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. | 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. | ||
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| + | ===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 | ||
Revision as of 01:41, 8 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.
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
