Difference between revisions of "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=== |
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| + | |||
| + | * Generalizing these constructions |
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| + | * Diagram and universal object mapping to (from) the diagram |
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| + | * Express product/coproduct as limit/colimit |
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| + | * Issues with Haskell |
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| + | ** No dependent types |
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| + | ** No compiler-enforced equational conditions |
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| + | ** Can be ''simulated'' but not enforced, e.g. using QuickCheck. |
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| + | |||
| + | ====Useful limits and colimits==== |
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| + | |||
| + | =====Equalizer, coequalizer===== |
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| + | |||
| + | * Kernels, cokernels, images, coimages |
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| + | ** connect to linear algebra: null spaces et.c. |
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| + | |||
| + | =====Pushout and pullback squares===== |
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| + | |||
| + | * Computer science applications |
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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.
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