INF 223, Spring 2017, Plan of Lectures (Status 04.04.2017)






Nr.DateWeekTopicsPages





1.17.01.17 3 - What is Category Theory?
- shift of paradigm
- informal discussion of products, dualization, sums
1 - 11
2.19.01.17 3 - graphs and graph homomorphisms: motivation, examples, definition
- opposite graphs
- discussion of isomorhisms between graphs
13 - 19





3.24.01.17 4 - composition of maps and identity maps
- composition of graph homomorphisms and identity graph homomorphisms
- associativity and identity law of composition
- definition of category
13 - 26
4.26.01.17 4 - categories Set and Graph
- 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
26 - 32





5.31.01.17 5 - other categories with sets as objects: Incl, Inj, Par
- subcategory: examples and definition
- Nat and Incl as pre-order categories
- partial-order categories
- discussion associations in class diagrams
- composition of relations
32 - 39
6.02.02.17 5 - category Rel
- association ends and multimaps
- category Mult
- monoids: examples and definition
- monoid morphisms: examples and definition
39 - 47





7.07.02.17 6 - monoid morphisms: examples and definition
- category Mon of monoids
- universal property of lists
- Boolean Algebras: motivation, definition, examples
- self-duality of Boolean algebras
48 - 60
8.09.02.17 6 - laws in Boolean algebras
- Boolean expressions: definition and evaluation
- universal porperty of Boolean expressions, substitution
60 - 67





9.14.02.17 7 - many-sorted signatures and algebras: definition, examples
- category of S-sets and category of Sigma-algebras
- terms, term-algebras and evaluation of terms
- universal property of the evaluation of terms
67 - 69
10.16.02.17 7 - subalgebras and their images and pre-images
- functors: motivation, definition
- functors: examples
- product graphs and product categories
- functors preserve isomorphisms
69 - 80





11.21.02.17 8 - subcategories, embeddings (examples graphs of maps)
- opposite category and contravariant functors
- identity functors and composition of functors
- categories of categories: Cat, CAT, SET, GRAPH
- pathes: motivation, examples, definition
- path graph and evaluation of paths
- categorical diagrams: motivation, definition, examples
- commutative diagram: definition and examples
39 - 42
12.23.02.17 8 - path categories and their universal property
- definition free constructions
- path categories and disjoint union as free construction
- any free construction induces a free functor
43 - 48





13.28.02.17 9 - cofree constructions introduced as dual of free constructions
- Cartesian products as cofree construction
- any cofree construction induces a cofree functor
- discussion data/structure as free constructions and behaviour as cofree constructions
- streams: motivation and definition
48 - 56
14.02.03.17 9 - category SET^loop and streams as cofree construction
- discussion: free construction ~ induction vs. cofree construction ~ coinduction
- examples coinduction
- free constructions induce cofree constructions and vice versa
- general informal discussion: algebras vs. colagebras
56 - 60





15.07.03.1710 - example: lists and streams together as cofree construction
- free constructions induce cofree constructions and vice versa
- informal discussion of adjunctions
- informal discussion algebra vs. coalgebra against free vs. cofree
- list vs. power set construction as a motivation for natural transformations
09.03.1710 no lecture (Fagkritisk dag)





16.14.03.1711 - general discussion about models and metamodels
- metamodel of concept graph
- graphs as interpretations and graph homomorphisms as natural transformations
- algebraic signatures as graphs
- definition of natural transformations and interpretation/functor categories
- indexed sets as interpretation category
- discussion indexed categories with model functor as an example
17.16.03.1711 - arrow categories
- category of E-graphs
- discussion of arrows between arrows
- path equations, satisfaction of path equations
- reflexive graphs (path categories as reflexive graphs)
- discussion indexed sets vs. typed sets as motivation for typing
- motivation of "typing" by ER-diagrams





18.21.03.1712 - motivation of "typing" by ER-diagrams and Petri nets
- type graphs and typed graphs and their morphisms
- definition slice category
- indexed sets vs. typed sets as motivation for equivalence of categories
- example typed E-graphs
- co-slice categories, example pointed sets
- equivalence of categories: definition and examples
19.23.03.1712 - functors between categories of typed graphs
- final definition of adjunctions
- satisfaction relation and corresponding Galois connection as an adjunction
- equivalences of categories are adjunctions
- hierarchy: isomorphism - equivalence - adjunction





20.28.03.1713 - monad of an adjunction and Kleisli category of a monad
- equivalence relations and equivalence classes
- quotient sets and natural maps
- unique factorization of maps
- equivalences as abstraction in mathematics
- quotient path categories/ quotient (term) algebras
21.30.03.1713 - 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
- jointly monic and jointly epic





22.04.04.1714 - initial objects: definition, examples in , Incl, Set, , Mult, Graph
- terminal objects: definition, examples in , Incl, Set, Mult, Graph
- freely generated objects as initial objects in a comma category
23.06.04.1714 - sum: definition, examples in , Incl, Set, Graph
- product: definition, examples in , Incl, Set, Graph
- sums and products as adjoints to the diagonal functor





11.04.1715 no lecture (Easter Holiday)
13.04.1715 no lecture (Easter Holiday)





18.04.1716 no lecture (conference)
20.04.1716 Consultancy 2nd compulsory exercise





24.25.04.1717 - exponentiation: example functions spaces and definition
- exponentiation as adjoint to product functor
- implication and deduction theorem as exponentiation
- Cartesian Closed Categories
- (maybe Local Cartesian Closed Categories)
- lambda-calculus and exponentiation
25.27.04.1717 - motivation pullbacks: inner join, synchronization, products of typed graphs, pre-images
- pullbacks: definition, examples in , Incl, Set, Graph
- preimages as pullbacks
- reduct functor between slice categories is given by pullbacks





26.02.05.1718 - 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
- synchronization is pullback
04.05.1718 no lecture (institute seminar)





27.09.05.1719 - motivation pushouts: sharing, decomposition of graphs, rule applications
- pushouts: definition, examples in , Incl, Set, Graph
- term rewriting and graph transformations as pushouts
- coequalizers, general construction of pushouts by sums and coequalizers
- discussion: two lines of constructions
28.11.05.1719 - cones and limits
- co-cones and colimits
- completeness and co-completeness
- free functors are based on colimit constructions and preserve colimits
- stepwise construction of limits and colimits





20 No more lectures





21 No more lectures





22 No more lectures





08.06.1723 Oral Exam (another day can be agreed)