We study a categorical generalization of the usual notion of basis. The chief concern is the exploration of the monoidal behaviour and transfer of these bases, and the connection with semisimplicity. Precisely, we construct a large strict monoidal Ab−category \(\textbf{Ba}(\mathcal {C})\) whose objects are bases on dualizable objects of a given strict monoidal Ab−category \(\mathcal {C}\) and show that it inherits the structures of a braiding and twist from \(\mathcal {C}\) . Moreover, we show that the dualizable objects of \(\textbf{Ba}(\mathcal {C})\) are precisely those defined by the usual duality maps on objects of \(\mathcal {C}\) and that this latter is naturally isomorphic to the full subcategory of dualizable objects of \(\textbf{Ba}(\mathcal {C})\) . In order to examine the transfer of bases, we determine sufficient conditions on a functor F between rigid Ab−categories, to ensure its extension to a functor \(\textbf{Ba}(F)\) between the corresponding categories of bases. Furthermore, this procedure reflects isomorphisms in the sense that F is an isomorphism if and only if so is \(\textbf{Ba}(F)\) .
