Both this post on mathoverflow and this wikipedia page claim that you can reconstruct a monoidal category from its Grothendieck ring and $6j$-symbols (or equivalently the associator).
Bruce Westbury in his post is more careful and asks that this category is semisimple. I can imagine that one can reconstruct a semisimple monoidal category from its Grothendieck ring and associator. But without semisimplicity it seems to me that you are missing a lot of information about the morphisms of this category. (So adding the Auslander algebra would probably suffice to reconstruct it.)
My question is twofold:
- Can I get some reference proving that you can reconstruct a semisimple monoidal category from its Grothendieck ring and associator?
- Is it true/false without semisimplicity?