Abstract. The Grothendieck K₀ functor sends a `ring like category' to a ring (the `fusion ring'). Can one add more structure on the fusion rings and appropriately `extend' K<sub>0</sub> so that the new functor will be able to distinguish nonequivalent ring like categories? Can one effectively reconstruct the `ring like' category? Can one effectively describe the fusion rings that are in the `image' of K₀? I cannot answers these questions but think they are important ones and should be addressed.

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