## 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 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) 17.04.14 16 no lecture (Easter Holiday) 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 22 No more lectures 3+4.06.13 23 Oral Exam (thus the day can be changed)