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...
1. The Yoneda lemma
and
String diagrams
Ray D. Sameshima
total 54 pages
1
2. Outlines
Category theory (categories, functors,
and natural transformations)
Examples
String diagrams
Diagrammatic proof Yoneda lemma
and more…
2
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. Categories
A Category is like
a network of
arrows with
identities and
associativity.
(We ignore the size
problem now!)
4
5. Functors
A functor is a
structure preserving
mapping between
categories
(homomorphisms of
categories).
5
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. 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 (g f)(a) = h(g(f(a))) = (h g) f(a)
1A : A ! A; a7! a
17. A class change
method
A class change
method: we can
always view an
arbitrary arrow as
a natural
transformation.
14
8f 2 C(A,B)
) 9 ¯ f 2 Nat( ¯ A, ¯B
)
where ¯ A, ¯B
2 Func(1,C)
18. Func(1,C)
This is just pointing mappings of
both objects and arrows in the
category that we consider.
¯ C 2 Func(1,C)
¯ C(⇤) := C 2 |C|
¯ C(1⇤) := 1C
So we can identify all objects as functors
from 1 to the category.
15
19. Nat(A,B 2 Func(1,C))
Under the
identifications, the
arrow in the
category can be
seen as the natural
transformation
between the objects.
16
8f 2 C(A,B)
¯ f 2 Nat(A,B) : ⇤7! ¯ f⇤ := f
This is, I call, a class change method.