This document defines categories, functors, and natural transformations in category theory. It begins by discussing the "size problem" in naively defining categories and introduces the concept of a universe to address this. Categories are then defined as classes of objects and sets of arrows between objects, satisfying composition and identity laws. Functors map categories to categories by mapping objects and arrows, preserving structure. Natural transformations relate functors by assigning morphisms between their actions on objects. The Yoneda lemma and Godement products of natural transformations are also introduced.

1. Yoneda lemma and string diagrams
Ray D. Sameshima
2014/09/06
2014/09/20

2. Contents
-1 Preface 3
-1.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
-1.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0 De

3. nitions 5
0.0 The size problem . . . . . . . . . . . . . . . . . . . . . . . . . 5
0.0.1 Naive de

4. nition of a category . . . . . . . . . . . . . . 5
0.0.2 De

5. nition of a universe . . . . . . . . . . . . . . . . . 7
0.0.3 Axiom (universe) . . . . . . . . . . . . . . . . . . . . . 9
0.0.4 Axiom (class) . . . . . . . . . . . . . . . . . . . . . . . 10
0.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
0.1.1 De

6. nition of categories . . . . . . . . . . . . . . . . . . 10
0.1.2 Examples of category . . . . . . . . . . . . . . . . . . 12
0.1.3 Some arrows . . . . . . . . . . . . . . . . . . . . . . . 12
0.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
0.2.1 De

7. nition of functors (covariant functors) . . . . . . . 13
0.3 Natural transformations . . . . . . . . . . . . . . . . . . . . . 15
0.3.1 De

8. nition of natural transformations . . . . . . . . . . 15
0.3.2 De

9. nition of functor categories . . . . . . . . . . . . . 16
1 Yoneda lemma 18
1.1 Representable functors . . . . . . . . . . . . . . . . . . . . . . 18
1.1.1 De

10. nition of representable functors . . . . . . . . . . . 18
1.1.2 The Yoneda lemma . . . . . . . . . . . . . . . . . . . . 19
2 Godement products of natural transformations 24
2.1 De

11. nition of Godement products . . . . . . . . . . . . . . . . 24
2.1.1 Check . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Proposition (The interchanging law) . . . . . . . . . . . . . . 26
1

12. 2.2.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 String diagrams 28
3.1 A class change method . . . . . . . . . . . . . . . . . . . . . . 28
3.2 String diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 The Godement product . . . . . . . . . . . . . . . . . . . . . 31
3.3.1 The interchanging law . . . . . . . . . . . . . . . . . . 33
3.3.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.3 Natural transformations . . . . . . . . . . . . . . . . . 34
3.4 The Yoneda lemma . . . . . . . . . . . . . . . . . . . . . . . . 35
2

13. Chapter -1
Preface
This is a rough note from my under progress work entitled Cat. I wish to
express my gratitude to Professor Azita Mayeli and Arthur Parzygnat for
their 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
8 : (for) all
9 : exists
3

14. 9! : 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 de

15. ned by (known) rhs.
4

16. Chapter 0
De

17. nitions
0.0 The size problem
We have to pay some attentions on the sizes, but let us start with some
intuitive de

18. nitions.
0.0.1 Naive de

19. nition of a category
A category C consists of the following date:
1. Objects: A;B;C; 2 Obj.
2. Arrows:
f!
;
g!
;
h!
; 2 Arr:
3. 8f 2 Arr; 9dom(f); cod(f) 2 Obj.
The notation
f : A ! B (1)
means that A = dom(f);B = cod(f).
4. (composition law) 8f : A ! B and g : B ! C with
cod(f) = B = dom(g) (2)
then 9 an arrow
g ◦ f : A ! C: (3)
5

20. 5. (9identity arrow as a unit) 8A 2 Obj; 9 an arrow
1A : A ! A (4)
s.t. if we compose it with 8 arrow from left and right, we get the same
arrow, 8f : A ! B,
f ◦ 1A = f = 1B ◦ f: (5)
Then the identity arrow is unique:
′
A = 1A ◦ 1
1
′
A = 1A: (6)
6. (associativity) 8f : A ! B; g : B ! C; h : C ! D,
h ◦ (g ◦ f) = (h ◦ g) ◦ f: (7)
We depict these in the following diagram:
1A f
A
/
@@ @@ @@ @
g◦f @
1B

21. B
@
@@ g
@@ @@
C
h◦g
@
h
/D
(8)
Now we can de

22. ne a category of sets and mappings. It is easy to check
the above conditions, for example
1A : A ! A; a7! 1A(a) := a (9)
and 8a 2 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 which
is 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 2 S , x is a set. (12)
6

