User:Michiexile/MATH198

Course overview

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.

Single unit course. 10 lectures.

• Category: Definition and examples.
• Concrete categories.
  • Set.
  • Various categories capturing linear algebra.
• Small categories.
  • Partial orders.
  • Monoids.
  • Finite groups.
• Special morphisms
  • Epimorphism.
  • Monomorphism.
  • Isomorphism.
  • Endomorphism.
  • Automorphism.
• Special objects
  • Initial.
  • Terminal.
  • Null.
• Functors.
• Natural transformations.
• Category of categories.
• The power of dualization.
• Limits, colimits.
• Products, coproducts.
• Equalizers, coequalizers.
• Exponentials.
• Power objects.
• Monads.
• Monoids.
• Triples.
• 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.