Personal tools

User:Michiexile/MATH198/Lecture 7

From HaskellWiki

< User:Michiexile | MATH198(Difference between revisions)
Jump to: navigation, search
 
Line 1: Line 1:
 
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.
 +
 +
===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)
 +
 +
 +
===Why should we care in CS?===
 +
 +
====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