User:Michiexile/MATH198/Lecture 4

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IMPORTANT NOTE: THESE NOTES ARE STILL UNDER DEVELOPMENT. PLEASE WAIT UNTIL AFTER THE LECTURE WITH HANDING ANYTHING IN, OR TREATING THE NOTES AS READY TO READ.

Product

Cartesian product in Set
Product of categories construction
Record types
Categorical formulation
- Universal X such that Y

Coproduct

Diagram definition
Disjoint union in Set
Coproduct of categories construction
Union types

Limits and colimits

Generalizing these constructions
Diagram and universal object mapping to (from) the diagram
Express product/coproduct as limit/colimit
Issues with Haskell
- No dependent types
- No compiler-enforced equational conditions
- Can be simulated but not enforced, e.g. using QuickCheck.

Useful limits and colimits

Equalizer, coequalizer

Kernels, cokernels, images, coimages
- connect to linear algebra: null spaces et.c.

Pushout and pullback squares

Computer science applications

   * The power of dualization.
   * Limits, colimits.
   * Products, coproducts.
   * Equalizers, coequalizers.