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.
Categories: math.CT