yoneda lemma and string diagrams
TRANSCRIPT
Yoneda lemma and string diagrams
Ray D. Sameshima
2014/09/06 ∼2014/09/20
Contents
-1 Preface 3-1.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-1.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0 Definitions 50.0 The size problem . . . . . . . . . . . . . . . . . . . . . . . . . 5
0.0.1 Naive definition of a category . . . . . . . . . . . . . . 50.0.2 Definition of a universe . . . . . . . . . . . . . . . . . 70.0.3 Axiom (universe) . . . . . . . . . . . . . . . . . . . . . 90.0.4 Axiom (class) . . . . . . . . . . . . . . . . . . . . . . . 10
0.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100.1.1 Definition of categories . . . . . . . . . . . . . . . . . . 100.1.2 Examples of category . . . . . . . . . . . . . . . . . . 120.1.3 Some arrows . . . . . . . . . . . . . . . . . . . . . . . 12
0.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130.2.1 Definition of functors (covariant functors) . . . . . . . 13
0.3 Natural transformations . . . . . . . . . . . . . . . . . . . . . 150.3.1 Definition of natural transformations . . . . . . . . . . 150.3.2 Definition of functor categories . . . . . . . . . . . . . 16
1 Yoneda lemma 181.1 Representable functors . . . . . . . . . . . . . . . . . . . . . . 18
1.1.1 Definition of representable functors . . . . . . . . . . . 181.1.2 The Yoneda lemma . . . . . . . . . . . . . . . . . . . . 19
2 Godement products of natural transformations 242.1 Definition of Godement products . . . . . . . . . . . . . . . . 24
2.1.1 Check . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Proposition (The interchanging law) . . . . . . . . . . . . . . 26
1
2.2.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 String diagrams 283.1 A class change method . . . . . . . . . . . . . . . . . . . . . . 283.2 String diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 The Godement product . . . . . . . . . . . . . . . . . . . . . 31
3.3.1 The interchanging law . . . . . . . . . . . . . . . . . . 333.3.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.3 Natural transformations . . . . . . . . . . . . . . . . . 34
3.4 The Yoneda lemma . . . . . . . . . . . . . . . . . . . . . . . . 35
2
Chapter -1
Preface
This is a rough note from my under progress work entitled ”Cat”. I wish toexpress my gratitude to Professor Azita Mayeli and Arthur Parzygnat fortheir advises.
-1.1 References
Handbook of Categorical Algebra 1 Basic Category Theory (Francis Borceux)
Category Theory (Steve Awodey)
An Introduction to Category Theory (Harold Simmons)
nLab (http://ncatlab.org)
http://d.hatena.ne.jp/m-hiyama/20130621/1371785971
http://hal.archives-ouvertes.fr/docs/00/69/71/15/PDF/csl-2008.pdf
http://www.pps.univ-paris-diderot.fr/~curien/categories-pl.ps
http://www.ma.kagu.tus.ac.jp/~abe/index.html
(underconstruction) Cat (Ray D. Sameshima)
-1.2 Notations
∀ : (for) all
∃ : exists
3
∃! : uniquely exists
S ⇒ T : If S, then T .
S ⇔ T : S iff (if and only if) T .
lhs := rhs or lhs :⇔ rhs : (unknown) lhs is defined by (known) rhs.
4
Chapter 0
Definitions
0.0 The size problem
We have to pay some attentions on the sizes, but let us start with someintuitive definitions.
0.0.1 Naive definition of a category
A category C consists of the following date:
1. Objects: A,B,C, · · · ∈ Obj.
2. Arrows:f→,
g→,h→, · · · ∈ Arr.
3. ∀f ∈ Arr,∃dom(f), cod(f) ∈ Obj.The notation
f : A → B (1)
means that A = dom(f), B = cod(f).
4. (composition law) ∀f : A → B and g : B → C with
cod(f) = B = dom(g) (2)
then ∃ an arrow
g ◦ f : A → C. (3)
5
5. (∃identity arrow as a unit) ∀A ∈ Obj,∃ an arrow
1A : A → A (4)
s.t. if we compose it with ∀ arrow from left and right, we get the samearrow, ∀f : A → B,
f ◦ 1A = f = 1B ◦ f. (5)
Then the identity arrow is unique:
1′A = 1A ◦ 1′A = 1A. (6)
6. (associativity) ∀f : A → B, g : B → C, h : C → D,
h ◦ (g ◦ f) = (h ◦ g) ◦ f. (7)
We depict these in the following diagram:
A
1A �� f //
g◦f ��@@@
@@@@
@ B
1B��
g
��
h◦g
@@@
@@@@
@
Ch
// D
(8)
Now we can define a category of sets and mappings. It is easy to checkthe above conditions, for example
1A : A → A; a 7→ 1A(a) := a (9)
and ∀a ∈ A,
h ◦ (g ◦ f)(a) = h (g (f(a))) = (h ◦ g) ◦ f(a). (10)
We denote this category as
Set (11)
We, however, face a problem: objects of Set runs through ”something whichis not a set!” This fact is a consequence of the following well-known paradox:
Russell’s paradox
There exists no set S s.t.,
x ∈ S ⇔ x is a set. (12)
6
Proof We use a contradiction argument. Let say there exists such S,define
R := {x ∈ S|x ∈ x}. (13)
R is well-defined and is a subset of S. By the law of excluded middle, eitherR ∈ R or R ∈ R, but from the definition of R itself,
R ∈ R ⇒ R ∈ R (14)
R ∈ R ⇒ R ∈ R. (15)
This leads us to a contradiction in each case.Or, we can prove it directly, let x be a set,
x ∈ R ⇔ x ∈ x (16)
From the axiom of extensionality, i.e., if every element of M is also anelement of N , and vice versa, then M = N , we get
R = x (17)
that is, R is not a set.■
Taking, intuitively, a set of sets, it is not a set, something ”bigger” thana set. In category theory, it is useful to pay some attention to the ”size.”In order to handle this size problem, there is a way to assume the axiom ofuniverses:
0.0.2 Definition of a universe
A universe is a set U with the following properties:
x ∈ y, y ∈ U ⇒ x ∈ U (18)
I ∈ U , ∀i ∈ I, xi ∈ U ⇒∪i∈I
xi ∈ U (19)
x ∈ U ⇒ P(x) ∈ U (20)
x ∈ U , f : x → y is surjective ⇒ y ∈ U (21)
N ∈ U (22)
where N denotes the set of finite ordinals and P(x) denotes the set of allsubsets of x (the power set of x).
