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User:Michiexile/MATH198/Lecture 7

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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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===Why should we care in CS?===
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====Monads====

Revision as of 22:20, 29 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 Some adjunctions we already know

  • initial/terminal are adjunctions.
  • (co)-products are adjunctions.
  • Actually, all (co)limits are adjunctions.


2 Some adjunctions we don't know yet

  • Existential and universal qualifiers as adjunctions.
  • Powersets and im(f) -| f^\inv

3 Properties of adjoints

3.1 RAPL: Right Adjoints Preserve Limits

3.2 Recognizing adjoints

Theorem (Freyd: The Adjoint Functor Theorem)


4 Why should we care in CS?

4.1 Monads