Abelian Categories

An abelian category is a mathematical framework that generalizes familiar algebraic structures like groups, vector spaces, and modules. It provides a setting where objects and their relationships can be studied systematically, with concepts such as kernels (generalizing solutions to equations), cokernels, and exact sequences (tracking how structures relate). In an abelian category, morphisms behave nicely—adding, subtracting, and composing them aligns with intuition. This structure allows mathematicians to analyze complex algebraic and geometric ideas using a unified language, making it fundamental in areas like homological algebra and algebraic geometry.