7
Corollary
The following results are immediate consequence of the definition of a uni-verse U .
x ∈ U , y ⊂ x ⇒ y ∈ U (23)
x, y ∈ U ⇒ {x, y} ∈ U (24)
x, y ∈ U ⇒ x× y ∈ U (25)
x, y ∈ U ⇒ yx ∈ U (26)
Proof Since
∅ ∈ N ⇒ ∅ ∈ U . (27)
Assume that x ∈ U , y ⊂ x with y = ∅. Pick z ∈ y and define f : x → y tobe
f(t) :=
{t t ∈ y
z t ∈ y(28)
then f is surjective and therefore y ∈ U .Then let us define
I := {1, 2} ∈ N, x1 = x, x2 = y (29)
then
{x, y} =∪i∈I
xi (30)
and we get {x, y} ∈ U .Since we can use x× y as its index, we have
x× y =∪x0∈x
x0 × y =∪y0∈y
∪x0∈x
x0 × y0 (31)
and hence x× y ∈ U .Finally,
yx =∪x0∈x
yx0 =∪y0∈y
∪x0∈x
yx0o (32)
and we get yx ∈ U .■
8
Axiom of choice
For any set X of nonempty sets, there exists a choice function f defined onX:
∀X, ∅ ∈ X ⇒ ∃f : X →∪
X;A 7→ f(A) ∈ A, (33)
eq.(21) should have been replaced precisely by eq.(23): (x ∈ U , y ⊂ x ⇒ y ∈U).
By the above, the existence of a universe axiom can be translated asbelow:
0.0.3 Axiom (universe)
Every set belongs to some universe.Because of the property in eq.(23), it sounds reasonable to think of the
elements of a universe as being ”sufficiently small sets.” If we choose to usethe theory of universes as a foundation for category theory, the followingconvention has to remain valid:
Convention
We fix a universe U and call ”small sets” the elements of U .Obviously we now have the following proposition:
Proposition
There exists a set S with the property
x ∈ S ⇔ x is a small set. (34)
Proof For the proof, it is sufficient to choose S = U .■
An alternative way to handle these size problem is to use the Godel-Bernays theory of sets and classes. In the Zermelo-Frankle theory, the prim-itive notions are ”set” and ”membership relation”. In the Godel-Bernaystheory, there is one more primitive notion called ”class” (think of it as ”abig set”); that primitive notion is related to the other two by the propertythat every set is a class and, more precisely:
9
0.0.4 Axiom (class)
A class is a set iff it belongs to some (other) class.The axioms concerning classes imply in particular the following ”com-
prehension scheme” for constructing classes:
Comprehension scheme
If
φ(x1, · · · , xn) (35)
is a formula where quantification just occurs on set variables, then there isa class A s.t.
(x1, · · · , xn) ∈ A ⇔ φ(x1, · · · , xn) (36)
Thus the ”class of all sets” is well defined:
the class of all sets := |Set|. (37)
When the axiom of universes is assumed and a universe U is fixed, onegets a model of the Godel-Bernays theory by choosing as ”sets” the elementsof U and as ”classes” the subsets of U ;
sets ∈ classes ⊂ a fixed universe U . (38)
It makes no relevant difference whether we base category theory on theaxiom of universes or on the Godel-Bernays theory of classes. We shall usethe terminology of the latter, thus using the words ”set” and ”class”.
0.1 Categories
0.1.1 Definition of categories
A category C consists of the following date:
1. (Objects) A class |C|, whose elements is called objects of C:
A,B,C,D, · · · ∈ |C|. (39)
10
2. (Arrows) ∀ pair of objects A,B, a set C(A,B), whose elements is calledarrows from A to B:
f ∈ C(A,B). (40)
We write
f : A → B or Af→ B (41)
to indicate that A = dom(f), B = cod(f). We sometimes call such acategory locally small whose arrows consist of a set. We may use
dom(f) = s(f) = source of f, (42)
cod(f) = t(f) = target of f. (43)
3. Let us abuse the notation; C also means all the arrows of category C.Then ∀f ∈ C,
∃dom(f), cod(f) ∈ |C|. (44)
4. (composition law) ∀f ∈ C(A,B) and g ∈ C(B,C) with
cod(f) = B = dom(g) (45)
then ∃ an arrow
g ◦ f ∈ C(A,C). (46)
5. (∃identity arrow as a unit) ∀A ∈ |C|, ∃ an arrow
1A ∈ C(A,A), (47)
s.t.
f ◦ 1A = f = 1B ◦ f. (48)
Then the identity arrow is unique, since
1′A = 1A ◦ 1′A = 1A. (49)
6. (associativity) ∀f ∈ C(A,B), g ∈ C(B,C), h ∈ C(C,D),
h ◦ (g ◦ f) = (h ◦ g) ◦ f. (50)
11
We depict these as a following diagram:
A
1A �� f //
g◦f ��@@@
@@@@
@ B
1B��
g
��
h◦g
@@@
@@@@
@
Ch // D
(51)
We stresses the fact that a category is a ”typed” monoid, that is, in order tomake a composition of arrows, we have to check the domain and codomain.
