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# The Yoneda lemma and String diagrams

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The Yoneda lemma and string diagrams

When we study the categorical theory, to check the commutativity is a routine work.
Using a string diagrammatic notation, the commutativity is replaced by more intuitive gadgets, the elevator rules.
I choose the Yoneda lemma as a mile stone of categorical theory, and will explain the equation-based proof using the string diagrams.

reference:
1: Category theory: a programming language-oriented introduction (Pierre-Louis Curien)
(especially in section 2.6)

You can get the pdf file in the below link:
http://www.pps.univ-paris-diderot.fr/~mellies/mpri/mpri-ens/articles/curien-category-theory.pdf

2: The Joy of String Diagrams (Pierre-Louis Curien)
http://hal.archives-ouvertes.fr/docs/00/69/71/15/PDF/csl-2008.pdf

3: (in progress) Cat (Ray D. Sameshima)

4: Physics, Topology, Logic and Computation: A Rosetta Stone (John C. Baez, Mike Stay)
http://math.ucr.edu/home/baez/rosetta.pdf
If you are physicist, this is a good introduction to category theory and its application on physics.
His string diagrams, however, differ from our one little.

5: Category Theory Using String Diagrams (Dan Marsden)
http://jp.arxiv.org/abs/1401.7220

outlines
1 Category, functor, and natural transformation
2 Examples
3 String diagrams
4 Yoneda lemma and string diagrams

5 and more...

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### The Yoneda lemma and String diagrams

1. 1. The Yoneda lemma and String diagrams Ray D. Sameshima total 54 pages 1
2. 2. Outlines Category theory (categories, functors, and natural transformations) Examples String diagrams Diagrammatic proof Yoneda lemma and more… 2
3. 3. References Handbook of Categorical Algebra (F. Borceux) The Joy of String Diagrams (P. L. Curien) Category theory (P. L. Curien) (in progress) Cat (R. D. Sameshima) 3
4. 4. Categories A Category is like a network of arrows with identities and associativity. (We ignore the size problem now!) 4
5. 5. Functors A functor is a structure preserving mapping between categories (homomorphisms of categories). 5
6. 6. Natural transformations A homotopy of categories. 6
7. 7. Natural transformations A natural transformation consists of a class (family, set, or collection) of arrows. 7 s.t.
8. 8. Natural transformations A natural transformation consists of a class (family, set, or collection) of arrows. 7 s.t.
9. 9. Natural transformations We call this commutativity the naturality of the natural transformations. 8
10. 10. Natural transformations We call this commutativity the naturality of the natural transformations. 8
11. 11. Outlines Category theory (categories, functors, and natural transformations) Examples String diagrams Diagrammatic proof Yoneda lemma and more… 9
12. 12. Outlines Category theory (categories, functors, and natural transformations) Examples String diagrams Diagrammatic proof Yoneda lemma and more… 9 ✔
13. 13. Examples 0 1 A category of sets and mappings A class change method Representable functors Natural transformations 10
14. 14. An empty category The empty category: No object and no arrow. 11
15. 15. A singleton category Discrete categories: objects with identities. E.g., the singleton (one-point set) can be seen as a discrete category 1. 12
16. 16. Set The mappings satisfy the associativity law. ! The identities are identity mappings. 13 f : A ! B; a7! f(a) g : B ! C; b7! g(b) h : C ! D; c7! h(c) h