中文摘要：

双线性扩张范畴f=sf[M，N，φ，φ]f的模范畴fMod是双扩张代数的自然推广，且等价于四元组范畴xf,作为应用，给出了2-循环复形范畴与特殊的双扩张范畴等价的证明，以及由范畴f或B中的某些AR-序列可得tF中的部分AR序列．

英文摘要：

The notion of the left module category over k-linear double-extension categories is a generalization of a double-extension algebra, and it is isomphie to the quadruple category ,eft.. As a main application of the result we use a special double-extension category to describe a 2-cycle complex category. And more, from some Austander-Reiten sequences in A or B we can obtain some Auslander- Reiten sequences in the corresponding quadruple category eF.