Difference between revisions of "User:Michiexile/MATH198/Lecture 8"
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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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+ | ===Some adjunctions we already know=== |
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+ | |||
+ | * initial/terminal are adjunctions. |
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+ | * (co)-products are adjunctions. |
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+ | * Actually, all (co)limits are adjunctions. |
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+ | |||
+ | |||
+ | |||
+ | ===Some adjunctions we don't know yet=== |
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+ | |||
+ | * Existential and universal qualifiers as adjunctions. |
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+ | * Powersets and im(f) -| f^\inv |
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+ | |||
+ | ===Properties of adjoints=== |
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+ | |||
+ | ====RAPL: Right Adjoints Preserve Limits==== |
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+ | |||
+ | ====Recognizing adjoints==== |
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+ | |||
+ | '''Theorem''' (Freyd: The Adjoint Functor Theorem) |
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+ | |||
+ | ===Yoneda's Lemma=== |
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+ | |||
+ | ===F-functors and algebraic data structures=== |
Revision as of 17:23, 4 November 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.
Some adjunctions we already know
- initial/terminal are adjunctions.
- (co)-products are adjunctions.
- Actually, all (co)limits are adjunctions.
Some adjunctions we don't know yet
- Existential and universal qualifiers as adjunctions.
- Powersets and im(f) -| f^\inv
Properties of adjoints
RAPL: Right Adjoints Preserve Limits
Recognizing adjoints
Theorem (Freyd: The Adjoint Functor Theorem)