Category Theory in ContextCategory theory has provided the foundations for many of the twentieth century's greatest advances in pure mathematics. This concise, original text for a one-semester course on the subject is derived from courses that author Emily Riehl taught at Harvard and Johns Hopkins Universities. The treatment introduces the essential concepts of category theory: categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads, and other topics. Suitable for advanced undergraduates and graduate students in mathematics, the text provides tools for understanding and attacking difficult problems in algebra, number theory, algebraic geometry, and algebraic topology. Drawing upon a broad range of mathematical examples from the categorical perspective, the author illustrates how the concepts and constructions of category theory arise from and illuminate more basic mathematical ideas. Prerequisites are limited to familiarity with some basic set theory and logic. |
Contents
Categories Functors Natural Transformations | 1 |
Universal Properties Representability and the Yoneda Lemma | 49 |
Limits and Colimits | 73 |
Adjunctions | 115 |
viii | 143 |
CONTENTS | 153 |
All Concepts are Kan Extensions | 189 |
Theorems in Category Theory | 217 |
| 225 | |
Glossary of Notation | 231 |
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Common terms and phrases
abelian group adjoint functors adjunction F axioms bifunctor bijection category of algebras category of elements category of sets category theory codomain coequalizer diagram colim comma category components composite construction contravariant functor coproduct Corollary counit covariant Definition derived functors diagram F directed graph domain dual Dually endomorphism epimorphism equivalence of categories Example Exercise exists finite forgetful functor function f functor F groupoid homomorphisms homotopy implies inclusion induced initial object inverse Kan extension left adjoint left Kan extension limit cone limits and colimits linear maps locally small category map f monad monomorphism morphism f natural isomorphism natural transformation numbers parallel pair pointwise poset preserves PROOF Proposition prove pullback quotient representable functors right adjoint right Kan extension subset surjective T-algebra terminal object Theorem topological space U-split underlying sets unique unital ring universal property vector space Yoneda lemma