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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