# 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. | ||

+ | |||

+ | ===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) | ||

+ | |||

+ | ===Yoneda's Lemma=== | ||

+ | |||

+ | ===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.

## 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)