Revision as of 16:13, 7 October 2009
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.
- Cartesian product in Set
- Product of categories construction
- Record types
- Categorical formulation
- Universal X such that Y
- 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.