INF 223, Spring 2013, Plan of Lectures

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

1. | 17.01.13 | 3 | - What is Category Theory? - shift of paradigm - informal discussion of products, dualization, sums | 5 - 8 | |

2. | 22.01.13 | 4 | - graphs and graph homomorphisms: motivation, examples, definition - opposite graphs - discussion of isomorhisms between graphs | 8 - 12 | |

3. | 24.01.13 | 4 | - composition of maps and identity maps - composition of graph homomorphisms and identity graph homomorphisms - associativity and identity law of composition - definition of category | 12 - 14 | |

4. | 29.01.13 | 5 | - categories Set and Graph - a universal definition of isomorphism - isomorphisms in Set are bijective maps - some finite categories | 14 - 16 | |

5. | 31.01.13 | 5 | - category Nat - other categories with sets as objects: Incl, Inj, Par, Rel - speciality of Nat and Incl? - pre-order categories - category Mult | 17 - 22 | |

6. | 05.02.13 | 6 | - monoids: examples and definition - monoid morphisms: examples and definition - category Mon of monoids - universal property of lists | 23 - 27 | |

7. | 07.02.13 | 6 | - functors: motivation, definition - functors: examples - functors preserve isomorphisms - opposite category and contravariant functors | 27 - 29 | |

8. | 12.02.13 | 7 | - identity functors and composition of functors - categories of categories: Cat, CAT, SET - pathes: motivation, examples, definition - path category: definition, examples | 29 - 32 | |

9. | 14.02.13 | 7 | - universal property of path categories - bijection between Graph(G,gr(C)) and Cat(P(G),C) - summary of the first lectures about "structures" | 32 - 33 | |

10. | 19.02.13 | 8 | - general discussion about models and metamodels - discussion of a "metamodel" MG of graphs - graphs as interpretations of the graph MG in Set - graph homomorphisms as natural transformations - definition of natural transformations - natural transformations: composition and identities | 35 - 39 | |

11. | 21.02.13 | 8 | - definition of interpretation and functor category - iso's in interpretation categories are the natural isomorphisms - indexed sets as functor category - motivation of "typing" by ER-diagrams and Petri nets - type graph and typed graphs and their morphisms - definition slice category | 39 - 42 | |

12. | 26.02.13 | 9 | - transformations between indexed and typed sets - discussion of equivalence of categories - generalization to interpretations of graphs - path equations, satisfaction of path equations | 43 - 48 | |

13. | 28.02.13 | 9 | - reflexive graphs and finite state machines - injective and surjective maps: definitions, examples, properties - equivalence relations and equivalence classes - quotient sets and natural maps - unique factorization of maps - equivalences as abstraction in mathematics - quotient path categories | 48 - 56 | |

14. | 05.03.13 | 10 | - 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
| 56 - 60 | |

| 07.03.13 | 10 | no lecture (Fagkritisk dag)
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15. | 12.03.13 | 11 | - initial objects: definition, examples in , Incl, Set, , Mult, Graph - terminal objects: definition, examples in , Incl, Set, Mult, Graph
| 60 - 63 | |

16. | 14.03.13 | 11 | - sum: definition, examples in , Incl, Set, Graph - product: definition, examples in , Incl, Set, Graph
| 63 - 69 | |

17. | 19.03.13 | 12 | - motivation pullbacks: inner join, synchronization, products of typed graphs, pre-images - pullbacks: definition, examples in , Incl, Set, Graph - 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 | 69 - 71 | |

18. | 21.03.13 | 12 | - motivation pushouts: sharing, decomposition of graphs, rule applications - pushouts: definition, examples in , Incl, Set, Graph - coequalizers, general construction of pushouts by sums and coequalizers - discussion: two lines of constructions | 71 | |

| 26.03.13 | 13 | no lecture (Easter Holiday)
| ||

| 28.03.13 | 13 | no lecture (Easter Holiday)
| ||

| 02.04.13 | 14 | no lecture (conference UNILOG) | ||

| 04.04.13 | 14 | no lecture (conference UNILOG) | ||

19. | 09.04.13 | 15 | - definition of diagrams - cones and limits - co-cones and colimits - completeness and co-completeness - stepwise construction of limits and colimits | 71 - 72 | |

| 11.04.13 | 15 | no lecture | ||

| 16.04.13 | 16 | no lecture (Meeting institute) | ||

20. | 18.04.13 | 16 | - discussion: formalization of ER diagrams - relations as jointly mono's (injections) - predicates "jointly epi (surjective)" (or "cover" respectively) and "disjoint" | ||

21. | 23.04.13 | 17 | - discussion: formalization of associations in class diagrams - predicate "inverse" (or "opposite" respectively) for multimaps - discussion: What is a "Diagrammatic Specification Technique" - example: information system - definition: signature, atomic constraint, specification | ||

22. | 25.04.13 | 17 | - discussion: indexed semantics vs. fibred semantics - instances of a graph and corresponding slice category - semantics of predicates | PhD 31 - 43 | |

23. | 30.04.13 | 18 | - instances of a specification and corresponding category - specification morphisms and corresponding category of specifications - pullback (reduction) functor - satisfaction condition | 31 - 43 | |

24. | 02.05.13 | 18 | - discussion: metamodelling and OMG's 4-layer hierarchy - definition, typed signatures, typed constraints, and typed specifications - conformant specifications - definition modelling formalism - discussion: reflexive metamodel and reflexive modelling formalism | 47 - 64 | |

25. | 07.05.13 | 19 | - specification entailments - universal constraints - discussion: logic and deduction | 44- 46, 64 - 70 | |

| 09.05.13 | 19 | no lecture (Ascension Day) | ||

26. | 14.05.13 | 20 | - course overview | ||

| 16.05.13 | 20 | No more lectures | ||

| 21 | No more lectures |
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| 22 | No more lectures |
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5+6.06.13 | 23 | Exam (most probably oral thus also the day can be changed) | |||