INF 223, Spring 2012, Plan of Lectures

Nr.DateWeekTopicsPages 3 - What is Category Theory?
- shift of paradigm
- informal discussion of products, dualization, sums 3 - graphs and graph homomorphisms with examples
- opposite graphs
- discussion of isomorhisms between graphs 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 - some finite categories
- category Nat
- other categories with sets as objects: Incl, Inj, Par, Rel
- speciality of Nat and Incl? - pre-order categories 5 - category Mult
- monoids: examples and definition
- monoid morphisms: examples and definition
- category Mon of momoids
- universal property of lists
- functors: motivation, definition 5 - functors: examples
- functors preserve isomorphisms
- opposite category and contravariant functors
- identity functors and composition of functors
- categories of categories: Cat, CAT, SET 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" 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 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 7 - Arrow categories
- disjoint union functor for arrow categories
- arrows between arrows
- reflexive graphs and finite state machines 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)

01.03.12 9 no lecture (Fagkritisk dag)
02.03.12 9 no lecture (Fagkritisk dag) - 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 - initial objects: definition, examples in , Incl, Set, , Mult, Graph
- terminal objects: definition, examples in , Incl, Set, , Mult, Graph - sum: definition, examples in , Incl, Set, Graph - product: definition, examples in , Incl, Set, Graph, RGraph - 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
- 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

29.03.1213 no lecture (conference ETAPS)
30.03.1213 no lecture (conference ETAPS)

05.04.1214 no lecture (Easter weak)
06.04.1214 no lecture (Easter week) - definition of diagrams
- cones and limits
- co-cones and colimits
- completeness and co-completeness
- stepwise construction of limits and colimits - discussion: formalization of ER diagrams
- relations as jointly mono's (injections)
- predicates "jointly epi (surjective)" (or "cover" respectively) and "disjoint"

19.04.1216 no lecture (Meeting institute)
20.04.1216 no lecture (Meeting institute) - 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 - 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 - 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 - specification entailments
- universal constraints
- discussion: logic and deduction
44-46, 64-70 - relating modelling formalisms
- joint modelling formalism
- constraint aware transformation rules
71-95 - course overview

17.05.1220 No more lectures

28.05.1223 oral Exam

04.06.1223 oral Exam