Post on 29-Jun-2015
description
The Yoneda lemma and
String diagrams
Ray D. Sameshima total 54 pages
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OutlinesCategory theory (categories, functors, and natural transformations)
Examples
String diagrams
Diagrammatic proof Yoneda lemma
and more…
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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)
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CategoriesA Category is like a network of arrows with identities and associativity.
(We ignore the size problem now!)
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Functors
A functor is a structure preserving mapping between categories (homomorphisms of categories).
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Natural transformations
A homotopy of categories.
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Natural transformations
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A natural transformation consists of a class (family, set, or collection) of
arrows.
s.t.
Natural transformations
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A natural transformation consists of a class (family, set, or collection) of
arrows.
s.t.
Natural transformations
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We call this commutativity the naturality of the natural transformations.
Natural transformations
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We call this commutativity the naturality of the natural transformations.
OutlinesCategory theory (categories, functors, and natural transformations)
Examples
String diagrams
Diagrammatic proof Yoneda lemma
and more…
9
OutlinesCategory theory (categories, functors, and natural transformations)
Examples
String diagrams
Diagrammatic proof Yoneda lemma
and more…
9
✔
Examples0
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A category of sets and mappings
A class change method
Representable functors
Natural transformations
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An empty categoryThe empty category: No object and no arrow.
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A singleton category
Discrete categories: objects with identities.
E.g., the singleton (one-point set) can be seen as a discrete category 1.
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The mappings satisfy the associativity law.
!
The identities are identity mappings.
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Setf : A ! B; a 7! f(a)
g : B ! C; b 7! g(b)
h : C ! D; c 7! h(c)
h � (g � f)(a) = h(g(f(a))) = (h � g) � f(a)
1A : A ! A; a 7! a
A class change method
A class change method: we can always view an arbitrary arrow as a natural transformation.
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8f 2 C(A,B)
) 9f 2 Nat(A, B)
where A, B 2 Func(1,C)
This is just pointing mappings of both objects and arrows in the category that we consider.
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Func(1,C)
C 2 Func(1,C)C(⇤) := C 2 |C|
C(1⇤) := 1CSo we can identify all objects as functors from 1 to the category.
Under the identifications, the arrow in the category can be seen as the natural transformation between the objects.
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Nat(A,B 2 Func(1,C))
8f 2 C(A,B)
f 2 Nat(A,B) : ⇤ 7! f⇤ := f
This is, I call, a class change method.
Representable functors
The functor represented by the object C.
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C(C,�) 2 Func(C, Set)
Now we ignore the size problems but…
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C(C,�) 2 Func(C, Set)
By definition
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↵ 2 Nat(C(C,�), F )
8B,C 2 |C|↵C � C(A, g) = Fg � ↵B
8f 2 C(A,B)
↵C � C(A, g)(f) = Fg � ↵B(f)
Let me see
Now we get all gadgets for the Yoneda lemma.
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Yoneda lemmaA milestone of category theory.
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Yoneda lemmaA milestone of category theory.
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An equation based proof
Basically, I traces the proof in this handbook ->.
See my notes.
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So many commutative diagrams
Diagram chasing are routine tasks in the category theory.
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OutlinesCategory theory (categories, functors, and natural transformations)
Examples
String diagrams
Diagrammatic proof Yoneda lemma
and more…
24
✔
OutlinesCategory theory (categories, functors, and natural transformations)
Examples
String diagrams
Diagrammatic proof Yoneda lemma
and more…
24
✔
✔
String diagrams
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Flipping the diagrams!
String diagrams
Two categories, two functors(objects), and a n.t. (an arrow.)
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Af! B
Point it
From above we can see…
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8f 2 C(A,B)
f 2 Nat(A,B) : ⇤ 7! f⇤ := f
f : ⇤ ! C(A,B) = C(A,�)B
Compositions
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These are good examples of vertical compositions.
Compositions
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These are good examples of horizontal compositions.
Basically, that’s all.
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No Standard Committees
… Enjoy!
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Category Theory Using String Diagrams (Dan Marsden)
OutlinesCategory theory (categories, functors, and natural transformations)
Examples
String diagrams
Diagrammatic proof Yoneda lemma
and more…
32
✔
✔
OutlinesCategory theory (categories, functors, and natural transformations)
Examples
String diagrams
Diagrammatic proof Yoneda lemma
and more…
32
✔
✔
✔
Diagrammatic proof
The basic gadget is the elevator rule.
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Yoneda lemmaA milestone of category theory.
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Yoneda lemmaA milestone of category theory.
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Choose wisely
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✓F,A(↵) := ↵A(1A)
Flip it
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⌧(a)(f) := Ff(a)
⌧ = �xy.Fy(x); a 7! �y.Fy(a); f 7! Ff(a)
Naturality of tau
The Adventure of the Dancing Men
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Step by step
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F is a functor
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by def. of tau
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a composition and the def. of tau for gf
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tricky part
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a representable
functor
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We have proved the
naturality of tau:
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⌧(a) 2 Nat (A(A,�), F )
The right inverse
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✓F,A � ⌧
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The left inverse
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⌧ � ✓F,A
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Finally, we have proved that theta and tau are the inverse
pair.
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⌧ � ✓F,A = 1Nat(A(A,�),F )
✓F,A � ⌧ = 1FA
String diagrams are fun!
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OutlinesCategory theory (categories, functors, and natural transformations)
Examples
String diagrams
Diagrammatic proof Yoneda lemma
and more…
53
✔
✔
✔
OutlinesCategory theory (categories, functors, and natural transformations)
Examples
String diagrams
Diagrammatic proof Yoneda lemma
and more…
53
✔
✔
✔
✔
Thank you!
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Godement products and elevator rules
Commutativity and elevator rules
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