Timeline for Characterising categories of vector spaces
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 11, 2013 at 21:52 | comment | added | Mike Shulman | @Martin: I don't know, whatever extra structure David had in mind. (-: | |
| Jan 7, 2013 at 23:16 | vote | accept | David Roberts♦ | ||
| Jan 7, 2013 at 19:39 | answer | added | John Baez | timeline score: 12 | |
| Jan 7, 2013 at 19:30 | comment | added | Martin Brandenburg | What happens if $1$ is not the only simple object? | |
| Jan 7, 2013 at 17:45 | comment | added | Martin Brandenburg | @Mariano: Indeed, or R could be Morita-equivalent to a commutative ring. A more precise (and correct) statement is that the monad corresponding to the forgetful functor $\mathsf{Mon}(R) \to \mathsf{Set}$ is monoidal if and only if $R$ is commutative. | |
| Jan 7, 2013 at 17:24 | comment | added | Mariano Suárez-Álvarez | @Martin:the claim that «Mod(R) is not monoidal» should be somewhat tamed by some hypotheses, as you could take $R$ to be a bialgebra, say. | |
| Jan 7, 2013 at 17:19 | answer | added | Mariano Suárez-Álvarez | timeline score: 3 | |
| Jan 7, 2013 at 11:52 | comment | added | Martin Brandenburg | @Mike: What is this extra structure? If $R$ is a non-commutative ring, then $\mathsf{Mod}(R)$ is not monoidal. And for commutative rings Morita-equivalence is just isomorphism. | |
| Jan 7, 2013 at 11:14 | answer | added | Martin Brandenburg | timeline score: 21 | |
| Jan 7, 2013 at 8:14 | comment | added | Bruce Westbury | There is a functor from finite sets which constructs the vector space with basis the finite set. This functor should have some good properties. | |
| Jan 7, 2013 at 8:11 | comment | added | Bruce Westbury | All finite limits and colimits exist. | |
| Jan 7, 2013 at 8:10 | comment | added | Bruce Westbury | It is rigid symmetric, is this the same as compact closed? | |
| Jan 7, 2013 at 8:07 | comment | added | David Roberts♦ | Really? If so, it's still good to know how close we can get. | |
| Jan 7, 2013 at 7:59 | comment | added | Mike Shulman | I think Morita equivalent rings should have equivalent module categories that preserve all the extra structure in sight as well. | |
| Jan 7, 2013 at 7:48 | comment | added | John Wiltshire-Gordon | Let f be an endomorphism of I. The object I is simple, so f has kernel 0 or all of I. If the kernel is 0, then f has a left inverse since every injection is split in a semisimple category. Using images instead of kernels gives right inverses as well. This shows that End(I) is a division ring. | |
| Jan 7, 2013 at 6:56 | comment | added | David Roberts♦ | I should point out this tangentially relevant paper: arxiv.org/abs/0807.2927 | |
| Jan 7, 2013 at 6:08 | history | asked | David Roberts♦ | CC BY-SA 3.0 |