S minaire 2020 IMD (4) .pdf

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Metric spaces, generalized logic, and closed
categories
Charly Finette

January 2020

Charly Finette

Seminar

1 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Table of contents

1

A brief introduction to the theory of category

2

Philosophical significance of category theory

3

Starting point and goal of this article

4

V-valued categories

5

Conclusion

Charly Finette

Seminar

2 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Introduction
An historical introduction to category theory

Created in the 40’s by Samuel Eilenberg and Saunders Mac
Lane for their work in algebraic topology.
Their goal was to classify topological spaces.
They transformed a topological problem (" difficult ") into
an algebraic one (" easy ").
In order to formalize this " transformation " they needed
the notion of functor and thus, the notion of category.

Charly Finette

Seminar

3 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Introduction
An historical introduction to category theory

Created in the 40’s by Samuel Eilenberg and Saunders Mac
Lane for their work in algebraic topology.
Their goal was to classify topological spaces.
They transformed a topological problem (" difficult ") into
an algebraic one (" easy ").
In order to formalize this " transformation " they needed
the notion of functor and thus, the notion of category.

Charly Finette

Seminar

3 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Introduction
An historical introduction to category theory

Created in the 40’s by Samuel Eilenberg and Saunders Mac
Lane for their work in algebraic topology.
Their goal was to classify topological spaces.
They transformed a topological problem (" difficult ") into
an algebraic one (" easy ").
In order to formalize this " transformation " they needed
the notion of functor and thus, the notion of category.

Charly Finette

Seminar

3 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Introduction
An historical introduction to category theory

Created in the 40’s by Samuel Eilenberg and Saunders Mac
Lane for their work in algebraic topology.
Their goal was to classify topological spaces.
They transformed a topological problem (" difficult ") into
an algebraic one (" easy ").
In order to formalize this " transformation " they needed
the notion of functor and thus, the notion of category.

Charly Finette

Seminar

3 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Introduction
A brief introduction to category theory

A category X is :
a class of elements called objects and denoted Ob(X)
for all a, b ∈ Ob(X), a set of elements called morphisms,
and denoted Hom X (a, b)
for all a, b, c ∈ Ob(X), a composition law
o : Hom X (a, b) × Hom X (b, c) → Hom X (a, c)
which is associative.
for all a ∈ Ob(X), an identity morphism 1a ∈ Hom X (a, a),
which is neutral for the composition law.
Charly Finette

Seminar

4 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Introduction
A brief introduction to category theory

Category

Objects

Morphisms

Composition law

Grp

groups

group
homomorphisms

usual composition
of functions

Top

topological spaces

continuous
functions

usual composition
of functions

Ord

partially ordered
sets

order-preserving
functions

usual composition
of functions

Table – Examples of “classical categories”

Charly Finette

Seminar

5 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Introduction
A brief introduction to category theory

A functor F from a category X to a category Y is :
For all a ∈ Ob(X), an object F a ∈ Ob(Y ).
For all a, b ∈ Ob(X) a map
F : Hom X (a, b) → Hom Y (F a, F b)
such that
For all a, b, c ∈ Ob(X), f ∈ Hom X (a, b) and
g ∈ Hom X (b, c),
F (g o f ) = F (g) o F (f )
For all a ∈ Ob(X), F (1a ) = 1F a
Charly Finette

Seminar

6 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Category theory : a third level of abstraction ?

1

objects

all mathematical structures of a given kind

2

morphisms

morphisms preserving this structure

3

functors

links between mathematical structures of another kind
Table – Form of “classical categories”

Charly Finette

Seminar

7 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

The category of an arbitrary group
A group is a category in which there is just one object and
in which every morphism is an isomorphism.
Each element of a group is represented by a morphism.
The fact that each morphism is an isomorphism stands
from the fact that each element has an inverse element.
0

objects

???

1

morphisms

elements of the group

2

functors

if restricted to groups : homomorphisms

Table – Form of “special categories”

Charly Finette

Seminar

8 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Implications of such a “special category”
functors from a group to a “classical category” are
important structure
for example, a functor from a group to the category of
vectorial spaces is a linear group representation
Category theory allows us to link “things” with different
levels of abstraction
“But the theory of categories actually penetrates much
more deeply than that attempted characterization [a third
level of abstraction] would suggest toward summing up the
essence of mathematics”

Charly Finette

Seminar

9 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Thesis : “ Every fundamental structure is in itself a
category”
A partially ordered set(Poset) X is a category in which for any
pair of objects a, b ∈ Ob(X), |Hom X (a, b)| ≤ 1.
|Hom X (a, b)| = 1 means a ≤ b.
What does this thesis says about structures we can’t represent
as a category ?
We cannot represent a metric space as a category (at least
not in a simple natural way).
isn’t a metric space a fundamental structure ?
is category theory “incomplete” ?

