[WANTED] your spicy comments.

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
1 Introduction1
2 Introduction2
3 Main contents Part1
4 Main contents Part2
5 Main contents Part3
6 Main contents Part4
7 Conclusion

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)
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. 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;
∀
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. 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;
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
GG 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
GG Y .
X
⟳
X × Y
πl
oo πr
GG Y
⟲
Z
∃!(f,g)
yy
f
‡‡
g
qq

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 pair C = (O, (C(a, b))(a,b)∈O2 , (◦(a,b,c))(a,b,c)∈O3 ) with the three
following 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. 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 a quiver or an oriented graph if
following conditions
O and M are sets;
s : M → O and t : M → O are maps;
are satisfied. An element of O is called a vertex, and that of M an
arrow. For an arrow f ∈ M, s(f) is called a source of f and t(f) is
called a target of f. Q is called a finite quiver if O and M are 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
O and M be sets.
s : M → O and t : M → O be maps.
Then, a quadruplet Q = (O, M, s, t) is called a quiver or an oriented
graph, where an element of
O is called a vertex;
M is called an arrow;
and the image
s(f) is called a source of f
t(f) is called a target of f
for an arrow f ∈ M.
Q is called a finite quiver if O and M are 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 n ≥ 2, Mn(Q)
a set defined by
Mn(Q) := {(f1, · · · , fn) ∈ Mn
; s(fi) = t(fi+1), 1 ≤ i ≤ n − 1},
where M0(Q) := O and M1(Q) := M.
Then, an element of Mn(Q) is called a path of length n in Q, and is
described as follows:
vn
fn
GG vn−1
fn−1
GG vn−2
fn−2
GG · · ·
f2
GG v1
f1
GG v0 .
Moreover,
a path (f1, · · · , fn) ∈ Mn
(Q) is denoted by f1 · · · fn.
Mn(Q) is denoted by Mn.

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)
Mn(Quiver(C)) is denoted by Mn or Mn(C)
◦ : M2 → M be a map, where ◦(f, g) is denoted by f ◦ g.
Then, a quintette C = (O, M, s, t, ◦) is called a category if
s(f ◦ g) = s(g) and t(f ◦ g) = t(f) hold for a path (f, g) ∈ M2
(associativity) (f ◦ g) ◦ h = f ◦ (g ◦ h) holds for a path
(f, g, h) ∈ M
(identity) there exists a map 1 : O → M both f ◦ 1s(f) = f and
1t(f) ◦ f = f hold, where 1(A) is denoted by 1A for a vertex A ∈ O
all satisfy C.
v3
h GG v2
g
GG v1
f GG 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 in O is called an object.
O is called a set of objects in C and denoted by Ob(C).
An element in M is called a morphism or an arrow.
M is called a set of morphisms in C and denoted by Mor(C).
For a morphism f ∈ M,
s(f) is called a source of f, or a domain of f.
t(f) is called a target of f, or a codomain of f.
source
f GG 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 a composition.
f ◦ g is called a composite of f and g.
A ∈ Ob(C) is denoted by A ∈ C for 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) of Mor(C) is called a set of morphisms
from A to B, and it is denoted by HomC(A, B).
an element in HomC(A, B) is called a morphism from A to B.
f ∈ HomC(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 ∈ O.
Then, GA is 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
∀
f, g, h ∈ UA , (h ◦ g) ◦ f = h ◦ (g ◦ f) (∵ the associativity on C).
putting on idGA
:= 1A , then
i ◦ idGA
= f ◦ 1A = f
and
idGA
◦ f = 1A ◦ f = f
hold for an arrow f ∈ UA (∵ the identity on C).
□

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 arrow 1A and the map 1 : O → M are
respectively unique, because of the uniqueness of the identity in a
monoid.
Definition
On the previous corollary, 1A is called the identity or the identity
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)∈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
.
Then, the triad C = (O, (C(a, b))(a,b)∈O2 , (◦(a,b,c))(a,b,c)∈O3 ) is called a
category if
{C(a, b)}(a,b)∈O2 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) is a map, and it is omitted
by ◦ for convinience from here onward
(identity) ∀
a ∈ O,∃
ida ∈ C(a, a) s.t. ∀
b ∈ O,∀
f ∈ C(b, a),∀
g ∈
C(a, b), ida ◦ f = f, g ◦ ida = g
(associativity) ∀
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.

