Timeline for Examples of strict monoidal categories and monoidal categories with nontrivial associators
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 29, 2020 at 19:00 | vote | accept | Jake Wetlock | ||
| Mar 29, 2020 at 15:42 | comment | added | Tim Campion | In this paper Mark Hovey studies the problem of classifying symmetric monoidal closed structure on the category $Mod_R$ of $R$-modules for a ring $R$. For example, when $R$ is a field, there is a unique such, and when $R = \mathbb F_2[C_2]$ there are exactly 7 up to equivalence. He doesn't cleanly separate out the problem of understanding non-symmetric monoidal structures, but I think his analysis should probably be enlightening. | |
| Mar 29, 2020 at 8:35 | answer | added | Bugs Bunny | timeline score: 3 | |
| Mar 28, 2020 at 21:44 | comment | added | Mike Shulman | It's hard to tell what you mean by "non-trivial" if you don't know any examples. One might go so far as to claim that the associator always consists of rewriting of brackets, essentially by definition. How about a 2-group constructed from a group cohomology class; would you regard its associator as always "trivial"? | |
| Mar 28, 2020 at 17:52 | comment | added | Tim Campion | Regarding (i), every monoidal category is monoidally equivalent to a strict monoidal category. An example of a "naturally occurring" strict monoidal category is the category $\Delta_+$ of finite ordinals, under ordinal sum. This is the free strict monoidal category on a monoid object. I find (ii) more interesting -- in particular, I believe there are interesting examples of categories equipped with a tensor bifunctor which admit more than one possible associator, and the resulting monoidal structures are not equivalent. I hope somebody can provide such examples. | |
| Mar 28, 2020 at 17:21 | history | asked | Jake Wetlock | CC BY-SA 4.0 |