean in some case it exists while in another it doesn't? Can you detail? | |
| Feb 12, 2012 at 1:14 | answer | added | Todd Trimble | timeline score: 13 | |
| Feb 12, 2012 at 1:06 | comment | added | Yemon Choi | I may be barking completely up the wrong tree, but I vaguely recall the construction of some kind of Abelian envelope for an additive category (Freyd?) - is something like that along the lines you seek? | |
| Feb 11, 2012 at 22:24 | comment | added | Will Sawin | $Hom(Ab,Ab)$ is already quite a complex category. Which subcategory is generated by the identity? | |
| Feb 11, 2012 at 22:21 | comment | added | Will Sawin | I very recently had a similar idea. The main subtlety that I found was with the concept of "generated". Suppose you generate an object by two different methods, but do not generate an isomorphism between them. In the subcategory, will the two objects be isomorphic? If no, are you sure that the category is well-defined? You might have to be careful. If yes, then which isomorphism? Secondarily, it seems that all functors in $Hom(C,Ab)$, allowed to act on the generated category, are exact. But shouldn't there be non-exact functors on the generated category sometimes? | |
| Feb 11, 2012 at 21:51 | comment | added | Fernando Muro | That depends on what you mean by the 2-category of abelian categories. | |
| Feb 11, 2012 at 21:33 | history | asked | Vanessa | CC BY-SA 3.0 |