# Week 1

## Schedule

Summary | Solutions | Time (min) |
---|---|---|

Lecture 1 | ||

* generative effects (11:13) | ||

* sets (14:25) | ||

* product sets (16:30) | ||

* subsets (19:58) | ||

* relations (21:50) | ||

* functions (25:40) | ||

* injections (28:52) | ||

* surjections (31:14) | ||

* partitions (31:43) | ||

* ordering on partitions (34:50) | ||

* preorders (43:12) | ||

Lecture 2 | ||

* preorders (1:04) | ||

* a preorder is a \(\mathbf{Bool}\)-enriched category (3:55) | ||

* meets & joins (5:16) | ||

* two meets are isomorphic (24:00) | ||

* monotone maps (29:20) | ||

* generative effects: join of two partitions, wrt monotone maps (34:30) | ||

* Galois connections & adjoints (43:03) | ||

* adjoint functor theorem for preorders (52:16) | ||

Introduction | ||

* motivation | 1/2 | |

* what is a category? what is a diagram? (AC0: Chapter I, 3.1) | 2 | |

SSC: Chapter 1 | * EX 1.69 | 5 |

* relations (D 1.12), functions (D 1.22), function composition (D 1.28), preorders (D 1.30) | * EX 1.94 | 5 |

* partition from surjection (EG 1.26) | * EX 1.125 | 15 |

* skelitality (R 1.35) | * PS1 Q1 | 5 |

* partition from preorder (EG 1.49) | * PS1 Q2 | 10 |

* order on partitions (EG 1.52) | * PS1 Q3 | 10 |

* monotone maps (D 1.59) | * PS1 Q4 | 5 |

* monotone map between partitions from surjection (EG 1.68) | ||

* isomorphisms (D 1.75) | ||

* meets & joins (D 1.81) | ||

* two meets are isomorphic (EG 1.84) | ||

* generative effects: monotone maps (D 1.93) | ||

* generative effects: join of two partitions (1.1.1) | ||

* Galois connections & adjoints (D 1.95) | ||

* adjoint functor theorem for preorders (T 1.115) | ||

* adjunctions from fixed points (EG 1.122) | ||

* level shifting (1.4.5) | ||

Total | 57.5 |

## Notes

- what is a category?
- "solutions from structures" (algebra course)
- "abstract nonsense" (AC0: Chapter I, 3.1)
- functions between sets, not elements in sets

- what is a diagram?
- unambiguously determinable (Lecture 3: 33:28)
- drawings restricted by proof rules, s.t. the diagram corresponds to a proof

- meets & joins (5:16)
- memorize the general symbols of a meet and a join of a preorder, by recalling that the symbols correspond to the special case where the preorder is \(\mathbf{Bool}\), where the meet is "and" is \(\wedge\) and the join is "or" is \(\vee\)

- universal properties and final objects (5.2)
- an object has universal property if unique morphism exists for all objects (remark after SSC D 3.79)

- natural projections, natural injections, canonical projection to quotient set (2.7)
- Two elements of a quotient \(S/~\) are equal iff the corresponding elements in \(S\) are related by \(~\). In other words, taking a quotient is a way to turn any equivalence relation into an equality (remark after AC0: Chapter I, EG 1.3)

- Galois connections & adjoints (D 1.95)
- Galois connections are relaxed isomorphisms (remark after SSC EX 1.109)