INF 223, Spring 2012, Plan of Lectures

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

1. | 19.01.12 | 3 | - What is Category Theory? - shift of paradigm - informal discussion of products, dualization, sums | ||

2. | 20.01.12 | 3 | - graphs and graph homomorphisms with examples - opposite graphs - discussion of isomorhisms between graphs | ||

3. | 26.01.12 | 4 | - A universal definition of isomorphism? - composition of maps and identity maps - associativity and identity law of composition - definition of category - categories Set and Graph
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4. | 27.01.12 | 4 | - some finite categories - category Nat - other categories with sets as objects: Incl, Inj, Par, Rel - speciality of Nat and Incl? - pre-order categories
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5. | 02.02.12 | 5 | - category Mult - monoids: examples and definition - monoid morphisms: examples and definition - category Mon of momoids - universal property of lists - functors: motivation, definition | ||

6. | 03.02.12 | 5 | - functors: examples - functors preserve isomorphisms - opposite category and contravariant functors - identity functors and composition of functors - categories of categories: Cat, CAT, SET
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7. | 09.02.12 | 6 | - pathes: motivation, examples, definition - path category: definition, examples - universal propertyof path categories - bijection between Graph(G,gr(C)) and Cat(P(G),C) - summary of the first lectures about "structures" | ||

8. | 10.02.12 | 6 | - 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 - definition of interpretation and functor category | ||

9. | 16.02.12 | 7 | - 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 - transformations between indexed and typed sets - discussion of equivalence of categories - generalization to interpretations of graphs | ||

10. | 17.02.12 | 7 | - Arrow categories - disjoint union functor for arrow categories - arrows between arrows - reflexive graphs and finite state machines | ||

11. | 23.02.12 | 8 | - 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 | ||

| 24.02.12 | 8 | no lecture (illness children)
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| 01.03.12 | 9 | no lecture (Fagkritisk dag)
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| 02.03.12 | 9 | no lecture (Fagkritisk dag)
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12. | 08.03.12 | 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
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13. | 09.03.12 | 10 | - initial objects: definition, examples in , Incl, Set, , Mult, Graph - terminal objects: definition, examples in , Incl, Set, , Mult, Graph
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14. | 15.03.12 | 11 | - sum: definition, examples in , Incl, Set, Graph
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15. | 16.03.12 | 11 | - product: definition, examples in , Incl, Set, Graph, RGraph
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16. | 22.03.12 | 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 | ||

17. | 23.03.12 | 12 | - monics are reflected by pullbacks, coding of monics by pullbacks - composition of pullbacks is a pullback and decomposition of pullbacks - 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 | ||

| 29.03.12 | 13 | no lecture (conference ETAPS) | ||

| 30.03.12 | 13 | no lecture (conference ETAPS) | ||

| 05.04.12 | 14 | no lecture (Easter weak) | ||

| 06.04.12 | 14 | no lecture (Easter week) | ||

18. | 12.04.12 | 15 | - definition of diagrams - cones and limits - co-cones and colimits - completeness and co-completeness - stepwise construction of limits and colimits | ||

19. | 13.04.12 | 15 | - discussion: formalization of ER diagrams - relations as jointly mono's (injections) - predicates "jointly epi (surjective)" (or "cover" respectively) and "disjoint" | ||

| 19.04.12 | 16 | no lecture (Meeting institute) | ||

| 20.04.12 | 16 | no lecture (Meeting institute) | ||

20. | 26.04.12 | 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 | ||

21. | 27.04.12 | 17 | - discussion: indexed semantics vs. fibred semantics - instances of a graph and corresponding slice category - semantics of predicates - instances of a specification and corresponding category - specification morphisms and corresponding category of specifications - pullback (reduction) functor | PhD 31-43 | |

22. | 03.05.12 | 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 | |

23. | 04.05.12 | 18 | - specification entailments - universal constraints - discussion: logic and deduction | 44-46, 64-70 | |

24. | 10.05.12 | 19 | - relating modelling formalisms - joint modelling formalism - constraint aware transformation rules | 71-95 | |

25. | 11.05.12 | 19 | - course overview | ||

| 17.05.12 | 20 | No more lectures |
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28.05.12 | 23 | oral Exam | |||

04.06.12 | 23 | oral Exam | |||