Timeline for Higher vector spaces
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 22, 2014 at 7:04 | comment | added | Chris Schommer-Pries | There is the following characterization which might be useful (k is a field): an abelian k-linear category is equivalent to the category of finite dimensional representations of a finite dimensional algebra iff (1) All homs are finite dimensional (2) Every object has finite length (3) there are enough projectives (e.g. if there is a projective generator) and (4) up to isomorphism there are finitely many simple objects. This is well-known in the right communities, and a proof can be found here arXiv:1406.4204. | |
| Oct 21, 2014 at 17:47 | comment | added | Zhen Lin | Well, there is a trivial observation: an idempotent-complete category $\mathcal{C}$ is (equivalent to) the category of finitely presented $R$-modules if and only if $\mathbf{Ind}(\mathcal{C})$ is (equivalent to) the category of $R$-modules. | |
| Oct 21, 2014 at 15:27 | comment | added | Julian Kuelshammer | @ChrisSchommer-Pries Thanks for pointing out this mistake. Of course the category should be cocomplete, otherwise there is the question what compact should mean if you don't have coproducts. I don't know of a theorem describing what properties describe finite dimensional vector spaces (or more general finitely generated, finitely presented, or coherent modules over a ring). | |
| Oct 21, 2014 at 15:24 | history | edited | Julian Kuelshammer | CC BY-SA 3.0 |
added the assumption of being cocomplete
|
| Oct 20, 2014 at 12:07 | comment | added | Chris Schommer-Pries | The category of finite dimensional $k$ vector spaces is Abelian and has a compact projective generator. How do you distinguish this from, say, the category of all vector spaces? | |
| Oct 18, 2014 at 5:19 | comment | added | Julian Kuelshammer | @Najib Idrissi Yes, I changed accordingly. | |
| Oct 18, 2014 at 5:18 | history | edited | Julian Kuelshammer | CC BY-SA 3.0 |
added 18 characters in body
|
| Oct 17, 2014 at 6:25 | comment | added | Najib Idrissi | Thanks. That first statement is true for abelian categories, right? (ie. if $C$ is abelian, then $C$ is a category of representations iff...). | |
| Oct 17, 2014 at 6:24 | vote | accept | Najib Idrissi | ||
| Oct 16, 2014 at 21:40 | history | answered | Julian Kuelshammer | CC BY-SA 3.0 |