Timeline for Can one characterize the category of finite-dimensional vector spaces? [duplicate]
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| Oct 23, 2013 at 14:23 | history | closed |
David Roberts♦ Daniel Moskovich Eric Wofsey Andrey Rekalo David White |
Duplicate of Characterising categories of vector spaces | |
| Oct 23, 2013 at 6:56 | comment | added | Fernando Muro | ... or you can add the condition that the center of your category, which in general is a commutative ring, is an algebra over the ground field $K$. That amounts to the category being $K$-linear. I think where speaking about the same thing, indeed. | |
| Oct 23, 2013 at 2:35 | comment | added | Fernando Muro | I see your point. Still you can add the simple condition that the center of the category be your fixed $K$. I only see a problem in the definition of Euler characteristic. If you take it to be the element in $K_0$ represented by a complex then any exact sequence would go to $0$ by definition, and that's like assuming 2 is always true. | |
| Oct 23, 2013 at 0:02 | review | Close votes | |||
| Oct 23, 2013 at 14:23 | |||||
| Oct 22, 2013 at 23:59 | comment | added | Qiaochu Yuan | @Fernando: that's not what I mean. The wording of the OP's question suggests that he wants simple conditions that pin down $\text{FinVect}_K$ for fixed $K$, and that seems unlikely to me unless we explicitly code in $K$, e.g. by working with $K$-linear categories. | |
| Oct 22, 2013 at 23:45 | comment | added | David Roberts♦ | See my question mathoverflow.net/questions/118246/…, which has a very satisfactory answer. | |
| Oct 22, 2013 at 20:46 | comment | added | Fernando Muro | $K$ could be recovered as the center of the category. | |
| Oct 22, 2013 at 20:03 | history | edited | Eric Wofsey |
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| Oct 22, 2013 at 19:28 | comment | added | Qiaochu Yuan | I'm also not sure how you expect to recover the field $K$. Are we secretly talking about $K$-linear categories or are you happy to characterize these categories as $K$ runs over all fields? | |
| Oct 22, 2013 at 19:20 | comment | added | Qiaochu Yuan | What do you mean by the Euler characteristic of a complex of objects in an abelian category? Do you mean the alternating sum of the images in the Grothendieck group, and if so, isn't this always equal to zero? | |
| Oct 22, 2013 at 19:16 | answer | added | Fernando Muro | timeline score: 4 | |
| Oct 22, 2013 at 19:08 | comment | added | Dag Oskar Madsen | mathoverflow.net/questions/118246/… | |
| Oct 22, 2013 at 19:07 | comment | added | Zhaoting Wei | Do you emphasis on how to characterize "finite dimensional" in a categorical way? | |
| Oct 22, 2013 at 18:59 | history | asked | Drew Armstrong | CC BY-SA 3.0 |