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User:Michiexile/MATH198/Lecture 5

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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.


Contents

1 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.

1.1 Useful limits and colimits

1.1.1 Equalizer, coequalizer
  • Kernels, cokernels, images, coimages
    • connect to linear algebra: null spaces et.c.
1.1.2 Pushout and pullback squares
  • Computer science applications