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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)

Author: koo

Created: 2020-01-13 Mon 14:36