Tentative plan of Lectures (status 05.01.2121)
Interesting links:
- https://www.schoolofhaskell.com/
- https://wiki.haskell.org/Category_theory
- https://ncatlab.org/nlab/show/extension+system (monads)
There are two important breaks that can be used to work on compulsory exercises: the winter break (first week of March, week 9) and an extended Easter break (week 13 and 14). There will be a third compulsory exercise after the Easter break.
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Lecture 21.01.21 (week 3)
- What is Category Theory?
- shift of paradigm
- informal discussion of products, dualization, sums
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Lecture 22.01.21 (week 3)
- graphs and graph homomorphisms: motivation, examples, definition
- opposite graphs
- discussion of isomorhisms between graphs
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Lecture 28.01.21 (week 4)
- composition of maps and identity maps
- composition of graph homomorphisms and identity graph homomorphisms
- associativity and identity law of composition
- definition of category
- categories Set and Graph
- a universal definition of isomorphism
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Lecture 29.01.21 (week 4)
- composition of isomorphisms is isomorphism
- isomorphisms in Set are bijective maps
- isomorphisms in Graph are componentwise bijective graph homomorphisms
- some finite categories
- representation of finite categories by pictorial diagrams
- other categories with sets as objects: Incl, Inj, Par
- Nat and Incl as pre-order categories
- pre-order categories and partial order categories
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Lecture 05.02.21 (week 5)
- subcategory: examples and definition
- associations in class diagrams
- composition of relations
- category Rel
- association ends and multimaps
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Lecture 06.02.21 (week 5)
- category Mult
- monoids: examples and definition
- monoid morphisms: examples and definition
- category Mon of monoids
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Lecture 12.02.21 (week 6)
- inductive definition of lists
- universal property of lists (free monoids)
- functors: motivation, definition
- functors: examples
- product graphs with finite example
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Lecture 13.02.21 (week 6)
- product categories
- functors preserve isomorphisms
- 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
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Lecture 19.02.21 (week 7)
- categorical diagrams: motivation, definition, examples
- commutative diagram: definition and examples
- path categories
- summary of the first lectures about "structures"
- general discussion about models and metamodels
- discussion of a "metamodel" MG of graphs
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Lecture 20.02.21 (week 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 categories
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Lecture 26.02.21 (week 8)
- indexed sets as functor category
- arrow categories
- category of E-graphs
- discussion of arrows between arrows
- path equations, satisfaction of path equations
- model interpretations
- reflexive graphs
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Lecture 27.02.21 (week 8, no lectures in week 9)
- motivation of "typing" by ER-diagrams and Petri nets
- type graph and typed graphs and their morphisms
- definition slice category
- example typed E-graphs
- indexed vs. typed sets
- equivalence of categories
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Lecture 11.03.21 (week 10)
- equivalence relations and equivalence classes
- quotient sets and natural maps
- unique factorization of maps
- equivalences as abstraction in mathematics
- representatives and normal forms
- quotient path categories
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Lecture 12.03.21 (week 10)
- 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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Lecture 18.03.21 (week 11)
- initial objects: definition, examples in Incl, Set, Mult, Graph
- terminal objects: definition, examples in Incl, Set, Mult, Graph
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Lecture 19.03.21 (week 11)
- sum: definition, examples in Incl, Set, Graph
- product: definition, examples in Incl, Set, Graph
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Lecture 25.03.21 (week 12)
- motivation pullbacks: intersection, inner join, products of typed graphs
- pullbacks: definition, examples in Incl, Set, Graph
- preimages as pullbacks
- equalizers: definition, example in Set
- kernel and graph of a map f:A->B as equalizers
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Lecture 26.03.21 (week 12, no lectures in week 13, 14)
- general construction of pullbacks by products and equalizers
- fibred products
- equalizers are mono
- monics are reflected by pullbacks, coding of monics by pullbacks
- composition of pullbacks is a pullback and decomposition of pullbacks
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Lecture (15.04.21, week 15)
Monads 1
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Lecture (16.04.21, week 15)
Monads 2
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Lecture (22.04.21, week 16)
Monads 3
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Lecture (23.04.21, week 16)
Monads 4
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Lecture (29.04.21, week 17)
- Coequalizers
- Limits and colimits
- Pushouts
- Finite limits from products and equalizers
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Lecture (30.04.21, week 17) No lecture
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Lecture (06.05.21, week 18)
Yoneda Lemma 1
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Lecture (07.05.21, week 18)
Yoneda Lemma 2
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No lectures from 13.05.21, Ascension Day