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…

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✔

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…

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✔

✔

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…

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✔

✔

OutlinesCategory theory (categories, functors, and natural transformations)

Examples

String diagrams

Diagrammatic proof Yoneda lemma

and more…

32

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✔

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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