中文摘要：

在Fuzzifying（模糊化）数学的框架下，建立了Fuzzifying闭包系统和Birkhoff型Fuzzifying闭包算子的概念；引入了Fuzzifying闭包空间范畴和Fuzzifying闭包系统空间范畴，并从范畴论的角度证明Birkhoff型Fuzzifying闭包算子与Fuzzifying闭包系统是协调的。最后文中还得到Fuzzifying闭包空间范畴和Fuzzifying闭包系统空间范畴可以嵌入到Birkhoff型L-闭包空间范畴这一重要结果。

英文摘要：

In this paper, in the framework of Fuzzifying mathematics, we propose the concepts of Fuzzifying closure system and its Birkhoff Fuzzifying closure operator; Then we introduce the category of Fuzzifying closure spaces and the category of Fuzzifying closure system spaces, furthermore, it is proved that Birkhoff Fuzzifying closure operator coordinates with Fuzzifying closure system in viewpoints of the category theory. At last, we obtain an important result that the category of Fuzzifying closure sapces and that of Fuzzifying closure system spaces can be both embedded in the category of Birkhoff I-closure spaces.