0.1.2 Examples of category
Empty category
The category 0 or simply ∅ looks like this:
(52)
It has neither objects nor arrows.
Discrete categories
Let us write
1 (53)
for the discrete category withe a single object ∗ and its identity arrow:
∗
id∗=1∗
(54)
0.1.3 Some arrows
Let us define some arrows here.
Commutative diagrams
If
f2 ◦ f1 = f4 ◦ f3 (55)
then we say that the following diagram is commutative:
f1 //
f3
��f2
��f4
//
(56)
12
Isomorphisms
Let f ∈ C(A,B), g ∈ C(B,A) be arrows,
f and g are isomorphisms :⇔ (g ◦ f = 1A and f ◦ g = 1B). (57)
We say that A and B are isomorphic to each other and denote
A ∼= B (f : A ∼= B : g) . (58)
If two arrows g, g′ satisfy eq.(57), then
g′ = g′ ◦ 1B = g′ ◦ f ◦ g = 1A ◦ g = g. (59)
i.e., g is unique. Thus we write the unique inverse
g = f−1(f : A ∼= B : f−1
), (60)
and denote it as a following diagram:
Ag◦f=1A 88
f++ B
g=f−1
kk 1B=f◦gww
(61)
0.2 Functors
Let us consider ”homomorphisms” i.e., structure preserving mappings, amongcategories.
0.2.1 Definition of functors (covariant functors)
A (covariant) functor
F ∈ Func(C,D) (62)
between categories C and D is a mapping between the classes of objects|C| → |D| and arrows C → D, such that,
1. (dom and cod) F preserves domains and codomains:
F (f ∈ C(A,B)) = F (f) ∈ D(FA,FB). (63)
2. (identity) F preserves identity arrow:
∀A ∈ |C|, F (1A) = 1FA. (64)
13
3. (composition) F preserves composition:
F (g ◦ f) = F (g) ◦ F (f). (65)
We depict it as follows:
C
F
��
Af //
g◦f ""FFF
FFFF
FFB
g
��C
D FAF (f) //
F (g◦f)=F (g)◦F (f) ""FFF
FFFF
F FB
F (g)��
FC
(66)
Given two functors
A F→ B G→ C, (67)
a point-wise composition produces a new functor
A G◦F→ C. (68)
This composition law is associative, and the identity functor on categoryA is clearly an identity for this composition law, since we can choose everymapping in the definition to be the identity.
A careless argument could thus lead to the conclusion that categoriesand functors constitute a new category... but this can easily be reduced toa contradiction using Russell’s paradox in §0.0.1. The point is that, in theaxioms for a category, it is required to have a set of morphisms betweenany two objects. And when the categories A and B merely have a classof objects, there is no way to force the functors from category A to B toconstitute a set.
Definition of small categories
A category C is called a small category when its class of object |C| is a set.The next result is then obvious (see Comprehension scheme in §0.0.4).
14
Proposition (Small categories)
Small categories and functors between them constitute a category Cat.
0.3 Natural transformations
Let us consider a ”homotopy” for categories and functors.
0.3.1 Definition of natural transformations
For categories C,D and functors
F,G ∈ Func(C,D), (69)
a natural transformation ϑ ∈ Nat(F,G):
f ∈ C(A,B) �F //
ϑ
��
Ff ∈ D(FA,FB)
f ∈ C(A,B) �G
// Gf ∈ D(GA,GB)
(70)
is a class of |C| indexed arrows in D, ∀f ∈ C(A,B),
(ϑC ∈ D(FC,GC))C∈|C| s.t. ϑB ◦ F (f) = G(f) ◦ ϑA, (71)
i.e., the following diagram is commutative:
FAF (f) //
ϑA
��
FB
ϑB
��GA
G(f)// GB
(72)
Tautologically, we call this commutativity as the naturality of the naturaltransformations.
Given such a natural transformation ϑ ∈ Nat(F,G), the arrow in D,
ϑC ∈ D(FC,GC) (73)
is called the component of ϑ at C or simply C-component of ϑ.Given two natural transformations
Fθ→ G
τ→ H, (74)
15
of functors between categories C and D, we can define a new natural trans-formation:
(τ ◦ θ)C := τC ◦ θC (75)
This composition law is associative, and the unit at each functor F is thenatural transformation 1F whose C-component is 1FC . We sometimes referthis composition as a vertical composition of natural transformations:
FAF (f) //
ϑA
��
FB
ϑB
��GA
G(f) //
τA��
GB
τB��
HAH(f) // HB
(76)
Again a careless argument would deduce the existence of a categorywhose objects are the functors from category C to D and whose arrows arethe natural transformations between them. But since C and D have merelyclasses of objects, there is in general no way to prove the existence of a setof natural transformations between two functors. But when C is a smallcategory, that problem disappears and we get the following result.
Proposition (Functor categories)
Let C be a small category and D be an arbitrary category. The functors fromcategory C to D and the natural transformations between them constitutea category; that category is small as long as D is small.
Let us define this new category;
0.3.2 Definition of functor categories
Let C be a small category. A functor category CFun(C,D) has the following.
