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1. Overview on
Category Theory
Naoto Agawa
Aim for today
Rough description
of this seminar
What is category
theory?
Another topic for
category thoery
Role of category
thoery
Origin of category
thoery
Apps of category
thoery
Definition of a
category
Examples of
categories
関係代数と位相
空間
形式的証明
Summary for
today
Overview on Category Theory
subtitle
Naoto Agawa
Tuesday, April 23rd, 2019

2. Overview on
Category Theory
Naoto Agawa
Aim for today
Rough description
of this seminar
What is category
theory?
Another topic for
category thoery
Role of category
thoery
Origin of category
thoery
Apps of category
thoery
Definition of a
category
Examples of
categories
関係代数と位相
空間
形式的証明
Summary for
today
1 Aim for today
2 Rough description of this seminar
3 What is category theory?
4 Another topic for category thoery
5 Role of category thoery
6 Origin of category thoery
7 Apps of category thoery
8 Definition of a category
9 Examples of categories
10 関係代数と位相空間
11 形式的証明
12 Summary for today

3. Overview on
Category Theory
Naoto Agawa
Aim for today
Rough description
of this seminar
What is category
theory?
Another topic for
category thoery
Role of category
thoery
Origin of category
thoery
Apps of category
thoery
Definition of a
category
Examples of
categories
関係代数と位相
空間
形式的証明
Summary for
today
Aim for this seminar
Proposition (M.Barr, 1970)
Top Rel(U)
† M. Barr, Relational algebra, Lecture Notes in Math., 137:39-55, 1970．
We try to formally prove his result with relational calculus.

4. Overview on
Category Theory
Naoto Agawa
Aim for today
Rough description
of this seminar
What is category
theory?
Another topic for
category thoery
Role of category
thoery
Origin of category
thoery
Apps of category
thoery
Definition of a
category
Examples of
categories
関係代数と位相
空間
形式的証明
Summary for
today
Tools
Categories (Of course!!)
Functors
Natural transformations
Vertical composites
”Quasi-” horizontal composites
Adjoint functors
Monads
Relational algebras
Filters

5. Overview on
Category Theory
Naoto Agawa
Aim for today
Rough description
of this seminar
What is category
theory?
Another topic for
category thoery
Role of category
thoery
Origin of category
thoery
Apps of category
thoery
Definition of a
category
Examples of
categories
関係代数と位相
空間
形式的証明
Summary for
today
What is category theory?
Definition of category as one thoery in math

6. Overview on
Category Theory
Naoto Agawa
Aim for today
Rough description
of this seminar
What is category
theory?
Another topic for
category thoery
Role of category
thoery
Origin of category
thoery
Apps of category
thoery
Definition of a
category
Examples of
categories
関係代数と位相
空間
形式的証明
Summary for
today
Another topic for category thoery
Beck’s theorem
Required tools
Fundamental ideas on the previous slide
Universality
The comparison functor
Coequalizers
Coequalizer creators
Implemented FORGETTING types (· · · a variable absorbs
everything)
categories
associativity
identity
→
functors
Law of operators-preservation
natural transformations

7. Overview on
Category Theory
Naoto Agawa
Aim for today
Rough description
of this seminar
What is category
theory?
Another topic for
category thoery
Role of category
thoery
Origin of category
thoery
Apps of category
thoery
Definition of a
category
Examples of
categories
関係代数と位相
空間
形式的証明
Summary for
today
Aim00 for category thoery
Areas of mathematics
Set theory
Linear algebra
Group theory
Ring theory
Module theory
Topology
Algebraic geometry