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. Then |A| < |P(A)| holds, where P(A) is the power set
of A.
Proof.
If A = ∅, then we have P(A) = {∅}, which yields |A| < |P(A)|.
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) = {x},
then this map is an injection, which yields |A| ≤ |P(A)|. Thus, we
only have to verify that f is 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)
Assuming |A| = |P(A)| holds 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 of A because g(a) ∈ P(A) for all a ∈ A.
Thus, we have
a ∈ g(a) ∨ a g(a),
so let R be a set defined by
R = {x ∈ A; x g(x)},
then we have R ∈ P(A). Note that g is a surjection, we find α ∈ A
satisfied with g(α) = R.
If α ∈ R, we have α g(α) = R by the definition of R.
Conversely, given α R, we have α ∈ g(α) = R.
By this contradiction, we get |A| |P(A)|, which presents the desired
equation |A| ⪇ |P(A)|. □

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)i∈I be a family of sets with the index set I.
Then there exists a set not isomorphic to any of the set Aj for an
index j ∈ I.
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 that O and M
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 universe U such that X ∈ U holds for a set X, where
a (Grothendieck) universe is a set U with the following properties:
N = {0, 1, 2, · · · } ∈ U.
∀
x, y, x ∈ y, y ∈ U ⇒ x ∈ U.
I ∈ U, f : I → U :a map ⇒
∪
i∈I f(i) ∈ U.
x ∈ U ⇒ P(x) ∈ U, where P(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 take C, an unique pair of sets
defined by
Ob(C) = ∅
Mor(C) = ∅,
and this is called the empty 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
77 GG 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.
··
33
id
ÒÒ
·
cc
GGid
55
· · · 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, then C, a pair of sets, defined by
Ob(C) = X
Mor(C) = {1x; x ∈ X}
is a category, and it is called a descrete category. In fact,
C(x, x) = {1x}
C(x, y) = ∅ (x y)
hold for an element x ∈ X.
· · · · · · x
id
ÔÔ
y
id
ÔÔ
z
id
ÔÔ
· · · · · ·

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, then C, a pair of sets, is to be a category if
defined as follows:
Ob(C) := (the underlying set of M)
Mor(C) :=
{
∗
id∗
→ ∗
}
◦C := (the operator in M) (◦C is a map on category C).
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)
Let M with single object be a subcategory of Cat. Then, we have
Mon M.
∗
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 ◦X is composition in a category X.
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
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Part4
Conclusion
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Definition
functor
Definition
the composite
Definition
the identity 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 or equivalence isomorphic or equivalent

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
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Definition
isomorphic or equivalent
Definition
essential image isomorphism-dense or essentially 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
An equivalence between categories A and B consists of a pair
A
F
⇄
G
B of functors together with natural isomorphisms
η : 1A → G ◦ F,
ε : F ◦ G → 1B .
If there exists an equivalence between A and B, we say that A and B
are equivalent, and write A B. We also say that the functors F and
G are equivalences.
Definition
Let A be a category. A subcategory S of A consists of a sub ob(S)
of ob(A) together with, for each S, S′
∈ ob(S), a subclass S(S, S′
)
of A(S, S′
), such that S is closed under composition and identities. It
is a full subcategory if S(S, S′
) = A(S, S′
) for all S, S′
∈ ob(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 → D be a full and faithful functor. Then C is equivalent to
the full subcategory C′
of D whose objects are those of the form
F(C) for some C ∈ C.
Proposition (Prop 1.3.18 in ”Basic Category Theory”, T. Leinster)
essentially surjective on objects
Theorem
Let F : C → D be a functor. Then, the following propositions are
equivalent:
1 F is an equivalence.
2 F is full, faithful and essentially surjective.
3 F is a part of some adjoint equivalence (F, G, η, ε).

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 → D be a functor. Then, the following propositions are
equivalent:
1 F is an equivalence.
2 F is full, faithful and essentially surjective.
Proof.
By the previous lemma, we only have to take an equivalence
M
F
→ Mon; i.e.
∃
G : D → C : functor, F ◦ G IdC ∧ G ◦ F IdD
□