Charly Finette

Seminar

10 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Goals
During a conference, the author notices the following analogy
between the law of composition in a category X and the triangle
inequality.
Hom X (a, b) × Hom X (b, c) → Hom X (a, c)
d(a, b) ≤ d(b, c) + d(a, c)
→ We will define a natural generalization of category theory
within itself
→ Then we will show you that it is possible to regard a metric
space as a (generalized) category
→ Then we will talk about the consequences
Charly Finette

Seminar

11 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Closed categories
We are going to define what is a strong category valued in a
closed category. What is a closed category ?
Basically, closed categories are closed with respect to the
operation of forming the hom of two objects
i.e X is a closed category if for all a, b ∈ Ob(X),
Hom X (a, b) “ ∈00 Ob(X)
Example
Let S denote the category of abstract sets.
Ob(S) is the class of all sets.
If X and Y are abstract sets, Hom S (X, Y ) = Y X .

Charly Finette

Seminar

12 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Closed categories

Example
The category R of non-negative real quantities :
Ob(R) = [0, ∞] and if a, b ∈ [0, ∞]

b − a if b ≥ a
Hom R (a, b) =
0
if a ≥ b

Charly Finette

Seminar

(1)

13 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Formal definition
“Closed category” is short for “bicomplete symmetric monoidal
closed category”.
Definition
A monoidal structure in a category V is a given functor
⊗:V ×V →V
which is symmetric, associative and has a unit object k
satisfying
u⊗v ∼
=v⊗u
(u ⊗ v) ⊗ w ∼
= u ⊗ (v ⊗ w)
k⊗v ∼
=v∼
=v⊗k
Charly Finette

Seminar

14 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Examples of a monoidal structure

Example
For the category S of abstract set, the monoidal structure is
given by the cartesian product × and the unit object is any
one-element set.
X ×Y ∼
=Y ×X
(X × Y ) × Z ∼
= X × (Y × Z)
{0} × X ∼
=X∼
= X × {0}

Charly Finette

Seminar

15 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Examples of a monoidal structure

Example
For the category R of non-negative real quantities, the monoidal
structure is given by the sum + and the unit object is 0.
a+b∼
=b+a
(a + b) + c ∼
= a + (b + c)
0+a∼
=a∼
=a+0

Charly Finette

Seminar

16 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Closed monoidal structure
Definition
That a monoidal structure is closed means that we are given a
functor

and two natural transformations

Charly Finette

Seminar

17 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Closed monoidal structure
Definition
such that the two processes

are inverse bijections

Charly Finette

Seminar

18 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Exemples

Example
For our category S, our inverse bijections are
B → CA
A×B →C
The rule of lambda conversion !

Charly Finette

Seminar

19 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Exemples

Example
For our category R, our inverse bijections are
a+u≥v
u≥v−a

Charly Finette

Seminar

20 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

V-valued categories
Definition
Given a closed category V, a strong category X valued in V or
simply a V-valued category is any structure consisting of :
a set of X-objects a, b, c...
for every ordered pair of X-objects a, b, X(a, b) ∈ Ob(V)
for every ordered triple a, b, c of X-objects, the assignement
of a V-morphism

Charly Finette

Seminar

21 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

V-valued categories
Definition
To every X-objects a, a V-morphism

To the condition that the following diagrams always commute

Charly Finette

Seminar

22 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

V-valued categories

Definition

Charly Finette

Seminar

23 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Examples
Example
What is a S-valued category X ?
It is a category whose morphisms take values in the
abstract sets
The laws satisied by X are :
X(a, b) × X(b, c) → X(a, c)
1 → X(a, a)
it’s a regular category !

Charly Finette

Seminar

24 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Examples
Example
What is a R-valued category ?
It is a category whose morphisms take values in R
The laws satisfied by X are :
X(a, b) + X(b, c) ≥ X(a, c)
X(a, a) ≥ 0
it is a (generalized) metric space !
if a, b are X-objects, X(a, b) represents the distance
between a and b.
Charly Finette

Seminar

25 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Generalized metric space
The usual axioms for a metric space are :
X(a, b) = 0 ⇒ a = b
X(a, b) < ∞
X(a, b) = X(b, a)
Metric spaces where X(a, b) 6= X(b, a) are interesting
(Work required to travel from a point a to a point b in
mountainous regions, non symmetric Hausdorff spaces)
You can symmetrize your metric spaces by the following
procedures :
X(a, b) + X(b, a)
M ax(X(a, b), X(b, a))
Charly Finette

Seminar

26 / 27

A brief introduction to the theory of category
Philosophical significance of category theory
Starting point and goal of this article
V-valued categories
Conclusion

Conclusion

Enriched categories can suggest new directions in research
in metric spaces theory, and conversely, which is unusual for
two subjects so old ( 1966 and 1906)
Enriched categories were developped to deal with more
complicated subjects such as functorial semantics and
theory of topoi

Charly Finette

Seminar

27 / 27