The objects are functors F ∈ Func(C,D):
|CFun(C,D)| = Func(C,D). (77)
The arrows are natural transformations ϑ ∈ Nat(F,G):
CFun(C,D)(F,G) = Nat(F,G). (78)
16
For each object F , the natural transformation 1F has components (C-components)
(1F )C = 1FC ∈ D(FC,FC) (79)
and the composite natural transformation of
Fθ→ G
τ→ H (80)
has components (the vertical composition)
(τ ◦ ϑ)C = τC ◦ ϑC . (81)
17
Chapter 1
Yoneda lemma
1.1 Representable functors
1.1.1 Definition of representable functors
Given a category C and a fixed object C ∈ |C|, let us define a covariantfunctor
C(C,−) ∈ Func(C, Set) (1.1)
from C to the category of sets by first putting
C(C,−)(A) := C(C,A). (1.2)
Now if f ∈ C(A,B) is an arrow of C, the corresponding mapping of arrows
C(C,−)(f) := C(C, f) (1.3)
is defined by the formula, ∀g ∈ C(C,A),
C(C, f)(g) := f ◦ g (1.4)
C(C,A)C(C,f)
//
∈��
C(C,B)
∈��
C
g
��
f◦g
��===
====
=
g �C(C,f)
// C(C, f)(g) = f ◦ g Af
// B
(1.5)
Such a functor is called a ”representable functor” because the functor is”represented” by the object C. Since C has a set of arrows (i.e., is locallysmall), the objects |C(C,−)(C)| consist a set. And the arrows of C(C,−)(C),i.e., the codomain of the functor C(C,−), are just mappings between sets,that is, C(C,−)(C) is a category of sets and mappings: Set.
18
1.1.2 The Yoneda lemma
Let A be an arbitrary category. Consider a representable functor of A ∈ |A|and arbitrary functor from A to Set:
A(A,−), F ∈ Func(A, Set). (1.6)
There exists a bijective correspondence
θF,A : Nat(A(A,−), F ) ∼= FA (1.7)
between the natural transformations from A(A,−) to F and the elementsof the set FA; in particular, those natural transformations constitute aset. The bijections θF,A constitute a natural transformation in the variableA ∈ |A|; when A is small, the bijections θF,A also constitute a naturaltransformation in the variable F .
Proof
(bijective) ∀α ∈ Nat(A(A,−), F ), let us define
θF,A(α) := αA(1A). (1.8)
∀a ∈ FA,B ∈ |A|, let us define a mapping as follows, ∀f ∈ A(A,B),
τ(a)B : A(A,B) → FB; f 7→ τ(a)B(f) := F (f)(a). (1.9)
Then τ(a) is a natural transformation, because, ∀g ∈ A(B,C),
F (g) ◦ τ(a)B(f) = F (g) ◦ F (f)(a) ∵ eq.(1.9) (1.10)
= F (g ◦ f)(a) ∵ F is a functor (1.11)
= τ(a)C(g ◦ f) ∵ eq.(1.9) (1.12)
= τ(a)C ◦ A(A, g)(f) ∵ eq.(1.4) (1.13)
Thus we get
F (g) ◦ τ(a)B = τ(a)C ◦ A(A, g) (1.14)
that is, τ(a) : A(A,−) → F is a natural transformation:
τ(a) ∈ Nat(A(A,−), F ) (1.15)
19
A(A,B)
τ(a)B��
A(A,g)// A(A,C)
τ(a)C��
FBF (g)
// FC
(1.16)
In order to prove that θF,A and τ are inverse to each other, let us firstconsider, ∀a ∈ FA,
θF,A ◦ τ(a) = τ(a)A(1A) ∵ eq.(1.8) (1.17)
= F (1A)(a) ∵ eq.(1.9) (1.18)
= 1FA(a). (1.19)
And ∀f ∈ A(A,B),
(τ ◦ θF,A(α))B (f) = τ (θF,A(α))B (f) (1.20)
= τ(αA(1A))B(f) ∵ eq.(1.8) (1.21)
= F (f)(αA(1A)) ∵ eq.(1.9) (1.22)
= αB ◦ A(A, f)(1A) ∵ α ∈ Nat(A(A,−), F )(1.23)
= αB(f ◦ 1A) ∵ eq.(1.4) (1.24)
= αB(f). (1.25)
Thus
θF,A ◦ τ = 1FA (1.26)
τ ◦ θF,A = 1Nat(A(A,−),F ) (1.27)
and θF,A : Nat(A(A,−), F ) ∼= FA is bijective.(naturality) ∀f ∈ A(A,B), define a natural transformation
A(f,−) ∈ Nat(A(B,−),A(A,−)) (1.28)
as follows:
∀C ∈ |A|, g ∈ A(B,C),A(f,−)C(g) = A(f, C)(g) := g ◦ f, (1.29)
then the following diagram is commutative, h ∈ A(C,D),
A(B,C)A(f,C)//
A(B,h)��
A(A,C)
A(A,h)��
A(B,D)A(f,D)
// A(A,D)
(1.30)
20
since
g � A(f,C) //_
A(B,h)��
A(f, C)(g) = g ◦ f_
A(A,h)��
A(B, h)(g) = h ◦ g �A(f,D)
// A(A, h)(g ◦ f) = h ◦ g ◦ f = A(f,D)(h ◦ g)
(1.31)
is commutative.Let us consider a functor N ∈ Func(A, Set), and define
N(Af→ B) = NA
N(f)→ NB (1.32)
:= Nat(A(A,−), F )N(f)→ Nat(A(B,−), F ) (1.33)
s.t., ∀α ∈ Nat(A(A,−), F ),
αN(f)7→ N(f)(α) := α ◦ A(f,−). (1.34)
Then ∃ a natural transformation η ∈ Nat(N,F ) defined by
ηA := θF,A. (1.35)
Indeed, ∀α ∈ Nat(A(A,−), F ),
(ηB ◦N(f)(α)) = (θF,B ◦N(f)(α)) (1.36)
= θF,B (α ◦ A(f,−)) ∵ eq.(1.33) (1.37)
= (α ◦ A(f,−))B (1B) ∵ eq.(1.8) (1.38)
= αB(1B ◦ f) ∵ eq.(1.29) (1.39)
= αB(f ◦ 1A) (1.40)
= (αB ◦ A(A, f)) (1A) ∵ eq.(1.4) (1.41)
= F (f)(αA(1A)) ∵ α ∈ Nat(A(A,−), F ) (1.42)
= (F (f) ◦ θF,A)(α) ∵ eq.(1.8) (1.43)
= (F (f) ◦ ηA)(α). (1.44)
That is, the following diagram is commutative:
NA
ηA=θF,A
��
N(f) // NB
ηB=θF,B
��FA
F (f)// FB
(1.45)
21
Thus η is a natural transformation, i.e., θF,A is a natural transformation inthe variable A.
