User:Michiexile/MATH198/Lecture 7
From HaskellWiki
(Difference between revisions)
| 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)
