# 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)
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)
* 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).

Created: 2020-01-13 Mon 14:36