Timeline for locally finitely presentable tensor categories
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 6, 2017 at 3:16 | vote | accept | Martin Brandenburg | ||
| Jun 1, 2017 at 17:04 | comment | added | Martin Brandenburg | 1) For locally finitely presentable we need that every object is a filtered colimit of finitely presentable objects. 2) When $I$ is an infinite set, the unit object in $\mathsf{Mod}(k)^I$ is just $(k)_{i \in I}$ and therefore an infinite direct sum of copies of $k$ placed in degree $i$. This is not finitely generated. | |
| May 31, 2017 at 12:11 | comment | added | Greg Stevenson | Concerning the second comment, perhaps I'm missing something but won't the representable functors always give you a set of finitely presented projective generators? Concerning the first comment, isn't the unit object in that example actually finitely presented? The colimits and maps are all pointwise, so it seems to me the fact that $I$ has lots of objects doesn't matter much. One should be able to obtain examples where the unit isn't finitely presented by looking at something like categories of complete modules over an adically complete ring though. | |
| May 31, 2017 at 9:09 | comment | added | Martin Brandenburg | In general, $[I,\mathsf{Mod}(k)]$ might not be locally finitely presentable? Anyway, it is the case if $I$ is discrete or has only one object. | |
| May 31, 2017 at 6:01 | comment | added | Martin Brandenburg | Thank you. One can also take $k[t]$ for instance. One gets more example by considering functor categories $[I,\mathsf{Mod}(k)]$ for suitable small categories $I$, where the tensor product is defined pointwise? For instance, when $I$ is infinite discrete, the unit object is not finitely presentable, right? | |
| May 30, 2017 at 15:32 | history | answered | Greg Stevenson | CC BY-SA 3.0 |