When A is small, it makes sense to consider the functor category
CFun(A,Set) (1.46)
of functors from A to Set and natural transformations between them. Letus fix an object A ∈ |A|, and consider a functor
M ∈ Func(CFun(A, Set), Set). (1.47)
∀F,G ∈ CFun(A, Set), γ ∈ Nat(F,G), define
MF := Nat(A(A,−), F ), (1.48)
and
M(γ) : Nat(A(A,−), F ) → Nat(A(A,−), G);α 7→ M(γ)(α) := γ ◦ α.(1.49)
Let us define an evaluation functor in A ∈ |A|,
evalA ∈ Func (CFun(A, Set),Set) (1.50)
by, ∀F ∈ CFun(A, Set), γ ∈ Nat(F,G),
evalAF := FA (1.51)
evalA(γ) := γA (1.52)
evalA(Fγ→ G) := FA
γA→ FG. (1.53)
Let us define µ ∈ Nat(M, evalA) as follows,
µF := θF,A (1.54)
then
µG ◦M(γ)(α) = θG,A ◦M(γ)(α) (1.55)
= θG,A(γ ◦ α) ∵ eq.(1.49) (1.56)
= (γ ◦ α)A(1A) ∵ eq.(1.8) (1.57)
= γA ◦ αA(1A) (1.58)
= γA ◦ θF,A(α) ∵ eq.(1.8) (1.59)
= γA (θF,A(α)) (1.60)
= evalA(γ) ◦ θF,A(α) ∵ eq.(1.52). (1.61)
22
Thus
µG ◦M(γ) = evalA(γ) ◦ µF (1.62)
and the following diagram is commutative:
Nat(A(A,−), F )
µF=θF,A
��
M(γ) // Nat(A(A,−), G)
µG=θG,A
��FA
evalA(γ)=γA
// FB
(1.63)
That is, µ is a natural transformation, i.e., θF,A is a natural transformationsin the variable F (when A is small).■
23
Chapter 2
Godement products ofnatural transformations
In eq.(81), we have used a first composition law for natural transforma-tions, a vertical composition. In fact, there exists another possible type ofcomposition for natural transformations, a horizontal composition.
2.1 Definition of Godement products
Let A,B,C be categories, F,H ∈ Func(A,B), G,K ∈ Func(B,C), α ∈Nat(F,H), β ∈ Nat(G,K):
AF
""
H
>>BG
""
K
>>C (2.1)
Fα→ H,G
β→ K (2.2)
The formula, ∀A ∈ |A|,
(β ∗ α)A := βHA ◦G(αA) = K(αA) ◦ βFA (2.3)
defines a natural transformation
β ∗ α : G ◦ F → K ◦H (2.4)
called the ”Godement product” of two natural transformations α and β.
