# 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 PresentationTRANSCRIPT

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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