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Categories of Functors

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.

hiram-williams
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Categories of Functors

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  1. Categories of Functors http://cis.k.hosei.ac.jp/~yukita/

  2. Functors

  3. Ex. 1. Group Homomorphisms

  4. Ex. 2.

  5. Category Cat

  6. Ex. 3. Functors are ConstructionsBasis  Vector Space

  7. Ex. 4. Functors are ConstructionsX Stacks of X

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

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

  10. Def. Faithful/Full Functor

  11. Ex. 8. Giving a functor is specifying three sets and two maps in Sets. Category A 0 Sets 1 2

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

  13. Ex. 9-11

  14. Ex. 13. Directed Graphs 1 0 d0 X Y d1

  15. Ex. 13. continued

  16. Ex. 13. continued b a b g a c e e d

  17. Ex. 14. Deterministic Automata

  18. The Functor Category from A to B

  19. Ex. 15. A natural transformation from F to G is acommutative squares in B.

  20. Composites of Natural Transformations

  21. Def. Functor Catetories

  22. The category of directed graphs Grphs=SetsA 1 0 d0 X0 X1 d1

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

  24. Def. Regular Languages

  25. Remark

  26. Ex. 21. A regular grammar

  27. Note. A morphism of graphs

  28. Def. Regular Languages

  29. Ex. 28. The Union Operation

  30. Ex. 30. The Star Operation

  31. Ex. 31. The Concatenation Operation

  32. Ex. 32. Pumping a regular language … … …

  33. Theorem. (Kleene)

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

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