24
2.1.1 Check
Since
αA ∈ B(FA,GA) (2.5)
β ∈ Nat (G,K ∈ Func(B,C)) (2.6)
we have the following commutative diagram:
B
��
FAαA // GA
C GFAGαA //
βFA
��
GHA
βHA
��KFA
KαA
// KHA
(2.7)
Thus the definition
βHA ◦GαA =: (β ∗ α)A := KαA ◦ βFA (2.8)
is consistent:
B
��
FAαA // GA
C GFAGαA //
βFA
��(β∗α)A
((
GHA
βHA
��KFA
KαA
// KHA
(2.9)
Let us consider the following diagram:
A Af // A′
C GFAGFf // GFA′
KHAKHf // KHA′
(2.10)
25
Using the 1st equality of the definition,
GFAGFf //
GαA
��
GFA′
GαA′��
GHAHGf //
βHA
��
GHA′
βHA′��
KHAKGf // KHA′
= GFAGFf //
βHA◦GαA=(β∗α)A��
GFA′
βHA′◦GαA′=(β∗α)A′��
KHAKGf // KHA′
(2.11)
Similarly, we can use the 2nd equality:
GFAGFf //
βFA
��
GFA′
βFA′��
KFAKFf //
KαA
��
KFA′
KαA′��
KHAKHf // KHA′
= GFAGFf //
KαA◦βFA=(β∗α)A��
GFA′
KαA′◦βFA′=(β∗α)A′��
KHAKGf // KHA′
(2.12)
In summary, the Godement product (β ∗ α) is
(β ∗ α)A = βcod(α)A ◦ dom(β)(αA) = cod(β)(αA) ◦ βdom(α)A. (2.13)
■
2.2 Proposition (The interchanging law)
AF
""H //L >>B
G""
K //M >>C (2.14)
Fα→ H
γ→ L,Gβ→ K
δ→ M, (2.15)
⇒ (δ ∗ γ) ◦ (β ∗ α) = (δ ◦ β) ∗ (γ ◦ α) (2.16)
2.2.1 Proof
Since
β ∗ α = βH ◦Gα = Kα ◦ βF (2.17)
δ ∗ γ = δL ◦Kγ = Mγ ◦ δH (2.18)
26
We have the following commutative diagrams, ∀A ∈ |A|,
GFAGαA //
(β∗α)A((
βFA
��
GHA
βHA
��
KHA
δHA
��(δ∗γ)A
((
KγA // KLA
δLA
��KFA
KαA
// KHA MHAMγA
// MLA
(2.19)
Thus we get the following square diagram:
GFAGαA //
(β∗α)A((
βFA
��
GHA
βHA
��
GγA // GLA
βLA
��KFA
δFA
��
KαA
// KHA
δHA
��(δ∗γ)A
((
KγA // KLA
δLA
��MFA
MαA
// MHAMγA
// MLA
(2.20)
From the diagonal composition of the square, ∀A ∈ |A|,
(δ ∗ γ)A ◦ (β ∗ α)A = {(δ ∗ γ) ◦ (β ∗ α)}A , (2.21)
and, since the outer square is commutative, ∀A ∈ |A|, from the upper side,
{(δ ◦ β) ∗ (γ ◦ α)}A = (δ ◦ β)LA ◦G(γ ◦ α)A (2.22)
= δLA ◦ βLA ◦GγA ◦GαA (2.23)
= (δ ∗ γ)A ◦ (β ∗ α)A (2.24)
= MγA ◦MαA ◦ δFA ◦ βFA (2.25)
= M(γ ◦ α)A ◦ (δ ◦ β)FA (2.26)
= {(δ ◦ β) ∗ (γ ◦ α)}A (2.27)
i.e., we get
(δ ∗ γ) ◦ (β ∗ α) = (δ ◦ β) ∗ (γ ◦ α) (2.28)
■
27
Chapter 3
String diagrams
We now introduce a graphical language of string diagrams in category the-ory. Our, or at least my motivation to use such diagrammatic notions is tosimplify some of routine works in category theory with a pencil and papers.We have to save notations, because on 2 dimensional papers we can use atmost 3 types of shapes; dots(nodes), lines, and areas. How to do it?
3.1 A class change method
We can always view ∀ arrow f ∈ C(A,B) in a category C as a naturaltransformation. That is, we can view objects and arrows as functors andnatural transformations:
∀f ∈ C(A,B) ⇒ ∃f ∈ Nat(A, B ∈ Func(1,C)). (3.1)
Let us define, ∀C ∈ |C|,
C ∈ Func(1,C) (3.2)
as follows:
C(∗) := C (3.3)
C(id∗) := idC = 1C . (3.4)
∀f ∈ C(A,B), define a natural transformation
f ∈ Nat(A, B) : ∗ 7→ f∗ := f (3.5)
28
1(∗ id∗ // ∗) A //
f
��
C(A 1A // A)
1(∗ id∗ // ∗)B
// C(B 1B // B)
(3.6)
See the following diagram:
A1A
f∗��
A
f∗��
B1B
B
= A
f��B
(3.7)
Now we have a correspondence between arrows and natural transforma-tions:
¯: C(A,B) → Nat(A, B); f 7→ f , (3.8)
that is
(Af→ B) 7→ (A
f→ B). (3.9)
We may omit upper bar for simplicity, and we can view ∀f ∈ C(A,B)as f ∈ Nat(A,B), and A,B ∈ |C| as ∀A,B ∈ Func(1,C). This is quitecategorical view of points, i.e., the objects are also arrows!
3.2 String diagrams
We represent 5 elements in the category theory, that is, objects, arrows,categories, functors, and natural transformations, with string diagrams. Inour graphical representation, functors are 1 dimensional lines, natural trans-formations are 0 dimensional nodes, and categories are 2-dimensional areas.This situation is thus Poincare dual to that of usual diagrams in categorytheory, see Table 3.1.
We represent a natural transformation α between functors F,G of cate-gories C,D,
α ∈ Nat(F,G), F,G ∈ Func(C,D) (3.10)
29
Table 3.1: Poincare dual
categories functors n.t.
diagrams 0 1 2
string diagrams 2 = 2-0 1 = 2-1 0 = 2-2
as the following diagram:
F
C /.-,()*+α
G
D
(3.11)
From §3.1, we can always see f ∈ C(A,B) as a natural transformationf ∈ Nat(A,B ∈ Func(1,C)):
A
1 /.-,()*+f
B
C
(3.12)
We can also view f as a correspondence 1 → C(A,−)(B) := C(A,B):
1
1 /.-,()*+f
B��������� C(A,−)
AAAA
AAAA
AASet
C
(3.13)
30
Then the (vertical) composition of f ∈ C(A,B), g ∈ C(B,C) is
A/.-,()*+f
B'&%$ !"#gC
=
A
g ◦ fC
(3.14)
or
1/.-,()*+f
B�������
C(A,−)
..............'&%$ !"#g
C
=
1
g ◦ f
Czzzzzzzz C(A,−)
DDDD
DDDD
(3.15)
Notice that, we may omit the label of category for each areas. And forFf ∈ D(FA,FB),
A
F/.-,()*+f
B
=ACC
CCCC
C
F{{{{{{{GFED@ABCFf
B
{{{{{{{{
F CCCC
CCCC
(3.16)
3.3 The Godement product
With our graphical notation, Godement product is the following identity,i.e. a good example of so called the ”elevator” rule: we can freely move the
31
nodes up or down in the diagram as long as they have no overlaps.
