Algebraic Topology is a branch of mathematics that combines techniques from algebra and topology to study spaces and their properties. It seeks to understand the fundamental structure and properties of spaces by using algebraic tools. In Algebraic Topology, algebraic invariants are assigned to [[Topological Space|topological spaces]] in order to classify them up to certain equivalence relations. These invariants often arise from various algebraic structures, such as groups, rings, modules, or categories. By studying these invariants, Algebraic Topology aims to reveal the underlying geometric and topological features of spaces. Category Theory, on the other hand, is a branch of mathematics that provides a framework for studying mathematical structures and relationships between them. It focuses on the abstract properties of objects and morphisms within categories, which are mathematical structures consisting of objects and arrows (morphisms) that link these objects. The relationship between [[Algebraic Topology]] and [[Category Theory]] lies in their common focus on understanding mathematical structures. Category Theory provides a powerful language for describing mathematical concepts and relationships in a general way, making it applicable to various branches of mathematics including [[Algebraic Topology]]. [[Allen Hatcher]] has an online book on this subject matter. In fact, many concepts and techniques in Algebraic Topology can be understood more abstractly using Category Theory. For example, the concept of a homotopy group can be formulated categorically using higher category theory. Furthermore, Category Theory provides tools for studying functors between categories, which play an important role in relating different aspects of Algebraic Topology. Overall, Category Theory provides a formalism that helps organize and understand the underlying structure and relationships within Algebraic Topology. It allows for a more abstract approach to studying spaces and their properties by providing a unified framework for analyzing various algebraic structures associated with topology. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Topological Space") or contains(subject, "Algebraic Topology") ```