User:Michiexile/MATH198/Lecture 4

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Contents 1 Product 2 Coproduct 3 Limits and colimits 3.1 Useful limits and colimits 3.1.1 Equalizer, coequalizer 3.1.2 Pushout and pullback squares

1 Product

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

2 Coproduct

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

3 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.

3.1 Useful limits and colimits

3.1.1 Equalizer, coequalizer

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

3.1.2 Pushout and pullback squares

• Computer science applications

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