F G/.-,()*+α
H
/.-,()*+β
K
=FEE
EEEE
EE
GzzzzzzzzWVUTPQRSβ ∗ α
Hzzzzzzzzz K
EEEE
EEEE
(3.17)
=
F
G/.-,()*+α
H /.-,()*+β
K
=FCC
CCCC
CC
G{{{{{{{{GFED@ABCGα
H GGFED@ABCβH
H|||||||| K
BBBB
BBBB
(3.18)
=
F
G/.-,()*+β
K/.-,()*+α
H
=FBB
BBBB
BB
G||||||||GFED@ABCβF
F KGFED@ABCKα
H{{{{{{{{ K
CCCC
CCCC
(3.19)
that is
β ∗ α = βH ◦Gα = Kα ◦ βF. (3.20)
Thus we get ∀A ∈ |A|,
(β ∗ α)A = βHA ◦G(αA) = K(αA) ◦ βFA. (3.21)
See eq.(2.3).
32
3.3.1 The interchanging law
Using the elevator rule, we can easily derive the interchanging law. Considerthe following diagram:
F G/.-,()*+α
H
/.-,()*+β
K/.-,()*+γ
L
'&%$ !"#δM
(3.22)
then we have two expression of it:
FEE
EEEE
EE
Gzzzzzzzz
β ∗ α
H K
δ ∗ γ
Lzzzzzzzzz M
EEEE
EEEE
=
F G
γ ◦ αL
δ ◦ βM
(3.23)
3.3.2 Functors
Using our graphical notation and the Godement product, we have anotherrepresentation of Ff . From eq.(3.16) and eq.(3.17),
ACC
CCCC
C
F{{{{{{{GFED@ABCFf
B
{{{{{{{{
F CCCC
CCCC
=
A
F/.-,()*+f
B
=
A F/.-,()*+f
B
?>=<89:;1F
F
=AGG
GGGG
GG
Fwwwwwwww_^]\XYZ[1F ∗ f
B
wwwwwwwww
F GGGG
GGGG
G
(3.24)
i.e., we get
Ff = 1F ∗ f. (3.25)
33
3.3.3 Natural transformations
Using our graphical notation, we can show that the Godement products area kind of generalization of the natural transformations. Let us consider anatural transformation
θ ∈ Nat(F,G), F,G ∈ Func(C,D) (3.26)
∀f ∈ C(A,B), the commutative diagram
FAFf //
ϑA
��
FB
ϑB
��GA
Gf// GB
(3.27)
is represented by the following ”identity”:
A F/.-,()*+f
B
'&%$ !"#θG
=ADD
DDDD
DD
FzzzzzzzzWVUTPQRSθ ∗ f
Bzzzzzzzz G
DDDD
DDDD
(3.28)
=
A
F/.-,()*+f
B '&%$ !"#θG
=FAA
AAAA
AA
F}}}}}}}}GFED@ABCFf
B F?>=<89:;θB
B}}}}}}}} G
AAAA
AAAA
(3.29)
=
A
F'&%$ !"#θG/.-,()*+f
B
=ABB
BBBB
BB
F}}}}}}}}?>=<89:;θA
A GGFED@ABCGf
B}}}}}}}} G
AAAA
AAAA
(3.30)
34
Thus the naturality of the natural transformations are represented by theelevator rule, and we get another representation of the naturality
θB ◦ Ff = θ ∗ f = Gf ◦ θA. (3.31)
Equivalently, we can say that θ : F → G is a natural transformation iff∀C ∈ |C|,
C@@
@@@@
@@
F~~~~~~~~?>=<89:;θC
G
@@@@
@@@@
C~~~~~~~~
= C
F'&%$ !"#θG
(3.32)
In general, the commutativity of the category theory is naturally repre-sented as the elevator rule in our string diagrams.