8. Overview on
Category Theory
Naoto Agawa
Aim for today
Rough description
of this seminar
What is category
theory?
Another topic for
category thoery
Role of category
thoery
Origin of category
thoery
Apps of category
thoery
Definition of a
category
Examples of
categories
関係代数と位相
空間
形式的証明
Summary for
today
Origin of category thoery
ORIGIN
Sprout
(PAPER) S. Eilenberg and S. MacLane, Natural Isomorshisms in
Group Theory, Proceedings of the National Academy of Sciences,
28(1942), 537-543.
”Frequently in modern mathematics there occur phenomena of
”naturality”: a ”natural” isomorphism between two groups or between
two complexes, a ”natural” homeomorphism of two spaces and the
like. We here propose a precise definition of the ”naturality” of such
correspondences, as a basis for an appropriate general theory.”
→ They might want to formulize ”naturality” between one mathematical
flamework and another flamework; i.e. a NATURAL ISOMORPHISM
between two functors in the current category theory.
Ref: https://qiita.com/snuffkin/items/ecda1af8dca679f1c8ac
Topology (Homology)
(PAPER) Samuel Eilenberg and Saunders Mac Lane, General theory
of natural equivalences. Transactions of the American Mathematical
Society 58 (2) (1945), pp.231-294.
They must find it important to DEVELOP an ALGEBRAIC
FLAMEWORK focused on the feature of homomorphisms or
mappings, by CALCULATION of the TOPOLOGICAL INVARIANT
from a series of GROUP HOMOMORPHISMs.
Ref: Book of Proffesor Y. Kawahara

9. Overview on
Category Theory
Naoto Agawa
Aim for today
Rough description
of this seminar
What is category
theory?
Another topic for
category thoery
Role of category
thoery
Origin of category
thoery
Apps of category
thoery
Definition of a
category
Examples of
categories
関係代数と位相
空間
形式的証明
Summary for
today
Definition of natural isomorphism

10. Overview on
Category Theory
Naoto Agawa
Aim for today
Rough description
of this seminar
What is category
theory?
Another topic for
category thoery
Role of category
thoery
Origin of category
thoery
Apps of category
thoery
Definition of a
category
Examples of
categories
関係代数と位相
空間
形式的証明
Summary for
today
Apps of category theory
APPLIED AREAS:
Quantum topology
Happy outcomes:
→ Tangles have a great interaction with various algebraic properties for
their invariants, which allows us to have more deep study for
substantial properties of links.
→ Helps us to see the quantum invariants as the functors from the
category of tangles to a category, where a tangle is a subset of links
(, in intuition, where a link is a collection of multiple knots and a knot
is one closed string).
→ We can generate a invariant for a tangle every time you choose a
special category (called ribbon category) and its object, where in
most cases we choose ribbon category with myriads of elements.
”A polynomial invariant for knots via non Neumann algebras”,
Bulletin of American Mathematical Society (N. S.) 12 (1985), no. 1,
pp.103-111.
Awarded the fields medal on 1990 at Kyoto with ”For the proof of
Hartshorne’s conjecture and his work on the classification of
three-dimensional algebraic varieties.”
cf. At the same meeting a Japanese proffesor Shigefumi Mori was
awarded with ”For the proof of Hartshorne’s conjecture and his work
on the classification of three-dimensional algebraic varieties.”

11. Overview on
Category Theory
Naoto Agawa
Aim for today
Rough description
of this seminar
What is category
theory?
Another topic for
category thoery
Role of category
thoery
Origin of category
thoery
Apps of category
thoery
Definition of a
category
Examples of
categories
関係代数と位相
空間
形式的証明
Summary for
today
Apps of cateory
Denotational semantics for programming languages
Group theory
Mathematical physics (especially, quantum physics) based on
operator algebras
Galois theory and physics
Logic
Algebraic geometry
Algebraic topology
Representation theory
System biology

12. Overview on
Category Theory
Naoto Agawa
Aim for today
Rough description
of this seminar
What is category
theory?
Another topic for
category thoery
Role of category
thoery
Origin of category
thoery
Apps of category
thoery
Definition of a
category
Examples of
categories
関係代数と位相
空間
形式的証明
Summary for
today
Features on cateory theory
set theory · · · point-oriented;
∀
x, x′
∈ s(f), f(x) = f(x′
) ⇒ x = x′
;
∀
y ∈ t(f),∃
x ∈ s(f)s.t.f(x) = y;
∅X ;
{a} ∈ X;
category thoery · · · arrow-oriented;
∀
g1, g2 : W → s(f), f ◦ g1 = f ◦ g2 ⇒ g1 = g2
( assuming W is a set with W = s(g1), W = s(g2));
∀
g1, g2 : t(f) → Z, g1 ◦ f = g2 ◦ f ⇒ g1 = g2
( assuming Z is a set with Z = t(g1), W = t(g2));
∀
X,∃!
f : X → ∅X ;
∀
Y,∃!
f : {a} → Y;

