| Nr. | Date | Week | Topics | Pages | |
| 1. | 15.01.20 | 3 | - What is Category Theory? - shift of paradigm - informal discussion of products, dualization, sums | ||
| 2. | 16.01.20 | 3 | - graphs and graph homomorphisms: motivation, examples, definition - opposite graphs - discussion of isomorhisms between graphs | ||
| 3. | 22.01.20 | 4 | - composition of maps and identity maps - composition of graph homomorphisms and identity graph homomorphisms - associativity and identity law of composition - definition of category - categories Set and Graph - a universal definition of isomorphism | ||
| 4. | 23.01.20 | 4 | - composition of isomorphisms is isomorphism - isomorphisms in Set are bijective maps - isomorphisms in Graph are componentwise bijective graph homomorphisms - some finite categories - representation of finite categories by pictorial diagrams - other categories with sets as objects: Incl, Inj, Par - Nat and Incl as pre-order categories - pre-order categories and partial order categories | ||
| 5. | 29.01.20 | 5 | - subcategory: examples and definition - discussion associations in class diagrams - composition of relations - category Rel - association ends and multimaps | ||
| 6. | 30.01.20 | 5 | - category Mult - monoids: examples and definition - monoid morphisms: examples and definition - category Mon of monoids | ||
| 7. | 05.02.20 | 6 | - inductive definition of lists - universal property of lists (free monoids) - functors: motivation, definition - functors: examples - product graphs with finite example | ||
| 8. | 06.02.20 | 6 | - product categories - functors preserve isomorphisms - 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 | ||
| 9. | 12.02.20 | 7 | - categorical diagrams: motivation, definition, examples - commutative diagram: definition and examples - path categories - summary of the first lectures about "structures" - general discussion about models and metamodels - discussion of a "metamodel" MG of graphs | ||
| 10. | 13.02.20 | 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 categories | ||
| 11. | 19.02.20 | 8 | - indexed sets as functor category - arrow categories - category of E-graphs - discussion of arrows between arrows - path equations, satisfaction of path equations - model interpretations - reflexive graphs | ||
| 12. | 20.02.20 | 8 | - motivation of "typing" by ER-diagrams and Petri nets - type graph and typed graphs and their morphisms - definition slice category - example typed E-graphs - indexed vs. typed sets - equivalence of categories | ||
| 26.02.20 | 9 | no lecture (Winter holiday) | |||
| 27.02.20 | 9 | no lecture (Winter holiday) | |||
| 13. | 04.03.20 | 10 | - equivalence relations and equivalence classes - quotient sets and natural maps - unique factorization of maps - equivalences as abstraction in mathematics - representatives and normal forms - quotient path categories | ||
| 14. | 05.03.20 | 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 | ||
| 15. | 11.03.20 | 11 | - initial objects: definition, examples in , Incl, Set, , Mult, Graph - terminal objects: definition, examples in , Incl, Set, Mult, Graph | ||
| 16. | 12.03.20 | 11 | - sum: definition, examples in , Incl, Set, Graph - product: definition, examples in , Incl, Set, Graph | ||
| 17. | 18.03.20 | 12 | - motivation pullbacks: intersection, inner join, products of typed graphs - pullbacks: definition, examples in Incl, Set, Graph - preimages as pullbacks - equalizers: definition, example in Set - kernel and graph of a map f:A->B as equalizers | ||
| 18. | 19.03.20 | 12 | - general construction of pullbacks by products and equalizers - fibred products - equalizers are mono - monics are reflected by pullbacks, coding of monics by pullbacks - composition of pullbacks is a pullback and decomposition of pullbacks | ||
| 19. | 25.03.20 | 13 | - 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 | ||
| 26.03.20 | 13 | - definition of diagrams - cones and limits - co-cones and colimits - completeness and co-completeness - stepwise construction of limits and colimits | |||
| 01.04.20 | 14 | no lecture (buffer/lecture monads?) | |||
| 02.04.20 | 14 | no lecture (buffer/lecture monads?) | |||
| 08.04.20 | 15 | no lecture (Easter Holiday) | |||
| 09.04.20 | 15 | no lecture (Easter Holiday) | |||
| 20. | 15.04.20 | 16 | - research project: flexible and universal diagrammatic formalism - categorical sketches: example binary relations - criticism of sketch approach | ||
| 21. | 16.04.20 | 16 | - relation = jointly injective/monic - dualization = jointly surjective/epic (cover), example subclasses - ER diagrams in DPF - key = injective map - composite attributes = products - formalization of associations: predicate [opp] | ||
| 22. | 22.04.20 | 17 | - diagrammatic signature: definition and examples - atomic constraints - diagrammatic specification - discussion of two variants of EER models - semantic interpretations in a "semantic universe" U - indexed semantics = interpretation categories - satisfaction of atomic constraints - specification morphisms: definition and example; category of specifications | ||
| 23. | 23.04.20 | 17 | - revised type graph for ER diagrams - semantics-as-instances - discussion extending type graphs to metamodels - informal discussion of modelling hierarchies | ||
| 24. | 28.04.20 | 18 | - typed signatures, typed atomic constraints, typed specifications - conformant specification - modelling formalism and modelling hierarchies | ||
| 25. | 29.04.20 | 18 | - discussion model transformation - joined modelling formalism - example object-oriented models joined with relational data models - transformation rules: definition - model transformation = application of transformation rule = pushout - example from oo models to relational data models - discussion control of rule applications: negative application conditions, stratification - discussion: extracting the constructed relational data model by pullback | ||
| 26. | 06.05.20 | 19 | TBA | ||
| 27. | 07.05.20 | 19 | - course summary | ||
| 20 | No more lectures | ||||
| 21 | No more lectures | ||||
| 22 | No more lectures | ||||
| 03.06.20 | 23 | Oral Exam (Plan) | |||
| 04.06.20 | 23 | Oral Exam (Plan) | |||