Category Theory Using Haskell

Preface
	References
Contents
1 Category, Functor, Natural Transformation
	1.1 Sets and Functions
	1.2 Category, Object, Morphism
	1.3 Data Structures of Categories
	1.4 Functor and Contravariant Functor
	1.5 Faithful Functors and Full Functors
	1.6 Natural Transformations
	1.7 Subcategories of `3́9`42`\"̇613A``45`47`\"603AHask
2 Equivalence of Categories
	2.1 Functor Category
	2.2 Equivalence of Categories
3 Universality and Limits
	3.1 Initial and Terminal Objects
	3.2 Products
	3.3 Coproducts
	3.4 Limits
	3.5 Colimits
	3.6 Existence of Limits
	3.7 Existence of Colimits
4 Functors and Limits
	4.1 Functors Map Cones to Cones
	4.2 `3́9`42`\"̇613A``45`47`\"603AHom Functors and Limits
	4.3 `3́9`42`\"̇613A``45`47`\"603AHom Functors and Colimits
	4.4 Executable Examples
	4.5 Limit Functors
		4.5.1 Limits as a Functor
		4.5.2 Colimits as a Functor
5 Adjoints
	5.1 Adjunctions
	5.2 Unit and Counit
	5.3 Triangle Identities
	5.4 Universal Arrows and Adjunctions
	5.5 Equivalent Formulations of Adjunction
	5.6 Adjunction and Category Equivalence
	5.7 Global Naturality of Adjunction
	5.8 Adjunctions and Limits
6 Monads
	6.1 Kleisli Triples
	6.2 Moggi\' Theory
		6.2.1 Non-determinism and the List Functor
		6.2.2 Computation with Output
		6.2.3 Continuation Passing
		6.2.4 Computation that May Fail and the Maybe Functor
	6.3 Monads
	6.4 Monad Instances in Haskell
		6.4.1 Computation with Output(revisited)
		6.4.2 Computation with Continuation(revisited)
	6.5 Functor, Applicative, and Monad
	6.6 Monads and Adjoints
		6.6.1 Adjoints to Monads
		6.6.2 List Monad as Adjoint
7 Representable Functors
	7.1 Representation of Covariant Functors
	7.2 Representation of Contravariant Functors
	7.3 The Yoneda Lemma
		7.3.1 Covariant Case
		7.3.2 Contravariant Case (Presheaves)
	7.4 Milewski\'s ``Understanding Yoneda\'\'
		7.4.1 Specification Determines Implementation
		7.4.2 A Function That Has the Same Signature as the Identity Function
	7.5 Reverse Engineering by the Yoneda Lemma
		7.5.1 A Machine That Has a Hidden Parameter
		7.5.2 A Machine That Hides a List Inside
		7.5.3 A Machine That Has a Secret Morphism
		7.5.4 The Design Pattern with Yoneda
	7.6 Adjoints Preserve Limits
8 Monoidal Categories and Coherence
	8.1 Categories with Tensor Products
	8.2 Coherence—Part One
	8.3 Monoidal Functors
	8.4 Coherence—Part Two
Epilogue
Appendix A Sets and Universe
Appendix B Calculations with Output
	B.1 Functor Instance
	B.2 Applicative Instance
	B.3 Monad Instance
Appendix C Continuation Passing
	C.1 Functor Instance
	C.2 Applicative Instance
	C.3 Monad Instance
Index