[SEMINAR at my uni] Tuesday, 14th May, 2019

This document discusses categories of topological spaces and their isomorphism to categories of relational algebras for a monad. It begins with introductions to the topic and tools used, including categories, functors, natural transformations, monads, and relational algebras. The main content is divided into multiple parts, exploring the proposition that the category of topological spaces is isomorphic to the category of relational algebras. It concludes by restating the aim to formally prove this result using relational calculus.

1. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Categories of topological spacies isomorphic
to categories of relational algebras for a
monad
Naoto Agawa
Tuesday, 14 May, 2019

2. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
1Introduction1
2Introduction2
3Main contents Part1
4Main contents Part2
5Main contents Part3
6Main contents Part4
7Conclusion

3. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
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. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Tools in this seminar
Categories
Functors
Natural transformations
Vertical composites
”Quasi-” horizontal composites
Adjoint functors
Monads
Relational algebras
Filters

5. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
What is category theory?
Definition of category as one thoery in math

6. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
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. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Aim00 for category thoery
Areas of mathematics
Set theory
Linear algebra
Group theory
Ring theory
Module theory
Topology
Algebraic geometry

8. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
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. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Definition of natural isomorphism

10. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
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. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
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. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
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. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
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. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Definition of categories
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 three
following conceptsO: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. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
GO THROUGH THIS SLIDE
Definition (Quivers)
A pair Q= (O;M; s;t)is called aquiveror anoriented graphif
following conditions
OandMare sets;
s
:M ! Oand t:M ! Oare maps;
are satisfied. An element of
Ois called avertex, and that ofMan
arrow. For an arrow f2 M, s(f)is called asourceof f and t(f)is
called a
targetof f. Q is called afinite quiverifOandMare finite.
Figure:an oriented graph

16. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Definition of categories
Definition (Quivers)
Let
OandMbe sets.
s
:M ! Oand t:M ! Obe maps.
Then, a quadruplet Q
= (O;M; s;t)is called aquiveror anoriented
graph
, where an element of
Ois called avertex;
Mis called anarrow;
and the image
s
(f)is called asourceof f
t
(f)is called atargetof f
for an arrow f
2 M.
Q is called a
finite quiverifOandMare finite.

17. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Definition of categories
Figure:an oriented graph

18. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Definition of categories
Definition (Paths in a quiver)
Let a quadruplet Q= (O;M; s;t)be a quiver, and, for n2,Mn(Q)
a set defined by
Mn(Q) :=f(f1; ;fn)2 M
n
;s(fi) =t(fi+1);1 in1g;
whereM0(Q) :=O andM1(Q) :=M .
Then, an element of
Mn(Q)is called apathof length n in Q, and is
described as follows:
vn
fn
//
vn1
fn1
//vn2
fn2
//
f2
//
v1
f1
//
v0:
Moreover,
a path
(f1; ;fn)2 M
n
(Q)is denoted by f1 fn.
Mn(Q)is denoted byMn.

19. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Definition of categories
Definition (Categories)
Let
(O;M; s;t)be a quiver, where
it is denoted by
Quiver(C)
M
n(Quiver(C)) is denoted byMnorMn(C)
◦:M2! M
be a map, where◦(f;g)is denoted by f◦g.
Then, a quintette
C= (O;M; s;t;◦)is called acategoryif
s
(f◦g) =s(g)and t(f◦g) =t(f)hold for a path(f;g)2 M2
(associativity)(f◦g)◦h=f◦(g◦h)holds for a path
(f;g;h)2 M
(identity) there exists a map1:O ! M both f◦1
s(f)=f and
1
t(f)◦f=f hold, where1(A)is denoted by1Afor a vertex A2 O
all satisfyC.
v
3
h
//
v2
g
//
v1
f
//
v0:

20. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Definition of categories
Definition (from the previous slide (other helpful words))
An element inOis called anobject.
Ois calleda set of objectsinCand denoted byOb(C).
An element in
Mis called amorphismor anarrow.
Mis calleda set of morphismsinCand denoted byMor(C) .
For a morphism f
2 M,
s
(f)is called asourceof f, or adomainof f.
t
(f)is called atargetof f, or acodomainof f.
source
f
//
target

21. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Definition of categories
Definition (from the previous slide (other helpful words))
◦
is called acomposition.
f
◦g is called acompositeof f and g.
A
2Ob(C) is denoted by A2 Cfor simplicity as long as there is
no risk of confusion.
For sets A and B in
C,
a subset s
1
(
A)\t
1
(
B)ofMor(C) is called aset of morphisms
from A to B, and it is denoted byHomC( A;B).
an element in
HomC( A;B)is called amorphismfrom A to B.
f
2HomC( A;B)is denoted by f:A!B.

22. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Definition of categories
Lemma
Let
C:= (O;M; s;t;◦)be a category,
and let
UA
:= HomC( A;A)be a set
GA
:= (UA;◦)the diad
for an object A
2 O.
Then, GAis a monoid with the identity element
1A.

23. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Proof.
We have that
8
f;g;h2UA;(h◦g)◦f=h◦(g◦f) (∵the associativity onC).
putting on id
GA:=1A, then
i◦idGA=f◦1A=f
and
id
GA◦f=1A◦f=f
hold for an arrowf2UA(∵the identity onC).
□

24. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Corollary
On the previous lemma, the arrow1Aand the map1:O ! M are
respectively
unique, because of the uniqueness of the identity in a
monoid.
Definition
On the previous corollary,1Ais called theidentityor theidentity
morphism
on A.

25. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Definition of categories
Definition (Categories (traditional))
Let
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.
Then, the triad
C= (O;(C( a;b))
(
a;b)2O
2;(◦( a;b;c))
(a;b;c)2O
3)is called a
categoryif
fC(a;b)g
(
a;b)2O
2is 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)is a map, and it is omitted
by
◦for convinience from here onward
(identity)
8
a2 O;
9ida2 C(a;a)s:t:
8b2 O;
8f2 C(b;a);
8g2
C(
a;b);ida◦f=f;g◦ida=g
(associativity)
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.

26. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Definition of categories
Lemma (Cantor)
Let A be a set. ThenjAj<jP(A)jholds, whereP(A)is the power set
of A.
Proof.
If
A=∅, then we haveP( A) =f∅g, which yieldsj Aj<jP(A)j.
If
A,∅, we only have to take an injection but is a bijection. Let
f:A! P(A)be a map defined by
f(x) =fxg;
then this map is an injection, which yieldsj
Aj jP(A)j. Thus, we
only have to verify that
fis not a bijection.
(GO TO THE NEXT SLIDE.)
□

27. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Definition of categories
Proof.
(FROM THE PREVIOUS SLIDE)
Assumingj Aj=jP( A)jholds in order to use the proof of
contradiction, then we have a bijection
g:A! P(A)by definition.
By the way,
g(a)is a subset ofAbecauseg(a)2 P(A)for alla2A.
Thus, we have a2g(a)_a<g(a),
so let
Rbe a set defined by
R=fx2A;x<g(x)g;
then we have
R2 P(A). Note thatgis a surjection, we find2 A
satisfied withg() =R.
If2
R, we have< g() =Rby the definition ofR.
Conversely, given<
R, we have2 g() =R.
By this contradiction, we getj
Aj,jP(A)j, which presents the desired
equationj
Aj⪇jP(A)j.
□

28. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Definition of categories
Proposition
Let(Ai)i2Ibe a family of sets with the index set I.
Then there exists a set not isomorphic to any of the set Ajfor an
index j
2I.Proof.
By the previous lemma, the proof completes when you take the
power set of
Aj.
□
!A collection of all sets is too large to be a set, and is neither a
category.
(In this seminar, we do not use a category such thatOandM
are too large to be sets.)
A conept ”universe” was yielded.
An universe is a set among which we can consider any
operations.

29. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Definition of categories
Axiom (Universes)
There exists an universeUsuch that X2 Uholds for a set X, where
a
(Grothendieck) universeis a setUwith the following properties:
N=f0;1;2; g 2 U .
8
x;y;x2y;y2 U )x2 U.
I
2 U;f:I! U: a map)
∪
i2If(i)2 U.
x
2 U ) P( x)2 U, whereP(x)is the power set of x.

30. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Example
By the defininition of quivers, we can takeC, an unique pair of sets
defined by
Ob(C) =∅
Mor(C) =∅;
and this is called theempty category.
Example
There exists a category with single object and single arrow (the
identity), and is denoted by
1.
id
{{

31. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Example
There exists a category with two objects a;b and just one arrow not
the identity, and is denoted by
2.
aid
%%
//
b id
yy
Example
There exists a category with three objects, non-identity arrows of
which are arranged as the following traiangle, and it is denoted by
3.

!!
id
fifi

??
//
id
##

id
{{

32. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Example
Let X be a set, thenC, a pair of sets, defined by
Ob(C) = X
Mor(C) =f1 x;x2Xg
is a category, and it is called adescrete category. In fact,
C(x;x) =f1 xg
C(
x;y) =∅( x,y)
hold for an element x2X.
x
id

y
id

z
id

33. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Example (a category with single object)
Given a monoid M, thenC, a pair of sets, is to be a category if
defined as follows:
Ob(C) := (the underlying set of M)
Mor(C) :=
{
fl
idfl
! fl
}
◦C:= (
the operator inM) (◦C is a map on categoryC):
Thus, we can construct a category with single object.

34. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Proposition (Categories with single object)
LetMwith single object be a subcategory ofCat. Then, we have
MonM .
fl
zz
!I’m going to strictly show this fact later, because we have to have
more concepts in category theory e.g.
functors
natural transformations
the isomorphic-density for a functor
other more concepts...

35. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
GO THROUGH THIS SLIDE
Definition
The denotation◦Xis composition in a categoryX.
Definition
subcategory
Definition
full subcategory

36. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
GO THROUGH THIS SLIDE
Definition
functor
Definition
thecomposite
Definition
theidentity functor
Definition
full functor faithfull functor

37. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
GO THROUGH THIS SLIDE
Definition
natural transformation
Definition
natural isomorphism
orequivalence isomorphicorequivalent

38. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
GO THROUGH THIS SLIDE
Definition
isomorphic
orequivalent
Definition
essential image isomorphism-dense
oressentially surjective

39. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
GO THROUGH THIS SLIDE
Definition
Anequivalencebetween categoriesAandBconsists of a pair
A
F
⇄
G
Bof functors together with natural isomorphisms
:1A!G◦F;
":
F◦G!1B:
If there exists an equivalence between A and B, we say that A and B
are
equivalent, and write AB. We also say that the functors F and
G are
equivalences.
Definition
LetAbe a category. AsubcategoryS ofAconsists of a subob(S)
ofob(A) together with, for each S;S
′
2ob(S)
, a subclassS(S;S
′
)
ofA(S;S
′
)
, such thatSis closed under composition and identities. It
is a
full subcategoryifS(S;S
′
) =A(
S;S
′
)
for all S;S
′
2ob(S)
.

40. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
GO THROUGH THIS SLIDE
Lemma (Corollary 1.3.19 in ”Basic Category Theory”, T. Leinster)
Let F:C ! Dbe a full and faithful functor. ThenCis equivalent to
the full subcategory
C
′ofDwhose objects are those of the form
F
(C)for some C2 C.
Proposition (Prop 1.3.18 in ”Basic Category Theory”, T. Leinster)
essentially surjective on objects
Theorem
Let F:C ! Dbe a functor. Then, the following propositions are
equivalent:
1F is an equivalence.
2F is full, faithful and essentially surjective.
3F is a part of some adjoint equivalence(F;G; ; ").

41. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
GO THROUGH THIS SLIDE
Lemma
Let F:C ! Dbe a functor. Then, the following propositions are
equivalent:
1F is an equivalence.
2F is full, faithful and essentially surjective.
Proof.
By the previous lemma, we only have to take an equivalence
M
F
!Mon; i.e.
9
G:D ! C: functor;F◦GIdC^ G◦FIdD
□

42. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
GO THROUGH THIS SLIDE
Proof.
(!)Given a monoidM2Mon, thenC, a pair of sets, is a category
when defined as follows:
Ob(M) :=fflg
Mor(M) :=
M
◦M:=◦Mon:
( )Let
Sbe a category ofM, then Sis a monoid when defined as
follows:
Ob(
M) :=Ob(C)
Mor(
M) :=EndC(fl) = M
◦Mon:=◦M:
□

43. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Definition (pseudo-orders)
A relationon a set P is called apseudo-orderor apreorderif it is
reflexive and transitive; i.e. for all a
;b;c2P, we have that:
(reflexivity) a
a
(transitivity) a
b;bc)ac.
A set that is equipped with a preorder is called a
preordered set(or
proset).
Example
partial orders
total orders, or linear orders
equivalence relations

44. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Example (the category with preordered sets)
Let P be a poset, and a;b elements in P. Then,C, a pair of sets,
defined by
Ob(C) := (the underlying set of P)
C(
a;b) :=
{
(b;a)
}
(if ab)
C(
a;b) :=
{
(b;a)
}
(otherwise)
(
c3;c2)◦C(c2;c1) := (c3;c1) (
8c1;
8c2;
8c32P)(, where◦CinC)
is a category.

45. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Example (the category with totally ordered sets)
Let P be an ordered set, and a;b elements in P. Then,C, a pair of
sets, defined by
Ob(C) :=f the underlying set of Pg
C(
a;b) :=
{
(b;a)
}
(if ab)
C(
a;b) :=
{
(b;a)
}
(otherwise)
(
c3;c2)◦C(c2;c1) := (c3;c1) (
8c1;
8c2;
8c32P)(, where◦CinC)
is a category.

46. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Example (special case for the above)
For n0, a countable finite totally orderd set
Sn
:= (f0;1; ; n1g; P)is a category.
1
//
2
//
3
//

!We already had nearly the same chain as follows:
1 2 3
!The most upside is thesameas descrete categories.

47. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Definition (Zermelo-Fraenkel set theory (ZFC))
(Axiom of extensionality)
Two sets are equal (are the same set) if they have the same
elements:
8x8y[8z(z2x,z2y))x=y]:
(Axiom of regularity(also called theAxiom of foundation))
Every non-empty set x contains a member y such that x and y
are disjoint sets:
8x(x,∅! 9y2x(y\x=∅))
This implies, for example, that no set is an element of itself and
that every set has an ordinal rank.
(
Axiom schema of specification(also called theaxiom
schema of separation
orof restricted comprehension))

48. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Definition (Zermelo-Fraenkel set theory (ZFC))
(Axiom of pairing)
(
Axiom of union)
(
Axiom schema of replacement)

49. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Definition
(Axiom of infinity)
(
Axiom of power set)
For any set x, there is a set y that contains every subset of x:
8x9y8z[zx)z2y]:
(Well-ordering theorem)

50. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Axiom (HERE IN USE)
An universeUis fixed.
Definition
An element ofUis called asmall set.
This expression ”small” does not refer to how small its cardinality is.
Proposition
fUg
is a finite set with some single element, howeverfUg<U holds.
Proof.
First we haveU 2 fUgby definition. AssumingfUg 2 Uholds, by
using the definition of universes, we haveU 2 Uin contradiction to
the axiom of regularity. □

51. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Example
For small sets a;b2 U, a categoryCdefined by
Ob(C) :=U
C(
a;b) :=fmaps from a to bg
ordinary composite of maps
is called a category of (small) sets, and is denoted by
Set
.
!We want to consider a category ofentiresets, however we have
difficulty using that category because that is not a set. Therefore, we
compose a category ofsmallsets, which is a really set.
Definition
A structured set with a small underlying set is called asmall
structured set
.

52. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Example
Grpis a category, where its objects are all small groupsfGigi2Ifor an
index set I, its arrow is a group homomorphism of GSfor a set S, and
its composition is the operator in GS.
Grp
is called acategory of
(small) groups
.
Example
Monis a category, where its objects are all small monoidsfMigi2Ifor
an index set I, its arrow is a monoid homomorphism of MSfor a set
S, and its composition is the operator in MS.
Mon
is called a
category of (small) monoids.
Example
Abis a category, where its objects are all small Abelian groupsfAigi2I
for an index set I, its arrow is an Abelian group homomorphism of AS
for a set S, and its composition is the operator in AS.
Ab
is called a
category of (small) abelian groups.

53. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Example
Ringis a category, where its objects are all small ringsfRigi2Ifor an
index set I, its arrow is a ring homomorphism of RSfor a set S, and
its composition is the operators in RS.
Ring
is called acategory of
(small) rings
.
Example
CRingis a category, where its objects are all small commutative
fRigi2Ifor an index set I, its arrow is a ring homomorphism restericted
to RSfor a set S, and its composition is the operators in RS.
CRing
is
called a
category of (small) commutative rings.

54. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Example
RModis a category, where its objects are all small left R-modules, its
arrows are all linear maps. R
Mod
is called acategory of (small)
left
R-modules.
Example
ModRis a category, where its objects are all small right R-modules,
its arrows are all linear maps. R
Mod
is called acategory of (small)
right
R-modules.
Example
Ordis a category, where its objects are all small ordered sets, its
arrows are all preserving maps, and its composition is regular one of
maps.
Ord
is called acategory of (small) ordered sets.

55. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Example
Topis a category, where its objects are all small topological spaces,
its arrows are all continuous maps, and its composition is the usual
composition of maps.
Top
is called acategory of (small)
topological spaces
.
Example
Tophis a category, where its objects are all small topological spaces,
its arrows are all homotopy classes of continuous maps.
Toph
is
called a
category of (small) topological spaces.
Example
Top
fl
is a category, where its objects are topological spaces with
selected base point, its arrows are all base point-preserving maps.
Top
fl
is called acategory of (small) topological spaces.

56. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Example
C
rMfd
is a category, where its objects are all small C
r
-manifolds, its
arrows are all C
r
-maps. C
r
Mfd
is called acategory of (small)
C
r-manifolds.
Example
Schis a category, where its objects are all small schemes, its arrows
are all morphisms of schemes, and its composition is the usual
composition of maps.
Sch
is called acategory of (small) schemes.

57. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Example
Matr
K
for a fixed field K is a category, where its objects are all
positive integers m
;n; , and its arrow is a mfin matrix A (which is
regarded as a map A
:m!n), and its composition is the usual
matrix product.
MatrK
is called a
category of (small) vector spaces.
Example
Vct
K
for a fixed field K is a category, where its objects are all small
vector spaces over K, its arrows are all linear transformations, and its
composition is usual composition of maps.
VctK
is called a
category
of (small) vector spaces
.

58. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Example
Euclidis a category, where its objects are all small Euclidean spaces,
its arrows are all orthogonal transformations.
Euclid
is called a
category of (small) Euclidean spaces.
Example
Ses
A
is a category, where its objects are all small short exaxt
sequences of A-modules.
SesA
is called a
category of (small)
A-modules.

59. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Example
Setflis a category, where its objects are all small sets each with a
selected base-point, its arrows are all base-point preserving maps.
Setfl
is called acategory of (small) base points.
Example
Smgrpis a category, where its objects are all small semigroups, its
arrows are all semigroup morphisms.
Smgrp
is called acategory of
(small) semigroups
.

60. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Example
Metis a category, where its objects are all small metric spaces
X
;Y; , its arrows X!Y those functions which preserve the
metric, and its composition is usual multiplication of real numbers.
Met
is called acategory of (small) metric spaces.

61. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Definition
A categoryCis small if it is small as a set; i.e.OandMare small.
Example (small categories)
Set;Grp;Ab;Top
Counterexample (small categories)
Set;Grp;Ab;Top

62. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Examples of categories
Definition
A categoryCis called anUcategory if
C(a;b)2 U
holds for objects a;b2Ob(C) .
Example (Ucategories)
Set;Grp;Ab;Top

63. Categories of
topological
spacies
isomorphic to
categories of
relational algebras
for a monad
Naoto Agawa
Introduction1
Introduction2
Main contents
Part1
Main contents
Part2
Main contents
Part3
Main contents
Part4
Conclusion
Conclusion
the defintion of categories
a ”directed graph” together with
associative composite regarding arrows
the identity arrow
the examples of categories
Topof topological spaces and homeomorphisms.
Vect
Kof vector spaces over a fieldKand homomorphisms.
Monof monoids and hoomorphisms restricted to them.
more other examples...
we make sure to verifyMMon(Mis a subcategory with
single object, of a category)