Categories for the working mathematician

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Category Theory has seen rapid development, and this book presents ideas and methods now effectively utilized by Mathematicians across various fields of research. It begins by introducing categories as a conceptual language, focusing on key notions such as category, functor, natural transformation, contravariance, and functor category in Chapters I and II. The book then delves into the fundamental concept of adjoint pairs of functors, which manifest in various equivalent forms, including universal construction, direct and inverse limits, and pairs of functors with natural isomorphisms between corresponding sets of arrows. These interrelated forms are explored in Chapters III to V, emphasizing the idea that "Adjoint functors arise everywhere." Another core concept is the monoid, defined as a set with an associative binary operation and a unit; categories can be viewed as generalized monoids. Chapters VI and VII investigate this idea and its generalizations, highlighting its connection to adjoint functors and universal algebra, culminating in Beck's theorem that characterizes categories of algebras. Furthermore, categories with a monoidal structure, defined by a tensor product, lead to the exploration of more convenient categories of topological spaces.