The book "Basic Concepts of Enriched Category Theory" by G.M. Kelly provides a comprehensive treatment of enriched category theory. It introduces the concept of a monoidal category, which is a category equipped with a tensor product and a unit object, along with natural isomorphisms that ensure associativity and unit properties. The text then explores the 2-category V-CAT of V-categories, V-functors, and V-natural transformations, and discusses the forgetful 2-functor from V-CAT to CAT. The book also delves into symmetric monoidal categories, which are monoidal categories with a symmetry isomorphism that allows the tensor product to be commutative. It examines closed and biclosed monoidal categories, where the tensor product has a right adjoint, the internal hom, and the category is equipped with both left and right adjoints. The text discusses the tensor product of V-categories, the Yoneda lemma, and the concept of representable V-functors. The book covers various topics such as ends, functor categories, indexed limits and colimits, Kan extensions, density, and essentially-algebraic theories. It provides a detailed analysis of these concepts, their properties, and their applications in enriched category theory. The text also includes a bibliography and index for reference. The book is structured to provide a self-contained account of enriched category theory, assuming only the most elementary categorical concepts and treating both ordinary and enriched cases. The authors acknowledge the contributions of various individuals and institutions in the production and publication of the book.The book "Basic Concepts of Enriched Category Theory" by G.M. Kelly provides a comprehensive treatment of enriched category theory. It introduces the concept of a monoidal category, which is a category equipped with a tensor product and a unit object, along with natural isomorphisms that ensure associativity and unit properties. The text then explores the 2-category V-CAT of V-categories, V-functors, and V-natural transformations, and discusses the forgetful 2-functor from V-CAT to CAT. The book also delves into symmetric monoidal categories, which are monoidal categories with a symmetry isomorphism that allows the tensor product to be commutative. It examines closed and biclosed monoidal categories, where the tensor product has a right adjoint, the internal hom, and the category is equipped with both left and right adjoints. The text discusses the tensor product of V-categories, the Yoneda lemma, and the concept of representable V-functors. The book covers various topics such as ends, functor categories, indexed limits and colimits, Kan extensions, density, and essentially-algebraic theories. It provides a detailed analysis of these concepts, their properties, and their applications in enriched category theory. The text also includes a bibliography and index for reference. The book is structured to provide a self-contained account of enriched category theory, assuming only the most elementary categorical concepts and treating both ordinary and enriched cases. The authors acknowledge the contributions of various individuals and institutions in the production and publication of the book.