  We firstly introduce some key concepts in category theory, such as quotient
category, completion of limits, $\mathrm{Mor}$ category, and so on; then give
the concept of topology algebras and sheaves, and discuss how to restore the
structue of sheaves from their stalks; lastly, we introduce the
sheaf-theoretical expression for topological spaces, and rediscribe some
essential items in topology and geometry by defining a kind of generally
existing category sheaves.
