Category Theory for Programmers

Preface
Part One
	Category: The Essence of Composition
		Arrows as Functions
		Properties of Composition
		Composition is the Essence of Programming
		Challenges
	Types and Functions
		Who Needs Types?
		Types Are About Composability
		What Are Types?
		Why Do We Need a Mathematical Model?
		Pure and Dirty Functions
		Examples of Types
		Challenges
	Categories Great and Small
		No Objects
		Simple Graphs
		Orders
		Monoid as Set
		Monoid as Category
		Challenges
	Kleisli Categories
		The Writer Category
		Writer in Haskell
		Kleisli Categories
		Challenge
	Products and Coproducts
		Initial Object
		Terminal Object
		Duality
		Isomorphisms
		Products
		Coproduct
		Asymmetry
		Challenges
		Bibliography
	Simple Algebraic Data Types
		Product Types
		Records
		Sum Types
		Algebra of Types
		Challenges
	Functors
		Functors in Programming
			The Maybe Functor
			Equational Reasoning
			Optional
			Typeclasses
			Functor in C++
			The List Functor
			The Reader Functor
		Functors as Containers
		Functor Composition
		Challenges
	Functoriality
		Bifunctors
		Product and Coproduct Bifunctors
		Functorial Algebraic Data Types
		Functors in C++
		The Writer Functor
		Covariant and Contravariant Functors
		Profunctors
		The Hom-Functor
		Challenges
	Function Types
		Universal Construction
		Currying
		Exponentials
		Cartesian Closed Categories
		Exponentials and Algebraic Data Types
			Zeroth Power
			Powers of One
			First Power
			Exponentials of Sums
			Exponentials of Exponentials
			Exponentials over Products
		Curry-Howard Isomorphism
		Bibliography
	Natural Transformations
		Polymorphic Functions
		Beyond Naturality
		Functor Category
		2-Categories
		Conclusion
		Challenges
Part Two
	Declarative Programming
	Limits and Colimits
		Limit as a Natural Isomorphism
		Examples of Limits
		Colimits
		Continuity
		Challenges
	Free Monoids
		Free Monoid in Haskell
		Free Monoid Universal Construction
		Challenges
	Representable Functors
		The Hom Functor
		Representable Functors
		Challenges
		Bibliography
	The Yoneda Lemma
		Yoneda in Haskell
		Co-Yoneda
		Challenges
		Bibliography
	Yoneda Embedding
		The Embedding
		Application to Haskell
		Preorder Example
		Naturality
		Challenges
Part Three
	It's All About Morphisms
		Functors
		Commuting Diagrams
		Natural Transformations
		Natural Isomorphisms
		Hom-Sets
		Hom-Set Isomorphisms
		Asymmetry of Hom-Sets
		Challenges
	Adjunctions
		Adjunction and Unit/Counit Pair
		Adjunctions and Hom-Sets
		Product from Adjunction
		Exponential from Adjunction
		Challenges
	Free/Forgetful Adjunctions
		Some Intuitions
		Challenges
	Monads: Programmer's Definition
		The Kleisli Category
		Fish Anatomy
		The do Notation
	Monads and Effects
		The Problem
		The Solution
			Partiality
			Nondeterminism
			Read-Only State
			Write-Only State
			State
			Exceptions
			Continuations
			Interactive Input
			Interactive Output
		Conclusion
	Monads Categorically
		Monoidal Categories
		Monoid in a Monoidal Category
		Monads as Monoids
		Monads from Adjunctions
	Comonads
		Programming with Comonads
		The Product Comonad
		Dissecting the Composition
		The Stream Comonad
		Comonad Categorically
		The Store Comonad
		Challenges
	F-Algebras
		Recursion
		Category of F-Algebras
		Natural Numbers
		Catamorphisms
		Folds
		Coalgebras
		Challenges
	Algebras for Monads
		T-algebras
		The Kleisli Category
		Coalgebras for Comonads
		Lenses
		Challenges
	Ends and Coends
		Dinatural Transformations
		Ends
		Ends as Equalizers
		Natural Transformations as Ends
		Coends
		Ninja Yoneda Lemma
		Profunctor Composition
	Kan Extensions
		Right Kan Extension
		Kan Extension as Adjunction
		Left Kan Extension
		Kan Extensions as Ends
		Kan Extensions in Haskell
		Free Functor
	Enriched Categories
		Why Monoidal Category?
		Monoidal Category
		Enriched Category
		Preorders
		Metric Spaces
		Enriched Functors
		Self Enrichment
		Relation to 2-Categories
	Topoi
		Subobject Classifier
		Topos
		Topoi and Logic
		Challenges
	Lawvere Theories
		Universal Algebra
		Lawvere Theories
		Models of Lawvere Theories
		The Theory of Monoids
		Lawvere Theories and Monads
		Monads as Coends
		Lawvere Theory of Side Effects
		Challenges
		Further Reading
	Monads, Monoids, and Categories
		Bicategories
		Monads
		Challenges
		Bibliography
Appendices
	Index
	Acknowledgments
	Colophon
	Copyleft notice