INF 223, Spring 2011, Plan of Lectures

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

1. | 18.01.11 | 3 | - What is Category Theory? - shift of paradigm - informal discussion of products, dualization, sums | ||

2. | 20.01.11 | 3 | - graphs and graph homomorphisms with examples - opposite graphs | ||

3. | 25.01.11 | 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. | 27.01.11 | 4 | - some finite categories - category Nat - other categories with sets as objects: Rel, Mult, Incl - speciality of Nat and Incl? - pre-orders
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5. | 01.02.11 | 5 | - category Mon of monoids, discussion of monoid morphisms - opposite category - functor: motivation, definition, examples | ||

6. | 03.02.11 | 5 | - some more examples of functors - functors preserve isomorphisms - identity functors and composition of functors - categories Cat, CAT, SET, and GRAPH - summary of the first lectures about "structures" | ||

7. | 08.02.11 | 6 | - discussion of a "metamodel" of graphs - graphs as functors into Set - graph homomorphisms as natural transformations - definition of natural transformations - composition of natural transformations, identity transformations - definition functor categoy and graphs as an example | ||

8. | 10.02.11 | 6 | - iso's in functor categories are the natural isomorphisms - indexed sets as functor category - reflexive graphs as functor category - (finite) state machines with idle actions as reflexive graphs | ||

9. | 15.02.11 | 7 | - "subsystem"-relation as morphisms between reflexive graphs - 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 | ||

10. | 17.02.11 | 7 | - 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 | ||

11. | 22.02.11 | 8 | - 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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| 24.02.11 | 8 | no lecture | ||

12. | 01.03.11 | 9 | - initial objects: definition, examples in , Incl, Set, , Mult, Graph - terminal objects: definition, examples in , Incl, Set, , Mult, Graph
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13. | 03.03.11 | 9 | - sum: definition, examples in , Incl, Set, Graph
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| 08.03.11 | 10 | no lecture (Fagkritisk dag) | ||

| 10.03.11 | 10 | no lecture (illness) | ||

14. | 15.03.11 | 11 | - product: definition, examples in , Incl, Set, Graph, RGraph - motivation pullbacks: inner join, synchronization, products of typed graphs, pre-images | ||

15. | 17.03.11 | 11 | - pullbacks: definition, examples in , Incl, Set, Graph - equalizers, general construction of pullbacks by products and equalizers - equalizers are mono | ||

16. | 22.03.11 | 12 | - monos are reflected by pullbacks - composition of pullbacks is a pullback and decomposition of pullbacks - motivation pushouts: deomposition of graphs, rule applications - pushouts: definition, examples in , Incl, Set, Graph - coequalizers, general construction of pushouts by sums and coequalizers | ||

| 24.03.11 | 12 | no lecture (conference ETAPS) | ||

17. | 29.03.11 | 13 | - discussion: two lines of constructions - definition of diagrams - cones and limits - co-cones and colimits - completeness and co-completeness - stepwise construction of limits and colimits | ||

18. | 31.03.11 | 13 | - discussion: formalization of ER diagrams - relations as jointly mono's (injections) - predicates "jointly epi (surjective)" (or "cover" respectively) and "disjoint" | ||

19. | 05.04.11 | 14 | - discussion: formalization of associations in class diagrams - predicate "inverse" (or "opposite" respectively) for multi functions - discussion: What is a "Diagrammatic Specification Technique" | ||

20. | 07.04.11 | 14 | - example: information system - definition: signature, atomic constraint, specification - specification morphisms and corresponding category of specifications | 31 - 36 | |

| 12.04.11 | 15 | no lecture (Course abroad) | ||

| 14.04.11 | 15 | no lecture (Course abroad) | ||

| 19.04.11 | 16 | no lecture (Easter weak) | ||

| 21.04.11 | 16 | no lecture (Easter week) | ||

| 26.04.11 | 17 | no lecture (Day after Easter Monday) | ||

| 28.04.11 | 17 | no lecture | ||

21. | 03.05.11 | 18 | - discussion: indexed semantics vs. fibred semantics - instances of a graph and corresponding slice category - semantics of predicates - instances of a specification and corresponding category | 34 - 41 | |

22. | 05.05.11 | 18 | - discussion: metamodelling and OMG's 4-layer hierarchy - definition, typed signatures, typed constraints, and typed specifications | 7 - 11, 47 - 49 | |

23. | 10.05.11 | 19 | - conformant specifications - definition modelling formalism - discussion: reflexive metamodel and reflexive modelling formalism | 50 - 54 | |

24. | 12.05.11 | 19 | - course overview | ||

| 17.05.11 | 20 | No more lectures |
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06.06.11 | 23 | Written Exam | |||