Study of monoidal categories

设A是一个张量范畴。我们定义范畴D（A）如下。D（A）的一个对象由以下数据组成：A的所有对象X和Y的向量空间F（X，Y），A的所有态射X '->X和Y->Y'的线性映射F（X，Y）-->F（X '，Y'），以及A的所有对象X，Y和Z的线性映射F（X，Y）--> F（ZX，ZY）和F（X，Y）-->F（XZ，YZ）。这些数据必须满足一定的条件。这里我们用ZX表示A中Z和X的张量积。当A是半单的时，范畴D（A）等价于A的中心。作为非半单张量范畴的一个有趣例子，我们取A为有限群G的Mackey范畴M。则D（M）等价于具有自同构的连通G-集范畴上的一类Mackey函子范畴。

Let A be a tensor category. We define the category D(A) as follows. An object of D(A) consists of the following data: vector spaces F(X,Y) for all objects X and Y ofA, linear maps F(X,Y)-->F(X',Y') for all morphisms X'-->X and Y->Y' of A, and linear maps F(X,Y) -->F(ZX,ZY) and F(X,Y) -->F(XZ,YZ) for all objects X, Y, and Z of A. These data should satisfy certain conditions. Here we denote by ZX the tensorproduct of Z and X in A. When A is semi-simple, the category D(A) is equivalent to the center of A. As an interesting example of non semi-simple tensor categories, we take A to be the Mackey category M of a finite group G. Then D(M) turns out to be equivalent to the category of a kind of Mackey functors over the category of connected G-sets equipped with automorphisms.