[Seminar at my uni] Tuesday, 23 April, 2019

1. The document provides an overview of category theory, outlining topics like what category theory is, its origins in topology, and applications in areas like quantum physics and programming languages.
2. It defines key category theory concepts like categories, functors, natural transformations and gives examples like sets and relations.
3. The seminar aims to formally prove a result of Barr (1970) relating the category of relations under an ultrafilter monad to the category of topological spaces using relational calculus.

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
E:qì
í
Ü$ ÂÌ
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
E:qì
í
Ü$ ÂÌ
Summary for
today
1Aim for today
2Rough description of this seminar
3What is category theory?
4Another topic for category thoery
5Role of category thoery
6Origin of category thoery
7Apps of category thoery
8Definition of a category
9Examples of categories
10E:qìí
11Ü$ ÂÌ
12Summary 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
E:qì
?
?$ ?◦?
Summary for
today
Aim for this seminar
Proposition (M:Barr;1970)
TopRel(U)
yM. Barr,
Relational algebra, Lecture Notes in Math., 137:39-55, 1970fi}
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
E:qì
í
Ü$ ÂÌ
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
E:qì
í
Ü$ ÂÌ
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
E:qì
?
?$ ?◦?
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
E:qì
í
Ü$ ÂÌ
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
E:qì
?
?$ ?◦?
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
E:qì
í
Ü$ ÂÌ
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
E:qì
?
?$ ?◦?
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
E:qì
í
Ü$ ÂÌ
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
E:qì
?
?$ ?◦?
Summary for
today
Features on cateory theory
set theory point-oriented;
8
x;x
′
2
s(f);f(x) =f(x
′
))
x=x
′
;
8
y2t(f);
9x2s(f)s:t:f(x) =y;
∅
X;
f
ag 2X;
category thoery arrow-oriented;
8
g1;g2:W!s(f);f◦g1=f◦g2)g1=g2
( assuming
Wis a set withW=s(g1);W=s(g2));
8
g1;g2:t(f)!Z;g1◦f=g2◦f)g1=g2
( assuming
Zis a set withZ=t(g1);W=t(g2));
8
X;
9!f:X! ∅X;
8
Y;
9!f:fag !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
E:qì
?
?$ ?◦?
Summary for
today
Features on cateory theory
set theory point-oriented;
XfiY=f(x;y);x2X;y2Yg;
category thoery arrow-oriented;
For sets
XandY, a setXfiYis called the cartesian product if the
following condition satisfies:
There exists arrows
X X fiY

l
oo

r
//
Ysuch that the
univarsality
9!
(
f;g) :Z!XfiY;s:t: l(f;g) =f^r(f;g) =g
holds for a setZand arrowsX Z
foo
g
//
Y.
X
⟳
XfiY

l
oo

r
//
Y
⟲
Z
9!
(
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
E:qì
?
?$ ?◦?
Summary for
today
Definition of a category
The definition of a category
A pairC= (O;(C(
a;b))
(
a;b)2O
2;(◦( a;b;c))
(a;b;c)2O
3)with the following
three concepts
O:a set;
(C(
a;b))
(
a;b)2O
2:a family of sets with the index setO
2
;
(◦
(
a;b;c))
(a;b;c)2O
3:a family of maps with the index setO
3
;
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)fi C(b;c)! C( c;a): a map; omitted by for
convinience from here onward;
8
a2 O;
9
id a2 C(a;a)s:t:
8b2 O;
8f2 C(b;a);
8g2
C(
a;b);ida◦f=f;g◦ida=g
8
a;b;c;d2 O;
8f2 C(a;b);
8g2 C(b;c);
8h2 C(c;d);(h◦g)◦f=
h◦(g◦f);
all satisfyC.

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
E:qì
?
?$ ?◦?
Summary for
today
Examples of categories
the definition of a category
A setC= (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 setM
n=M n(Q)is defined by
M
n(Q) :=f( f1; ;fn)2 M
n
;s(fi) =t(fi+1)fori=1;2; ; n1g
for
n2 and the setQ= (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
E:qì
?
?$ ?◦?
Summary for
today
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??wMTopw?x?^?oM??Barr;1970ޣ
yM. Barr,
Relational algebra, Lecture Notes in Math., 137:39-55,
1970fi}
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