Difference between revisions of "User:Michiexile/MATH198/Lecture 4"
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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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+ | |||
+ | ===Product=== |
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+ | |||
+ | * Cartesian product in Set |
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+ | * Product of categories construction |
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+ | * Record types |
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+ | * Categorical formulation |
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+ | ** Universal X such that Y |
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+ | |||
+ | ===Coproduct=== |
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+ | |||
+ | * Diagram definition |
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+ | * Disjoint union in Set |
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+ | * Coproduct of categories construction |
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+ | * Union types |
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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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+ | |||
+ | * The power of dualization. |
||
+ | * Limits, colimits. |
||
+ | * Products, coproducts. |
||
+ | * Equalizers, coequalizers. |
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.
Product
- Cartesian product in Set
- Product of categories construction
- Record types
- Categorical formulation
- Universal X such that Y
Coproduct
- Diagram definition
- Disjoint union in Set
- Coproduct of categories construction
- Union types
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
* The power of dualization. * Limits, colimits. * Products, coproducts. * Equalizers, coequalizers.