This text and reference book is aimed not only at mathematicians, but also researchers and students of computer science, logic, linguistics, cognitive science, philosophy, and any of the other fields in which the ideas are being applied. Containing clear definitions of the essential concepts, illuminated with numerous accessible examples, and providing full proofs of all important propositions and theorems, this book aims to make the basic ideas, theorems, and methods of category theory understandable to this broad readership. Although assuming few mathematical pre-requisites, the standard of mathematical rigour is not compromised. An extra topic of cartesian closed categories and the lambda-calculus is also provided - a must for computer scientists, logicians and linguists!

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A category C consists of the following three mathematical entities: A class ob C , whose elements are called objects; A class hom C , whose elements are called morphisms or maps or arrows.

Each morphism f has a source object a and target object b. The expression hom a, b — alternatively expressed as homC a, b , mor a, b , or C a, b — denotes the hom-class of all morphisms from a to b. From the axioms, it can be proved that there is exactly one identity morphism for every object.

Some authors deviate from the definition just given by identifying each object with its identity morphism. Morphisms can have any of the following properties. Every retraction is an epimorphism, and every section is a monomorphism. Furthermore, the following three statements are equivalent: f is a monomorphism and a retraction; f is an epimorphism and a section; f is an isomorphism.

Main article: Functor Functors are structure-preserving maps between categories. They can be thought of as morphisms in the category of all small categories. In other words, a contravariant functor acts as a covariant functor from the opposite category Cop to D. Main article: Natural transformation A natural transformation is a relation between two functors. Functors often describe "natural constructions" and natural transformations then describe "natural homomorphisms" between two such constructions.

Sometimes two quite different constructions yield "the same" result; this is expressed by a natural isomorphism between the two functors.

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## Category theory

A category C consists of the following three mathematical entities: A class ob C , whose elements are called objects; A class hom C , whose elements are called morphisms or maps or arrows. Each morphism f has a source object a and target object b. The expression hom a, b — alternatively expressed as homC a, b , mor a, b , or C a, b — denotes the hom-class of all morphisms from a to b. From the axioms, it can be proved that there is exactly one identity morphism for every object. Some authors deviate from the definition just given by identifying each object with its identity morphism.

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## Steve Awodey

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