# 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)

Created: 2020-01-13 Mon 14:36