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Week 5

Schedule

Summary Solutions Time (min)
Lecture 6    
* functors (2:07)    
* \(mathcacl{C}\)-instances (18:30)    
* functors preserve the "structure" of morphisms, specifically, they preserve the domain and codomain of morphisms (28:25)    
* natural transformations (34:30)    
* natural transformations are analogous to polymorphic functions, naturality corresponds to commutativity of this polymorphic function with any other function, see section 3.1 of Wadler's Theorems for Free (41:45)    
* functor categories (45:25)    
* adjoints (50:10)    
AC0: Chapter I    
* all final objects are unique up to unique isomorphism (P 5.4, SSC R 3.85)    
SSC: Chapter 3 * EX 3.98 10
* functors (D 3.35) * PS2 8 10
* \(mathcacl{C}\)-instances (D 3.44)    
* natural transformations (D 3.49)    
* diagrams and indexing categories (D 3.51), will be clearer in definition of cones over diagrams (D 3.92)    
* functor categories (D 3.54)    
* equivalence between categories (R 3.59)    
* instance homomorphisms (D 3.60)    
* pullback of data along functor (D 3.68)    
* adjoints, mates (D 3.70)    
* currying as adjunction, exponential objects, hom-elements (EG 3.72)    
* left and right pushforward adjoints of the pullback functor (3.4.3, 3.4.4)    
* final objects (D 3.79)    
* all final objects are unique up to unique isomorphism (R 3.85)    
* products (D 3.86)    
* cones over diagrams (D 3.92)    
* finite limit of set-valued diagram functor from an indexing category presented by a finite graph (T 3.95)    
* pullback as a limit of diagram indexed by a category presented as a cospan graph (EG 3.99, R 3.100)    
* coherence conditions, universal constructions (3.6)    
Total   20

Notes

  • diagrams and indexing categories (D 3.51), will be clearer in definition of cones over diagrams (D 3.92)
    • An indexing category is analogous to a database schema, and a diagram is analogous to instantiating that database schema (D 3.44)
    • See EG 3.93 and EG 3.94 for clarification.
  • products (D 3.86)
    • A product category is the product \(\mathcal{C} \times \mathcal{D}\) in the category \(\mathbf{Cat}\), where for any other category and two functors \(\mathcal{C} \leftarrow \mathcal{E} \rightarrow \mathcal{D}\), there exists a unique functor \(\mathcal{E} \rightarrow \mathcal{C} \times \mathcal{D}\). That is, if any category \(\mathcal{E}\) project its objects and morphisms onto the constituent categories \(\mathcal{C}\) and \(\mathcal{D}\), then we can embed \(\mathcal{E}\) in \(\mathcal{C} \times \mathcal{D}\), where the objects and morphisms of \(\mathcal{C} \times \mathcal{D}\) are defined in the usual sense (EG 3.89).

Author: koo

Created: 2020-01-13 Mon 14:36