One of the central highlights of this work is the exploration of the Yoneda lemma and its profound implications, during which intuitive explanations are provided, as well as detailed proofs, and specific examples. This book covers aspects of category theory often considered advanced in a clear and intuitive way, with rigorous mathematical proofs. It investigates universal properties, coherence, the relationship between categories and graphs, and treats monads and comonads on an equal footing, providing theorems, interpretations and concrete examples. Finally, this text contains an introduction to monoidal categories and to strong and commutative monads, which are essential tools in current research but seldom found in other textbooks.
Starting Category Theory serves as an accessible and comprehensive introduction to the fundamental concepts of category theory. Originally crafted as lecture notes for an undergraduate course, it has been developed to be equally well-suited for individuals pursuing self-study. Most crucially, it deliberately caters to those who are new to category theory, not requiring readers to have a background in pure mathematics, but only a basic understanding of linear algebra.
Readership: This book is primarily targeted towards undergraduate and graduate students in mathematics and related fields (physics, computer science, statistics, engineering), and is suitable for either course adoption for category theory and discrete mathematics, or for self-study. More broadly, this book can appeal to researchers in related fields and professionals working in technology (machine learning, etc.).
Author(s): Paolo Perrone
Edition: 1
Publisher: World Scientific
Year: 2024
Language: English
Pages: 464
Tags: Categories, Yoneda Lemma, Limits, Colimits, Adjunctions, Monads, Comonads, Monoidal Categories
Contents
Preface
Acknowledgments
About the Author
1 Basic Concepts
1.1 Categories
1.1.1 Categories defined by relations
1.1.2 Categories defined by operations
1.1.3 Categories defined by spaces and maps with extra structure
1.1.4 Set-theoretical considerations
1.1.5 Isomorphisms and groupoids
1.1.6 Diagrams: An informal definition
1.1.7 The opposite category
1.2 Mono and Epi
1.2.1 Monomorphisms
1.2.2 Split monomorphisms
1.2.3 Epimorphisms
1.2.4 Split epimorphisms
1.3 Functors
1.3.1 Functors as mappings preserving relations
1.3.2 Functors as mappings preserving operations
1.3.3 Functors defining induced maps
1.3.4 Functors and cocycles
1.3.5 Functors, mono and epi
1.3.6 What is not a functor?
1.3.7 Hom-functors, contravariant functors, and presheaves
1.3.8 Further particular functors
1.4 Natural Transformations
1.4.1 Natural transformations as systems of arrows
1.4.2 Natural transformations as structure-compatible mappings
1.4.3 Natural transformations as compatible systems of maps
1.4.4 What is not natural?
1.4.5 Functor categories and diagrams
1.4.6 Whiskering and horizontal composition
1.5 Studying Categories by Means of Functors
1.5.1 The category of categories
1.5.2 Subcategories
1.5.3 Full, faithful, and essentially surjective
1.5.4 Equivalences of categories
2 The Yoneda Lemma
2.1 Representable Functors and the Yoneda Embedding Theorem
2.1.1 Extracting sets from objects
2.1.2 Representable functors
2.1.3 The Yoneda embedding theorem
2.2 Statement and Proof of the Yoneda Lemma
2.2.1 Proof of the Yoneda lemma
2.2.2 Particular cases
2.3 Universal Properties
2.3.1 The universal property of the cartesian product
2.3.2 The universal property of the tensor product
3 Limits and Colimits
3.1 General Definitions
3.2 Particular Limits and Colimits
3.2.1 The poset case: Constrained optimization
3.2.2 The group case: Invariants and orbits
3.2.3 Products and coproducts
3.2.4 Equalizers and coequalizers
3.2.5 Pullbacks and pushouts
3.2.6 Initial and terminal objects, trivial cases
3.2.7 Sequential and (co)filtered limits and colimits
3.3 Functors, Limits and Colimits
3.3.1 The power set, probability functors, and complexity
3.3.2 Continuous functors and equivalence
3.3.3 The case of representable functors and presheaves
3.4 Limits and Colimits of Sets
3.4.1 Completeness of the category of sets
3.4.2 General proof of Theorem 3.3.16
4 Adjunctions
4.1 General Definitions
4.1.1 Free-forgetful adjunctions
4.1.2 Galois connections
4.2 Unit and Counit
4.2.1 Alternative definition of adjunctions
4.2.2 Example: Adjunction between categories and multigraphs
4.3 Adjunctions, Limits and Colimits
4.3.1 Right-adjoints and binary products
4.3.2 General proof of Theorem 4.3.1
4.3.3 Examples
4.4 The Adjoint Functor Theorem for Preorders
4.4.1 The case of convex subsets
4.4.2 Proof of Theorem 4.4.1
4.4.3 Further considerations and examples
5 Monads and Comonads
5.1 Monads as Extensions of Spaces
5.1.1 Kleisli morphisms
5.1.2 The Kleisli adjunction
5.1.3 Closure operators and idempotent monads
5.2 Monads as Theories of Operations
5.2.1 Algebras of a monad
5.2.2 Free algebras
5.2.3 The Eilenberg–Moore adjunction
5.3 Comonads as Extra Information
5.3.1 Co-Kleisli morphisms
5.3.2 The co-Kleisli adjunction
5.4 Comonads as Processes on Spaces
5.4.1 Coalgebras of a comonad
5.4.2 The adjunction of coalgebras
5.5 Adjunctions, Monads, and Comonads
5.5.1 The adjunction between categories and multigraphs is monadic
6 Monoidal Categories
6.1 General Definitions
6.1.1 Categories with multiplications and coherence
6.1.2 Parallel composition and string diagrams
6.1.3 More examples of monoidal categories
6.1.4 Points or states
6.2 Monoids and Comonoids
6.2.1 Internal monoids
6.2.2 Internal modules
6.2.3 Internal comonoids
6.2.4 Internal comodules
6.3 Monoidal Functors
6.3.1 Lax, colax, and strong-monoidal functors
6.3.2 Examples, interpretation, and more on complexity
6.3.3 Monoidal transformations and monoidal equivalences
6.3.4 Strictification
6.4 Monads on Monoidal Categories
6.4.1 Monoidal monads
6.4.2 Strong monads
6.4.3 Commutative monads
6.4.4 The tensor product of algebras and of Kleisli morphisms
6.5 Closed Monoidal Categories
6.5.1 Examples of internal homs
6.5.2 Functoriality of the internal hom
6.5.3 Evaluation and coevaluation
6.5.4 Strong monads on closed monoidal categories
6.5.5 The internal hom of algebras
Conclusion
Bibliography
Index
