An alternative description of braided monoidal categories (Q2350313)

An algebra (not necessarily unital or associative) \(A\) is called a b-algebra if its multiplication satisfies \(a(bc)=b(ac)\) for \(a,b,c\in A\). It is not hard to show that if a b-algebra \(A\) has a right unit then it is commutative and associative. The categorical version of an associative commutative algebra is a braided monoidal category. Therefore, if we can categorify the concept b-algebra then we will get another kind of description of braided monoidal category. This is indeed the main purpose of this paper. Precisely, the authors defined the concept b-category in this paper and showed that a b-category with right unit is the same as a braided monoidal category. Moreover, the authors defined b-bicategories and gave some applications of these new concepts.