A comprehensive guide to category theory, its concepts, applications, and role in mathematics and computer science.
Category Theory
Category theory is a unifying theory in mathematics that abstracts mathematical concepts and relationships between them into collections of objects and arrows.
Fundamental Concepts
Categories
Definition: Collection of objects and morphisms between them
Properties:
Composition of morphisms
Identity morphisms
Associativity
Unit laws
Functors
Definition: Structure-preserving maps between categories
Types:
Covariant functors
Contravariant functors
Endofunctors
Bifunctors
Natural Transformations
Definition: Morphisms between functors
Properties:
Naturality squares
Component maps
Vertical composition
Horizontal composition
Advanced Concepts
Universal Properties
Definition: Characterizations of mathematical structures through their relationships
Examples:
Products and coproducts
Limits and colimits
Adjoint functors
Kan extensions
Adjunctions
Definition: Pairs of functors with special relationships
Properties:
Unit and counit
Adjoint functor theorem
Examples in various categories
Applications
Monads and Comonads
Definition: Endofunctors with additional structure
Components:
Unit and multiplication
Kleisli category
Eilenberg-Moore category
Applications in programming
Applications
Mathematics
Algebraic Structures
Groups and group actions
Rings and modules
Vector spaces
Topological spaces
Homological Algebra
Chain complexes
Derived functors
Spectral sequences
Cohomology theories
Computer Science
Programming Language Theory
Type systems
Denotational semantics
Effect systems
Program verification
Functional Programming
Haskell's category theory foundations
Monadic programming
Type classes
Generic programming
Other Fields
Physics
Quantum mechanics
TQFT
String theory
Quantum computing
Logic and Foundations
Topos theory
Internal languages
Categorical logic
Proof theory
Learning Resources
Books
"Category Theory for Programmers" (Bartosz Milewski)
"Categories for the Working Mathematician" (Saunders Mac Lane)