23. Proof We use a contradiction argument. Let say there exists such S,
de

24. ne
R := fx 2 Sjx̸2 xg: (13)
R is well-de

25. ned and is a subset of S. By the law of excluded middle, either
R 2 R or R̸2 R, but from the de

26. nition of R itself,
R 2 R ) R̸2 R (14)
R̸2 R ) R 2 R: (15)
This leads us to a contradiction in each case.
Or, we can prove it directly, let x be a set,
x 2 R , x̸2 x (16)
From the axiom of extensionality, i.e., if every element of M is also an
element of N, and vice versa, then M = N, we get
R̸= x (17)
that is, R is not a set.
■
Taking, intuitively, a set of sets, it is not a set, something bigger than
a 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 of
universes:
0.0.2 De

27. nition of a universe
A universe is a set U with the following properties:
x 2 y; y 2 U ) x 2 U (18)
I 2 U; 8i 2 I; xi 2 U )
∪
i2I
xi 2 U (19)
x 2 U ) P(x) 2 U (20)
x 2 U; f : x ! y is surjective ) y 2 U (21)
N 2 U (22)
where N denotes the set of

28. nite ordinals and P(x) denotes the set of all
subsets of x (the power set of x).
7

29. Corollary
The following results are immediate consequence of the de

30. nition of a uni-
verse U.
x 2 U; y x ) y 2 U (23)
x; y 2 U ) fx; yg 2 U (24)
x; y 2 U ) x y 2 U (25)
x; y 2 U ) yx 2 U (26)
Proof Since
∅ 2 N ) ∅ 2 U: (27)
Assume that x 2 U; y x with y̸= ∅. Pick z 2 y and de

31. ne f : x ! y to
be
f(t) :=
{
t t 2 y
z t̸2 y
(28)
then f is surjective and therefore y 2 U.
Then let us de

32. ne
I := f1; 2g 2 N; x1 = x; x2 = y (29)
then
fx; yg =
∪
i2I
xi (30)
and we get fx; yg 2 U.
Since we can use x y as its index, we have
x y =
∪
x02x
x0 y =
∪
y02y
∪
x02x
x0 y0 (31)
and hence x y 2 U.
Finally,
yx =
∪
x02x
yx0 =
∪
y02y
∪
x02x
yx0
o (32)
and we get yx 2 U.
■
8

33. Axiom of choice
For any set X of nonempty sets, there exists a choice function f de

34. ned on
X:
8X; ∅̸2 X ) 9f : X !
∪
X;A7! f(A) 2 A; (33)
eq.(21) should have been replaced precisely by eq.(23): (x 2 U; y x ) y 2
U).
By the above, the existence of a universe axiom can be translated as
below:
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 use
the theory of universes as a foundation for category theory, the following
convention has to remain valid:
Convention
We

35. x 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 2 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-Bernays
theory, there is one more primitive notion called class (think of it as a
big set); that primitive notion is related to the other two by the property
that every set is a class and, more precisely:
9

36. 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 quanti

37. cation just occurs on set variables, then there is
a class A s.t.
(x1; ; xn) 2 A , φ(x1; ; xn) (36)
Thus the class of all sets is well de

38. ned:
the class of all sets := jSetj: (37)
When the axiom of universes is assumed and a universe U is

39. xed, one
gets a model of the Godel-Bernays theory by choosing as sets the elements
of U and as classes the subsets of U;
sets 2 classes a

40. xed universe U: (38)
It makes no relevant difference whether we base category theory on the
axiom of universes or on the Godel-Bernays theory of classes. We shall use
the terminology of the latter, thus using the words set and class.
0.1 Categories
0.1.1 De

41. nition of categories
A category C consists of the following date:
1. (Objects) A class jCj, whose elements is called objects of C:
A;B;C;D; 2 jCj: (39)
10

42. 2. (Arrows) 8 pair of objects A;B, a set C(A;B), whose elements is called
arrows from A to B:
f 2 C(A;B): (40)
We write
f : A ! B or A
f!
B (41)
to indicate that A = dom(f);B = cod(f). We sometimes call such a
category 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 8f 2 C,
9dom(f); cod(f) 2 jCj: (44)
4. (composition law) 8f 2 C(A;B) and g 2 C(B;C) with
cod(f) = B = dom(g) (45)
then 9 an arrow
g ◦ f 2 C(A;C): (46)
5. (9identity arrow as a unit) 8A 2 jCj; 9 an arrow
1A 2 C(A;A); (47)
s.t.
f ◦ 1A = f = 1B ◦ f: (48)
Then the identity arrow is unique, since
′
A = 1A ◦ 1
1
′
A = 1A: (49)
6. (associativity) 8f 2 C(A;B); g 2 C(B;C); h 2 C(C;D),
h ◦ (g ◦ f) = (h ◦ g) ◦ f: (50)
11