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Exploring Functors in Category Theory: Concepts, Examples, and Applications

This resource elaborates on functors in category theory, presenting key examples such as group homomorphisms, functors in the category of sets, and their applications in vector spaces and stacks. It highlights the significance of functors as constructions that represent various mathematical entities, including directed graphs and deterministic automata. The document also discusses natural transformations, faithful and full functors, and examines regular languages alongside crucial operations like union, star, and concatenation. Ideal for students and researchers seeking to understand functors more deeply.

hiram-williams
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Exploring Functors in Category Theory: Concepts, Examples, and Applications

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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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