# categories of functors

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Categories of Functors. http://cis.k.hosei.ac.jp/~yukita/. Functors. Ex. 1. Group Homomorphisms. Ex. 2. Category Cat. Ex. 3. Functors are Constructions Basis  Vector Space. Ex. 4. Functors are Constructions X  Stacks of X. - PowerPoint PPT Presentation

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• Categories of Functorshttp://cis.k.hosei.ac.jp/~yukita/

• Functors

• Ex. 1. Group Homomorphisms

• Ex. 2.

• Category Cat

• Ex. 3. Functors are ConstructionsBasis Vector Space

• Ex. 4. Functors are ConstructionsX Stacks of X

• Ex. 6. Functors are representations (or models) of categories

• Ex. 7. Functors are representations (or models) of categories

• Def. Faithful/Full Functor

• Ex. 8. Giving a functor is specifying three sets and two maps in Sets.012Category ASets

• RemarkIn Ex. 8, three sets and two arrows constitute a single functor.Complex entities can be thus represented.

• Ex. 9-11

• Ex. 13. Directed Graphs10XYd0d1

• Ex. 13. continued

• Ex. 13. continuedbaceabgde

• Ex. 14. Deterministic Automata

• The Functor Category from A to B

• Ex. 15. A natural transformation from F to G is a commutative squares in B.

• Composites of Natural Transformations

• Def. Functor Catetories

• The category of directed graphs Grphs=SetsA10X0d0d1X1

• A morphism in SetsA is a pair (f0,f1).

• Def. Regular Languages

• Remark

• Ex. 21. A regular grammar

• Note. A morphism of graphs

• Def. Regular Languages

• Ex. 28. The Union Operation

• Ex. 30. The Star Operation

• Ex. 31. The Concatenation Operation

• Ex. 32. Pumping a regular language

• Theorem. (Kleene)

• Repeat the procedure until we get to the following graph.

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