Categories of Functors

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Categories of Functors

• http//cis.k.hosei.ac.jp/yukita/

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Functors
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Ex. 1. Group Homomorphisms
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Ex. 2.
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Category Cat
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Ex. 3. Functors are ConstructionsBasis ? Vector
Space
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Ex. 4. Functors are ConstructionsX ? Stacks of X
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Ex. 6. Functors are representations (or models)
of categories
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Ex. 7. Functors are representations (or models)
of categories
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Def. Faithful/Full Functor
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Ex. 8. Giving a functor is specifying three sets
and two maps in Sets.
Category A
0
Sets
1
2
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Remark

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

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Ex. 9-11
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Ex. 13. Directed Graphs
1
0
d0
X
Y
d1
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Ex. 13. continued
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Ex. 13. continued
b
a
b
g
a
c
e
e
d
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Ex. 14. Deterministic Automata
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The Functor Category from A to B
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Ex. 15. A natural transformation from F to G is a
commutative squares in B.
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Composites of Natural Transformations
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Def. Functor Catetories
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The category of directed graphs GrphsSetsA
1
0
d0
X0
X1
d1
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A morphism in SetsA is a pair (f0,f1).
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Def. Regular Languages
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Remark
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Ex. 21. A regular grammar
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Note. A morphism of graphs
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Def. Regular Languages
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Ex. 28. The Union Operation
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Ex. 30. The Star Operation
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Ex. 31. The Concatenation Operation
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Ex. 32. Pumping a regular language



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Theorem. (Kleene)
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Repeat the procedure until we get to the
following graph.