User:Michiexile/MATH198/Lecture 5

From HaskellWiki

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.

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