13. Overview on
Category Theory
Naoto Agawa
Aim for today
Rough description
of this seminar
What is category
theory?
Another topic for
category thoery
Role of category
thoery
Origin of category
thoery
Apps of category
thoery
Definition of a
category
Examples of
categories
関係代数と位相
空間
形式的証明
Summary for
today
Features on cateory theory
set theory · · · point-oriented;
X × Y = {(x, y); x ∈ X, y ∈ Y};
category thoery · · · arrow-oriented;
For sets X and Y, a set X × Y is called the cartesian product if the
following condition satisfies:
There exists arrows X X × Y
πl
oo πr
// Y such that the
univarsality
∃!
(f, g) : Z → X × Y, s.t. πl(f, g) = f ∧ πr (f, g) = g
holds for a set Z and arrows X Z
f
oo g
// Y .
X
⟳
X × Y
πl
oo πr
// Y
⟲
Z
∃!(f,g)
OO
f
WW
g
GG

14. Overview on
Category Theory
Naoto Agawa
Aim for today
Rough description
of this seminar
What is category
theory?
Another topic for
category thoery
Role of category
thoery
Origin of category
thoery
Apps of category
thoery
Definition of a
category
Examples of
categories
関係代数と位相
空間
形式的証明
Summary for
today
Definition of a category
The definition of a category
A pair C = (O, (C(a, b))(a,b)∈O2 , (◦(a,b,c))(a,b,c)∈O3 ) with the following
three concepts
O:a set;
(C(a, b))(a,b)∈O2 :a family of sets with the index set O2
;
(◦(a,b,c))(a,b,c)∈O3 :a family of maps with the index set O3
;
is called a category if the following conditions
C(a, b) is disjoint i.e. (a, b) (a′, b′) ⇒ C(a, b) ∩ C(a′, b′) ∅;
◦(a,b,c):C(a, b) × C(b, c) → C(c, a): a map; omitted by for
convinience from here onward;
∀a ∈ O,∃ ida ∈ C(a, a) s.t. ∀b ∈ O,∀ f ∈ C(b, a),∀ g ∈
C(a, b), ida ◦ f = f, g ◦ ida = g
∀a, b, c, d ∈ O,∀ f ∈ C(a, b),∀ g ∈ C(b, c),∀ h ∈ C(c, d), (h ◦ g) ◦ f =
h ◦ (g ◦ f);
all satisfy C.

15. Overview on
Category Theory
Naoto Agawa
Aim for today
Rough description
of this seminar
What is category
theory?
Another topic for
category thoery
Role of category
thoery
Origin of category
thoery
Apps of category
thoery
Definition of a
category
Examples of
categories
関係代数と位相
空間
形式的証明
Summary for
today
Examples of categories
the definition of a category
A set C = (O, M, s, t, ◦) with the following five concepts
O :a set (an element in this set is called an object);
M :a set (an element in this set is called an arrow or a morphism);
s : M → O: a map;
t : M → O: a map;
◦ : M2 → M: a map.
is called a category, where the set Mn = Mn(Q) is defined by
Mn(Q) := {(f1, · · · , fn) ∈ Mn
; s(fi) = t(fi+1) for i = 1, 2, · · · , n − 1}
for n ≥ 2 and the set Q = (O, M, s, t).

16. Overview on
Category Theory
Naoto Agawa
Aim for today
Rough description
of this seminar
What is category
theory?
Another topic for
category thoery
Role of category
thoery
Origin of category
thoery
Apps of category
thoery
Definition of a
category
Examples of
categories
関係代数と位相
空間
形式的証明
Summary for
today
まとめ
関係計算を用いて形式的に検証可能な数学理論の構築例を示した
自動検証可能なプログラムの実装による数学の形式証明支援系の
整備につなげた
T 代数の拡張である関係 T 代数の形式化を関係式で定式化した
ウルトラフィルターモナド U による関係 T 代数の圏 Rel(U) と位
相空間の圏 Top の同型は示されている（Barr, 1970）
† M. Barr, Relational algebra, Lecture Notes in Math., 137:39-55,
1970．
→Barr の結果の関係計算による形式証明を行った．