INF 223, Spring 2011, Plan of Lectures






Nr.DateWeekTopicsPages





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
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





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
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
13.03.03.11 9 - sum: definition, examples in , Incl, Set, Graph





08.03.1110 no lecture (Fagkritisk dag)
10.03.1110 no lecture (illness)





14.15.03.1111 - product: definition, examples in , Incl, Set, Graph, RGraph
- motivation pullbacks: inner join, synchronization, products of typed graphs, pre-images
15.17.03.1111 - pullbacks: definition, examples in , Incl, Set, Graph
- equalizers, general construction of pullbacks by products and equalizers
- equalizers are mono





16.22.03.1112 - 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
24.03.1112 no lecture (conference ETAPS)





17.29.03.1113 - 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
18.31.03.1113 - discussion: formalization of ER diagrams
- relations as jointly mono's (injections)
- predicates "jointly epi (surjective)" (or "cover" respectively) and "disjoint"





19.05.04.1114 - discussion: formalization of associations in class diagrams
- predicate "inverse" (or "opposite" respectively) for multi functions
- discussion: What is a "Diagrammatic Specification Technique"
20.07.04.1114 - example: information system
- definition: signature, atomic constraint, specification
- specification morphisms and corresponding category of specifications
31 - 36





12.04.1115 no lecture (Course abroad)
14.04.1115 no lecture (Course abroad)





19.04.1116 no lecture (Easter weak)
21.04.1116 no lecture (Easter week)





26.04.1117 no lecture (Day after Easter Monday)
28.04.1117 no lecture





21.03.05.1118 - 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
22.05.05.1118 - discussion: metamodelling and OMG's 4-layer hierarchy
- definition, typed signatures, typed constraints, and typed specifications
7 - 11, 47 - 49





23.10.05.1119 - conformant specifications
- definition modelling formalism
- discussion: reflexive metamodel and reflexive modelling formalism
50 - 54
24.12.05.1119 - course overview





17.05.1120 No more lectures





06.06.1123 Written Exam