marcbezem / INF223

INF223 Spring 2021

Tentative plan of Lectures (status 05.01.2121)

Interesting links:

• https://www.schoolofhaskell.com/
• https://wiki.haskell.org/Category_theory
• https://ncatlab.org/nlab/show/extension+system (monads)

There are two important breaks that can be used to work on compulsory exercises: the winter break (first week of March, week 9) and an extended Easter break (week 13 and 14). There will be a third compulsory exercise after the Easter break.

1. Lecture 21.01.21 (week 3)

   1. What is Category Theory?
   2. shift of paradigm
   3. informal discussion of products, dualization, sums

2. Lecture 22.01.21 (week 3)

   1. graphs and graph homomorphisms: motivation, examples, definition
   2. opposite graphs
   3. discussion of isomorhisms between graphs

3. Lecture 28.01.21 (week 4)

   1. composition of maps and identity maps
   2. composition of graph homomorphisms and identity graph homomorphisms
   3. associativity and identity law of composition
   4. definition of category
   5. categories Set and Graph
   6. a universal definition of isomorphism

4. Lecture 29.01.21 (week 4)

   1. composition of isomorphisms is isomorphism
   2. isomorphisms in Set are bijective maps
   3. isomorphisms in Graph are componentwise bijective graph homomorphisms
   4. some finite categories
   5. representation of finite categories by pictorial diagrams
   6. other categories with sets as objects: Incl, Inj, Par
   7. Nat and Incl as pre-order categories
   8. pre-order categories and partial order categories

5. Lecture 05.02.21 (week 5)

   1. subcategory: examples and definition
   2. associations in class diagrams
   3. composition of relations
   4. category Rel
   5. association ends and multimaps

6. Lecture 06.02.21 (week 5)

   1. category Mult
   2. monoids: examples and definition
   3. monoid morphisms: examples and definition
   4. category Mon of monoids

7. Lecture 12.02.21 (week 6)

   1. inductive definition of lists
   2. universal property of lists (free monoids)
   3. functors: motivation, definition
   4. functors: examples
   5. product graphs with finite example

8. Lecture 13.02.21 (week 6)

   1. product categories
   2. functors preserve isomorphisms
   3. opposite category and contravariant functors
   4. identity functors and composition of functors
   5. categories of categories: Cat, CAT, SET, GRAPH
   6. pathes: motivation, examples, definition
   7. path graph and evaluation of paths

9. Lecture 19.02.21 (week 7)

   1. categorical diagrams: motivation, definition, examples
   2. commutative diagram: definition and examples
   3. path categories
   4. summary of the first lectures about "structures"
   5. general discussion about models and metamodels
   6. discussion of a "metamodel" MG of graphs

10. Lecture 20.02.21 (week 7)

    1. graphs as interpretations of the graph MG in Set
    2. graph homomorphisms as natural transformations
    3. definition of natural transformations
    4. natural transformations: composition and identities
    5. definition of interpretation categories

11. Lecture 26.02.21 (week 8)

    1. indexed sets as functor category
    2. arrow categories
    3. category of E-graphs
    4. discussion of arrows between arrows
    5. path equations, satisfaction of path equations
    6. model interpretations
    7. reflexive graphs

12. Lecture 27.02.21 (week 8, no lectures in week 9)

    1. motivation of "typing" by ER-diagrams and Petri nets
    2. type graph and typed graphs and their morphisms
    3. definition slice category
    4. example typed E-graphs
    5. indexed vs. typed sets
    6. equivalence of categories

13. Lecture 11.03.21 (week 10)

    1. equivalence relations and equivalence classes
    2. quotient sets and natural maps
    3. unique factorization of maps
    4. equivalences as abstraction in mathematics
    5. representatives and normal forms
    6. quotient path categories

14. Lecture 12.03.21 (week 10)

    1. monomorphisms: definition, examples in Set, Graph, Incl
    2. epimorphisms: definition, examples in Set, Graph, Incl
    3. split mono's and epi's
    4. in Set all epi's are split -> axiom of choice

15. Lecture 18.03.21 (week 11)

    1. initial objects: definition, examples in Incl, Set, Mult, Graph
    2. terminal objects: definition, examples in Incl, Set, Mult, Graph

16. Lecture 19.03.21 (week 11)

    1. sum: definition, examples in Incl, Set, Graph
    2. product: definition, examples in Incl, Set, Graph

17. Lecture 25.03.21 (week 12)

    1. motivation pullbacks: intersection, inner join, products of typed graphs
    2. pullbacks: definition, examples in Incl, Set, Graph
    3. preimages as pullbacks
    4. equalizers: definition, example in Set
    5. kernel and graph of a map f:A->B as equalizers

18. Lecture 26.03.21 (week 12, no lectures in week 13, 14)

    1. general construction of pullbacks by products and equalizers
    2. fibred products
    3. equalizers are mono
    4. monics are reflected by pullbacks, coding of monics by pullbacks
    5. composition of pullbacks is a pullback and decomposition of pullbacks

19. Lecture (15.04.21, week 15)

    Monads 1

20. Lecture (16.04.21, week 15)

    Monads 2

21. Lecture (22.04.21, week 16)

    Monads 3

22. Lecture (23.04.21, week 16)

    Monads 4

23. Lecture (29.04.21, week 17)

    1. Coequalizers
    2. Limits and colimits
    3. Pushouts
    4. Finite limits from products and equalizers

24. Lecture (30.04.21, week 17) No lecture

25. Lecture (06.05.21, week 18)

    Yoneda Lemma 1

26. Lecture (07.05.21, week 18)

    Yoneda Lemma 2

27. No lectures from 13.05.21, Ascension Day