关键词: 范畴理论，数据降维，线性降维范畴，非线性降维范畴

Abstract: The research of category theory is mainly about summary and abstraction of some specific mathematical objects and mapping. The dimensionality reduction problem based on category theory was discussed, and it could solve the problems of image analysis and image recognition. Besides, the process of dimensionality reduction based on category theory was listed. There are two examples,Principal Component Analysis Category and Isomap Category,which verified the correctness of the category theory applied to the dimensionality reduction problem.

Key words: Category theory, Data dimension reduction, Linear dimension reduction category, Nonlinear dimension reduction category