3.4 The Yoneda lemma
We will translate the Yoneda lemma in the string language here. The defi-nition θF,A(α) := αA(1A) (eq.(1.8)) is
1
θF,A(α)
Awwwwwwwww
F GGGG
GGGG
G
:=
1?>=<89:;1A
A
�������������� A(A,−)
AAAA
AAAA /.-,()*+α
F
(3.33)
and the definition τ(a)B(f) := F (f)(a) (eq.(1.9)) is
1/.-,()*+f
B A(A,−)
τ(a)BB
yyyyyyyyy F
EEEE
EEEE
E
:=
1
Ff(a)
Bxxxxxxxxx
F FFFF
FFFF
F =
1'&%$ !"#aA
��������
F
.............../.-,()*+f
B
(3.34)
35
Then the natulality of τ(a)B in the variable B is trivial; we can proveeq.(1.14)
F (g) ◦ τ(a)B = τ(a)C ◦ A(A, g) (3.35)
directly, ∀f ∈ C(A,B),
1/.-,()*+f
B A(A,−)
τ(a)B
B FGFED@ABCFgC
xxxxxxxxx F
FFFF
FFFF
F
=
1/.-,()*+f
B A(A,−)
τ(a)BB
zzzzzzzzz
F
2222
2222
2222
222'&%$ !"#g
C
=
1'&%$ !"#aA
��������
F
))))))))))))))))))))))/.-,()*+f
B'&%$ !"#gC
=
1
g ◦ f
C A(A,−)
τ(a)CC
yyyyyyyyy F
EEEE
EEEE
E
=
1/.-,()*+fB
yyyyyyyyy
A(A,−)'&%$ !"#gC DD
DDDD
DDD
τ(a)CC
wwwwwwwww F
EEEE
EEEE
E
=
1/.-,()*+f
B A(A,−)
A(A, g)
C A(A,−)
τ(a)CC
wwwwwwwww F
GGGG
GGGG
G
(3.36)
thus we get
τ(a) ∈ Nat (A(A,−), F ) (3.37)
36
and the definition eq.(3.34) becomes
1/.-,()*+f
B
��������������� A(A,−)
AAAA
AAAA ONMLHIJKτ(a)
F
=
1'&%$ !"#aA
��������
F
.............../.-,()*+f
B
(3.38)
Next, we show that θF,A and τ are inverse to each other. Using eq.(3.33)and eq.(3.38),
1'&%$ !"#aA F
θF,A ◦ τA
wwwwwwwww F
GGGG
GGGG
G
=
1
θF,A ◦ τ(a)A
tttttt
tttt F
JJJJJJ
JJJJ
=
1?>=<89:;1A
A
���������������
F
CCCC
CCCC ONMLHIJKτ(a)
F
=
1'&%$ !"#aA
}}}}}}}
F
...............?>=<89:;1A
A
=
1'&%$ !"#aA FONMLHIJK1FA
A
{{{{{{{{
F CCCC
CCCC
(3.39)
i.e., we get the same result of eq.(1.19),
θF,A ◦ τ = 1FA (3.40)
37
and
1/.-,()*+f
B
��������������� A(A,−)
IIII
IIII
II
τ(θF,A(α))
F
=
1
θF,A(α)
Axxxxxxxxx
F
3333
3333
3333
3333/.-,()*+f
B
=
1?>=<89:;1A
A�������� A(A,−)
????
????/.-,()*+f
B
/.-,()*+α
F
=
1/.-,()*+f
B
�������������� A(A,−)
????
???? /.-,()*+α
F
(3.41)
thus we get eq.(1.25)
τ (θF,A(α))B (f) = αB(f) (3.42)
= (τ ◦ θF,A(α))B (f) (3.43)
or simply
τ ◦ θF,A = 1Nat(A(A,−),F ). (3.44)
Let us consider the naturality of θF,A in the variable A. The defini-tion of N ∈ Func(A, Set) is as follows. ∀α ∈ Nat(A(A,−), F ),A(f,−) ∈Nat(A(B,−),A(A,−)),
A(B,−)
N(f)(α)
F
:=
A(B,−)
α ◦ A(f,−)
F
=
A(B,−)
A(f,−)
A(A,−)/.-,()*+α
F
(3.45)
38
Then, using eq.(3.33), (θF,B ◦N(f)(α)) is
1
θF,B ◦N(f)(α)
Bppp
pppppp
pp F
NNNNNN
NNNNN
=
1?>=<89:;1B
B
��������������� A(B,−)
IIII
IIII
I
N(f)(α)
F
=
1?>=<89:;1B
B
���������������������� A(B,−)
HHHH
HHHH
H
A(f,−)
A(A,−)/.-,()*+α
F
=
1?>=<89:;1B
B A(B,−)
A(f,B)
F
A(A,−)
HHHH
HHHH
H /.-,()*+α
F
(3.46)
Since
A(f,B)(1B) = 1B ◦ f = f, (3.47)
39
the above equation becomes
1?>=<89:;1B
B A(B,−)
A(f,B)
B
A(A,−)
HHHH
HHHH
H /.-,()*+α
F
=
1/.-,()*+f
B
�������������� A(A,−)
????
???? /.-,()*+α
F
=
1?>=<89:;1AA
�������� A(A,−)
????
????/.-,()*+f
B
/.-,()*+α
F
=
1
θF,A(α)
A FGFED@ABCFfB
vvvvvvvvv F
HHHH
HHHH
H
(3.48)
thus we get
(θF,B ◦N(f)(α)) = Ff ◦ θF,A(α). (3.49)
See eq.(1.45).In the case of A to be small, let us consider the naturality of θF,A in
the variable F . The definition of M ∈ Func(CFun(A, Set), Set) is as follows.∀α ∈ Nat(A(A,−), F ), γ ∈ Nat(F,G), M(γ)(α) := γ ◦ α:
A(A,−)
M(γ)(α)
G
:=
A(A,−)/.-,()*+α
F/.-,()*+γ
G
(3.50)
40
Then, using eq.(3.33), and the definition eq.(3.50),
1
θG,A(M(γ)(α))A
pppppp
ppppp G
NNNNNN
NNNNN
=
1?>=<89:;1A
A
��������������� A(A,−)
IIII
IIII
I
M(γ)(α)
G
=
1?>=<89:;1A
A
��������������������� A(A,−)
AAAA
AAAA /.-,()*+α
F/.-,()*+γ
G
=
1
θF,A(α)
A F?>=<89:;γAA
uuuuuuuuu
G
IIII
IIII
I
=
1
θF,A(α)
A F
evalA(γ)A
vvvvvvvvv G
HHHH
HHHH
H
(3.51)
thus
θG,A(M(γ)(α)) = evalA(γ) ◦ θF,A(α). (3.52)
See eq.(1.63).
41