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 monoid M ∈ Mon, then C, a pair of sets, is a category
when defined as follows:
Ob(M) := {∗}
Mor(M) := M
◦M := ◦Mon.
(←) Let S be a category of M, then S is a monoid when defined as
follows:
Ob(M) := Ob(C)
Mor(M) := EndC(∗) = 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 relation ≤ on a set P is called a pseudo-order or a preorder if it is
reflexive and transitive; i.e. for all a, b, c ∈ P, we have that:
(reflexivity) a ≤ a
(transitivity) a ≤ b, b ≤ c ⇒ a ≤ c.
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 a ≤ b)
C(a, b) :=
{
(b, a)
}
(otherwise)
(c3, c2) ◦C (c2, c1) := (c3, c1) (∀
c1, ∀
c2, ∀
c3 ∈ P) (, where ◦C in C)
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) := {the underlying set of P}
C(a, b) :=
{
(b, a)
}
(if a ≤ b)
C(a, b) :=
{
(b, a)
}
(otherwise)
(c3, c2) ◦C (c2, c1) := (c3, c1) (∀
c1, ∀
c2, ∀
c3 ∈ P) (, where ◦C in C)
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 n ≥ 0, a countable finite totally orderd set
Sn := ({0, 1, · · · , n − 1}, ≤P) is a category.
1 GG 2 GG 3 GG · · ·
→ We already had nearly the same chain as follows:
1 ≤ 2 ≤ 3 ≤ · · ·
→ The most upside is the same as 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:
∀x∀y[∀z(z ∈ x ⇔ z ∈ y) ⇒ x = y].
(Axiom of regularity (also called the Axiom of foundation))
Every non-empty set x contains a member y such that x and y
are disjoint sets:
∀x (x ∅ → ∃y ∈ x (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 the axiom
schema of separation or of 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:
∀x∃y∀z[z ⊆ x ⇒ z ∈ y].
(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 universe U is fixed.
Definition
An element of U is called a small set.
This expression ”small” does not refer to how small its cardinality is.
Proposition
{U} is a finite set with some single element, however {U} U holds.
Proof.
First we have U ∈ {U} by definition. Assuming {U} ∈ U holds, by
using the definition of universes, we have U ∈ U in 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, b ∈ U, a category C defined by
Ob(C) := U
C(a, b) := {maps from a to b}
ordinary composite of maps
is called a category of (small) sets, and is denoted by Set.
→ We want to consider a category of entire sets, however we have
difficulty using that category because that is not a set. Therefore, we
compose a category of small sets, which is a really set.
Definition
A structured set with a small underlying set is called a small
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
Grp is a category, where its objects are all small groups {Gi}i∈I for an
index set I, its arrow is a group homomorphism of GS for a set S, and
its composition is the operator in GS. Grp is called a category of
(small) groups.
Example
Mon is a category, where its objects are all small monoids {Mi}i∈I for
an index set I, its arrow is a monoid homomorphism of MS for a set
S, and its composition is the operator in MS. Mon is called a
category of (small) monoids.
Example
Ab is a category, where its objects are all small Abelian groups {Ai}i∈I
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
Ring is a category, where its objects are all small rings {Ri}i∈I for an
index set I, its arrow is a ring homomorphism of RS for a set S, and
its composition is the operators in RS. Ring is called a category of
(small) rings.
Example
CRing is a category, where its objects are all small commutative
{Ri}i∈I for an index set I, its arrow is a ring homomorphism restericted
to RS for 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
RMod is a category, where its objects are all small left R-modules, its
arrows are all linear maps. R−Mod is called a category of (small)
left R-modules.
Example
ModR is a category, where its objects are all small right R-modules,
its arrows are all linear maps. R−Mod is called a category of (small)
right R-modules.
Example
Ord is 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 a category 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
Top is 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 a category of (small)
topological spaces.
Example
Toph is 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∗ is a category, where its objects are topological spaces with
selected base point, its arrows are all base point-preserving maps.
Top∗ is called a category 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
Cr
Mfd is a category, where its objects are all small Cr
-manifolds, its
arrows are all Cr
-maps. Cr
Mfd is called a category of (small)
Cr
-manifolds.
Example
Sch is 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 a category 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
MatrK for a fixed field K is a category, where its objects are all
positive integers m, n, · · · , and its arrow is a m × n 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
VctK 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
Euclid is 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. Ses−A 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
Set∗ is a category, where its objects are all small sets each with a
selected base-point, its arrows are all base-point preserving maps.
Set∗ is called a category of (small) base points.
Example
Smgrp is a category, where its objects are all small semigroups, its
arrows are all semigroup morphisms. Smgrp is called a category 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
Met is 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 a category 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 category C is small if it is small as a set; i.e. O and M are 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 category C is called an U−category if
C(a, b) ∈ U
holds for objects a, b ∈ Ob(C).
Example (U−categories)
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
Top of topological spaces and homeomorphisms.
VectK of vector spaces over a field K and homomorphisms.
Mon of monoids and hoomorphisms restricted to them.
more other examples...
we make sure to verify M Mon (M is a subcategory with
single object, of a category)