Category theory : definitions and examples

Category theory was invented as an abstract language for describing certain structures and constructions which repeatedly occur in many branches of mathematics, such as topology, algebra, and logic. In recent years, it has found several applications in computer science, e.g., algebraic specification, type theory, and programming language semantics. In this paper, we collect definitions and examples of the basic concepts in category theory: categories, functors, natural transformations, universal properties, limits, and adjoints.