INF 223, Spring 2017, Plan of Lectures (Status 04.04.2017)

Nr. | Date | Week | Topics | Pages | |

1. | 17.01.17 | 3 | - What is Category Theory? - shift of paradigm - informal discussion of products, dualization, sums | 1 - 11 | |

2. | 19.01.17 | 3 | - graphs and graph homomorphisms: motivation, examples, definition - opposite graphs - discussion of isomorhisms between graphs | 13 - 19 | |

3. | 24.01.17 | 4 | - composition of maps and identity maps - composition of graph homomorphisms and identity graph homomorphisms - associativity and identity law of composition - definition of category | 13 - 26 | |

4. | 26.01.17 | 4 | - categories Set and Graph - universal definition of isomorphism - composition of isomorphisms is isomorphism - isomorphisms in Set are bijective maps - some finite categories - representation of finite categories by pictorial diagrams | 26 - 32 | |

5. | 31.01.17 | 5 | - other categories with sets as objects: Incl, Inj, Par - subcategory: examples and definition - Nat and Incl as pre-order categories - partial-order categories - discussion associations in class diagrams - composition of relations | 32 - 39 | |

6. | 02.02.17 | 5 | - category Rel - association ends and multimaps - category Mult - monoids: examples and definition - monoid morphisms: examples and definition | 39 - 47 | |

7. | 07.02.17 | 6 | - monoid morphisms: examples and definition - category Mon of monoids - universal property of lists - Boolean Algebras: motivation, definition, examples - self-duality of Boolean algebras | 48 - 60 | |

8. | 09.02.17 | 6 | - laws in Boolean algebras - Boolean expressions: definition and evaluation - universal porperty of Boolean expressions, substitution | 60 - 67 | |

9. | 14.02.17 | 7 | - many-sorted signatures and algebras: definition, examples - category of S-sets and category of Sigma-algebras - terms, term-algebras and evaluation of terms - universal property of the evaluation of terms | 67 - 69 | |

10. | 16.02.17 | 7 | - subalgebras and their images and pre-images - functors: motivation, definition - functors: examples - product graphs and product categories - functors preserve isomorphisms | 69 - 80 | |

11. | 21.02.17 | 8 | - subcategories, embeddings (examples graphs of maps) - opposite category and contravariant functors - identity functors and composition of functors - categories of categories: Cat, CAT, SET, GRAPH - pathes: motivation, examples, definition - path graph and evaluation of paths - categorical diagrams: motivation, definition, examples - commutative diagram: definition and examples | 39 - 42 | |

12. | 23.02.17 | 8 | - path categories and their universal property - definition free constructions - path categories and disjoint union as free construction - any free construction induces a free functor | 43 - 48 | |

13. | 28.02.17 | 9 | - cofree constructions introduced as dual of free constructions - Cartesian products as cofree construction - any cofree construction induces a cofree functor - discussion data/structure as free constructions and behaviour as cofree constructions - streams: motivation and definition | 48 - 56 | |

14. | 02.03.17 | 9 | - category SET^loop and streams as cofree construction - discussion: free construction ~ induction vs. cofree construction ~ coinduction - examples coinduction - free constructions induce cofree constructions and vice versa - general informal discussion: algebras vs. colagebras | 56 - 60 | |

15. | 07.03.17 | 10 | - example: lists and streams together as cofree construction - free constructions induce cofree constructions and vice versa - informal discussion of adjunctions - informal discussion algebra vs. coalgebra against free vs. cofree - list vs. power set construction as a motivation for natural transformations | ||

| 09.03.17 | 10 | no lecture (Fagkritisk dag)
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16. | 14.03.17 | 11 | - general discussion about models and metamodels - metamodel of concept graph - graphs as interpretations and graph homomorphisms as natural transformations - algebraic signatures as graphs - definition of natural transformations and interpretation/functor categories - indexed sets as interpretation category - discussion indexed categories with model functor as an example | ||

17. | 16.03.17 | 11 | - arrow categories - category of E-graphs - discussion of arrows between arrows - path equations, satisfaction of path equations - reflexive graphs (path categories as reflexive graphs) - discussion indexed sets vs. typed sets as motivation for typing - motivation of "typing" by ER-diagrams | ||

18. | 21.03.17 | 12 | - motivation of "typing" by ER-diagrams and Petri nets - type graphs and typed graphs and their morphisms - definition slice category - indexed sets vs. typed sets as motivation for equivalence of categories - example typed E-graphs - co-slice categories, example pointed sets - equivalence of categories: definition and examples | ||

19. | 23.03.17 | 12 | - functors between categories of typed graphs - final definition of adjunctions - satisfaction relation and corresponding Galois connection as an adjunction - equivalences of categories are adjunctions - hierarchy: isomorphism - equivalence - adjunction | ||

20. | 28.03.17 | 13 | - monad of an adjunction and Kleisli category of a monad - equivalence relations and equivalence classes - quotient sets and natural maps - unique factorization of maps - equivalences as abstraction in mathematics - quotient path categories/ quotient (term) algebras | ||

21. | 30.03.17 | 13 | - monomorphisms: definition, examples in Set, Graph, Incl - epimorphisms: definition, examples in Set, Graph, Incl - split mono's and epi's - in Set all epi's are split -> axiom of choice- jointly monic and jointly epic | ||

22. | 04.04.17 | 14 | - initial objects: definition, examples in , Incl, Set, , Mult, Graph - terminal objects: definition, examples in , Incl, Set, Mult, Graph - freely generated objects as initial objects in a comma category | ||

23. | 06.04.17 | 14 | - sum: definition, examples in , Incl, Set, Graph - product: definition, examples in , Incl, Set, Graph - sums and products as adjoints to the diagonal functor | ||

| 11.04.17 | 15 | no lecture (Easter Holiday)
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| 13.04.17 | 15 | no lecture (Easter Holiday)
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| 18.04.17 | 16 | no lecture (conference) | ||

| 20.04.17 | 16 | Consultancy 2nd compulsory exercise | ||

24. | 25.04.17 | 17 | - exponentiation: example functions spaces and definition - exponentiation as adjoint to product functor - implication and deduction theorem as exponentiation - Cartesian Closed Categories - (maybe Local Cartesian Closed Categories) - lambda-calculus and exponentiation | ||

25. | 27.04.17 | 17 | - motivation pullbacks: inner join, synchronization, products of typed graphs, pre-images - pullbacks: definition, examples in , Incl, Set, Graph - preimages as pullbacks - reduct functor between slice categories is given by pullbacks | ||

26. | 02.05.17 | 18 | - equalizers, general construction of pullbacks by products and equalizers - equalizers are mono - monics are reflected by pullbacks, coding of monics by pullbacks - composition of pullbacks is a pullback and decomposition of pullbacks - synchronization is pullback | ||

| 04.05.17 | 18 | no lecture (institute seminar) | ||

27. | 09.05.17 | 19 | - motivation pushouts: sharing, decomposition of graphs, rule applications - pushouts: definition, examples in , Incl, Set, Graph - term rewriting and graph transformations as pushouts - coequalizers, general construction of pushouts by sums and coequalizers - discussion: two lines of constructions | ||

28. | 11.05.17 | 19 | - cones and limits - co-cones and colimits - completeness and co-completeness - free functors are based on colimit constructions and preserve colimits - stepwise construction of limits and colimits |
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| 20 | No more lectures |
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| 21 | No more lectures |
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| 22 | No more lectures |
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08.06.17 | 23 | Oral Exam (another day can be agreed) | |||