User:Michiexile/MATH198

Course overview

Page is work in progress for background material for the Fall 2009 lecture course MATH198 on Category Theory with a view towards applications that I am planning to give at Stanford University.

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

• Exponentials.
• Power objects.
• Cartesian Closed Categories.
  • Categorical logic.
• Topoi.
  • Internal language and logic.
• Haskell-Curry isomorphism.
• Recursive categories.
• Recursion as fixed points of monad algebras.
• Recursion using special morphisms.
  • Hylo-
  • Zygo-
  • et.c.

• User:Michiexile/SU09 Lecture 1
  • Category: Definition and examples.
  • Concrete categories.
    • Set.
    • Various categories capturing linear algebra.
  • Small categories.
    • Partial orders.
    • Monoids.
    • Finite groups.

• User:Michiexile/SU09 Lecture 2
  • Special morphisms
    • Epimorphism.
    • Monomorphism.
    • Isomorphism.
    • Endomorphism.
    • Automorphism.
  • Special objects
    • Initial.
    • Terminal.
    • Null.

• User:Michiexile/SU09 Lecture 3
  • Functors.
  • Natural transformations.
  • Category of categories.

• User:Michiexile/SU09 Lecture 4
  • Adjunctions.
  • Free and forgetful.

• User:Michiexile/SU09 Lecture 5
  • The power of dualization.
  • Limits, colimits.
  • Products, coproducts.
  • Equalizers, coequalizers.

• User:Michiexile/SU09 Lecture 6
  • Monoids.
  • Monads.
  • Triples.
  • The Kleisli category.
  • Monad factorization.

• User:Michiexile/SU09 Lecture 7
  • Recursion as a categorical construction.

• User:Michiexile/SU09 Lecture 8
  • Topos.

• User:Michiexile/SU09 Lecture 9
  • Internal logic.

• User:Michiexile/SU09 Lecture 10
  • Review.