# User:Michiexile/MATH198

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==Course overview== |
==Course overview== |
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− | Page is work in progress for background material for the Fall 2009 lecture course on Category Theory with a view towards applications that I am planning to give at Stanford University. |
+ | Page is the background material for the Fall 2009 lecture course MATH198[http://coursework.stanford.edu/homepage/F09/F09-MATH-198-01.html] on Category Theory and Functional Programming that I gave at Stanford University. |

− | Single unit course. 10 lectures. |
+ | Single unit course. 10 lectures. Each lecture is Wednesday 4.15-5.05 in 380F. |

− | * Category: Definition and examples. |
+ | |

− | * Concrete categories. |
+ | * [[User:Michiexile/MATH198/Lecture 1]] |

− | ** Set. |
+ | ** Category: Definition and examples. |

− | ** Various categories capturing linear algebra. |
+ | ** Concrete categories. |

− | * Small categories. |
+ | *** Set. |

− | ** Partial orders. |
+ | *** Various categories capturing linear algebra. |

− | ** Finite groups. |
+ | ** Small categories. |

− | * Special morphisms |
+ | *** Partial orders. |

− | ** Epimorphism. |
+ | *** Monoids. |

− | ** Monomorphism. |
+ | *** Finite groups. |

− | ** Isomorphism. |
+ | ** Haskell-Curry isomorphism. |

− | ** Endomorphism. |
+ | |

− | ** Automorphism. |
+ | |

− | * Special objects |
+ | * [[User:Michiexile/MATH198/Lecture 2]] |

− | ** Initial. |
+ | ** Special morphisms |

− | ** Terminal. |
+ | *** Epimorphism. |

− | ** Null. |
+ | *** Monomorphism. |

− | * Functors. |
+ | *** Isomorphism. |

− | * Natural transformations. |
+ | *** Endomorphism. |

− | * Category of categories. |
+ | *** Automorphism. |

− | * The power of dualization. |
+ | ** Special objects |

− | * Limits, colimits. |
+ | *** Initial. |

− | * Products, coproducts. |
+ | *** Terminal. |

− | * Equalizers, coequalizers. |
+ | *** Null. |

− | * Exponentials. |
+ | |

− | * Power objects. |
+ | * [[User:Michiexile/MATH198/Lecture 3]] |

− | * Monads. |
+ | ** Functors. |

− | * Monoids. |
+ | ** Category of categories. |

− | * Triples. |
+ | ** Natural transformations. |

− | * Cartesian Closed Categories. |
+ | |

− | ** Categorical logic. |
+ | * [[User:Michiexile/MATH198/Lecture 4]] |

− | * Topoi. |
+ | ** Products, coproducts. |

− | ** Internal language and logic. |
+ | ** The power of dualization. |

− | * Haskell-Curry isomorphism. |
+ | ** The algebra of datatypes |

− | * Recursive categories. |
+ | |

− | * Recursion as fixed points of monad algebras. |
+ | |

− | * Recursion using special morphisms. |
+ | * [[User:Michiexile/MATH198/Lecture 5]] |

− | ** Hylo- |
+ | ** Limits, colimits. |

− | ** Zygo- |
+ | |

− | ** et.c. |
+ | * [[User:Michiexile/MATH198/Lecture 6]] |

+ | ** Equalizers, coequalizers. |
||

+ | ** Pushouts/pullbacks |
||

+ | ** Adjunctions. |
||

+ | ** Free and forgetful. |
||

+ | |||

+ | |||

+ | * [[User:Michiexile/MATH198/Lecture 7]] |
||

+ | ** Monoid objects. |
||

+ | ** Monads. |
||

+ | ** Triples. |
||

+ | ** Kleisli category. |
||

+ | ** Monad factorization. |
||

+ | |||

+ | * [[User:Michiexile/MATH198/Lecture 8]] |
||

+ | ** Algebras over monads |
||

+ | ** Algebras over endofunctors |
||

+ | ** Initial algebras and recursion |
||

+ | ** Lambek's lemma |
||

+ | |||

+ | * [[User:Michiexile/MATH198/Lecture 9]] |
||

+ | ** Catamorphisms |
||

+ | ** Anamorphisms |
||

+ | ** Hylomorphisms |
||

+ | ** Metamorphisms |
||

+ | ** Paramorphisms |
||

+ | ** Apomorphisms |
||

+ | ** Properties of adjunctions, examples of adjunctions |
||

+ | |||

+ | * [[User:Michiexile/MATH198/Lecture 10]] |
||

+ | ** Power objects |
||

+ | ** Classifying objects |
||

+ | ** Topoi |
||

+ | ** Internal logic |

## Latest revision as of 05:51, 24 July 2010

## [edit] Course overview

Page is the background material for the Fall 2009 lecture course MATH198[1] on Category Theory and Functional Programming that I gave at Stanford University.

Single unit course. 10 lectures. Each lecture is Wednesday 4.15-5.05 in 380F.

- User:Michiexile/MATH198/Lecture 1
- Category: Definition and examples.
- Concrete categories.
- Set.
- Various categories capturing linear algebra.

- Small categories.
- Partial orders.
- Monoids.
- Finite groups.

- Haskell-Curry isomorphism.

- User:Michiexile/MATH198/Lecture 2
- Special morphisms
- Epimorphism.
- Monomorphism.
- Isomorphism.
- Endomorphism.
- Automorphism.

- Special objects
- Initial.
- Terminal.
- Null.

- Special morphisms

- User:Michiexile/MATH198/Lecture 3
- Functors.
- Category of categories.
- Natural transformations.

- User:Michiexile/MATH198/Lecture 4
- Products, coproducts.
- The power of dualization.
- The algebra of datatypes

- User:Michiexile/MATH198/Lecture 5
- Limits, colimits.

- User:Michiexile/MATH198/Lecture 6
- Equalizers, coequalizers.
- Pushouts/pullbacks
- Adjunctions.
- Free and forgetful.

- User:Michiexile/MATH198/Lecture 7
- Monoid objects.
- Monads.
- Triples.
- Kleisli category.
- Monad factorization.

- User:Michiexile/MATH198/Lecture 8
- Algebras over monads
- Algebras over endofunctors
- Initial algebras and recursion
- Lambek's lemma

- User:Michiexile/MATH198/Lecture 9
- Catamorphisms
- Anamorphisms
- Hylomorphisms
- Metamorphisms
- Paramorphisms
- Apomorphisms
- Properties of adjunctions, examples of adjunctions

- User:Michiexile/MATH198/Lecture 10
- Power objects
- Classifying objects
- Topoi
- Internal logic