INF 223, Spring 2014, Plan of Lectures (Status 28.04.2014)

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

1. | 14.01.14 | 3 | - What is Category Theory? - shift of paradigm - informal discussion of products, dualization, sums | 5 - 8 | |

2. | 16.01.14 | 3 | - graphs and graph homomorphisms: motivation, examples, definition - opposite graphs - discussion of isomorhisms between graphs | 8 - 12 | |

3. | 22.01.14 | 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. | 24.01.14 | 4 | - categories Set and Graph - a 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 | 14 - 16 | |

5. | 28.01.14 | 5 | - category Nat - other categories with sets as objects: Incl, Inj, Par - subcategory: examples and definition - Nat and Incl as pre-order categories - discussion associations in class diagrams - category Rel | 17 - 22 | |

6. | 30.01.14 | 5 | - association ends and category Mult - monoids: examples and definition | 23 - 27 | |

7. | 04.02.14 | 6 | - monoid morphisms: examples and definition - category Mon of monoids - universal property of lists - functors: motivation, definition | 27 - 29 | |

8. | 06.02.14 | 6 | - functors: examples - functors preserve isomorphisms - opposite category and contravariant functors - identity functors and composition of functors - categories of categories: Cat, CAT, SET | 29 - 32 | |

9. | 11.02.14 | 7 | - pathes: motivation, examples, definition - path category: definition, examples - universal property of path categories - bijection between Graph(G,gr(C)) and Cat(P(G),C) - summary of the first lectures about "structures" - general discussion about models and metamodels - discussion of a "metamodel" MG of graphs | 32 - 33 | |

10. | 13.02.14 | 7 | - graphs as interpretations of the graph MG in Set - graph homomorphisms as natural transformations - definition of natural transformations - natural transformations: composition and identities - definition of interpretation and functor categories - iso's in interpretation categories are the natural isomorphisms - indexed sets as functor category | 35 - 39 | |

11. | 18.02.14 | 8 | - arrow categories - discussion of arrows between arrows - path equations, satisfaction of path equations - model interpretations - reflexive graphs - motivation of "typing" by ER-diagrams and Petri nets - type graph and typed graphs and their morphisms - definition slice category | 39 - 42 | |

12. | 20.02.14 | 8 | - transformations between indexed and typed sets - discussion and definition of equivalence of categories - arrow category vs. bipartite graphs - general interpretations in Set vs. typed graphs - discussion indexed vs. typed semantics | 43 - 48 | |

13. | 25.02.14 | 9 | - 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. | 27.02.14 | 9 | - 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 | |

15. | 04.03.14 | 10 | - initial objects: definition, examples in , Incl, Set, , Mult, Graph - terminal objects: definition, examples in , Incl, Set, Mult, Graph
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| 06.03.14 | 10 | no lecture (Fagkritisk dag)
| ||

16. | 11.03.14 | 11 | - sum: definition, examples in , Incl, Set, Graph | ||

17. | 13.03.14 | 11 | - product: definition, examples in , Incl, Set, Graph
| ||

18. | 18.03.14 | 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 | ||

19. | 20.03.14 | 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 | ||

| 25.03.14 | 13 | no lecture (illness) | ||

20. | 27.03.14 | 13 | - definition of diagrams - cones and limits - co-cones and colimits - completeness and co-completeness - stepwise construction of limits and colimits | ||

21. | 01.04.14 | 14 | - algebraic signatures, terms, equations - finite product and finite limit sketches - discussion Why "generalized sketches"? - discussion: formalization of ER diagrams - relations as jointly mono's (injections) - predicates "jointly epi (surjective)" (or "cover" respectively) and "disjoint" | ||

| 03.04.14 | 14 | no lecture (departmental seminar) | ||

| 08.04.14 | 15 | no lecture (conference ETAPS) | ||

| 10.04.14 | 15 | no lecture (conference ETAPS) | ||

| 15.04.14 | 16 | no lecture (Easter Holiday)
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| 17.04.14 | 16 | no lecture (Easter Holiday)
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22. | 22.04.14 | 17 | - discussion: formalization of associations in class diagrams - predicate "inverse" (or "opposite" respectively) for multimaps - discussion: What is a "Diagrammatic Specification Technique" | ||

23. | 24.04.14 | 17 | - example: information system - definition: signature, atomic constraint, specification - discussion: indexed semantics vs. fibred semantics - instances of a graph and corresponding slice category - semantics of predicates | ||

24. | 29.04.14 | 18 | - instances of a specification and corresponding category - specification morphisms and corresponding category of specifications - pullback (reduction) functor - satisfaction condition | ||

| 01.05.14 | 18 | no lecture (Labour Day) | ||

25. | 06.05.14 | 19 | - 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 | 44- 46, 64 - 70 | |

26. | 08.05.14 | 19 | - specification entailments - universal constraints - discussion: logic and deduction | ||

27. | 13.05.14 | 20 | - open | ||

28. | 15.05.14 | 20 | - course overview | ||

| 21 | No more lectures |
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| 22 | No more lectures |
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3+4.06.13 | 23 | Oral Exam (thus the day can be changed) | |||