Category Theory Modeled with Dependencies Theory

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Keywords: category theory, theory of categories, theory of dependencies, dependencies theory, morphism, morphisms, homomorphism, homomorphisms, isomorphism, isomorphisms, abstract mathematical theory, abstract theory, abstract math theory, abstract mathematics, abstract math, mathematical abstraction, math abstraction, model

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This document is draft.

Introduction to the Idea
Some Operations
Definition of Meta-Categories
Definition of Category
Relation between Categories and Meta-Categories
Postface

Introduction to the Idea

Declining from the way of research of trinity $X$ , $Y$ , $Z$ I have come back into the world of category theory and have modeled categories with ternary relations.

In this article I model categories (from category theory) using parametrized dependencies (systems of constructs) and reindexation operation. (See Theory of Dependencies and Theory of Formulas.)

In this way categories are modeled in such a way that algebraic operations on them are made possible. So I consider this my research as an attempt of algebraization of category theory.

Instead of the traditional (as I misspelled this word with additional letter o after first i) definition of a category (with single domain and range of every morphism), in this work I use modified variant of definition of a category with multiple domains and ranges of every morphism.

Some Operations

We will need the following operations:

Let $U$ is a class (multi-argument relation), $q$ is a property (an argument).

U |_{q = a} = {f | f \in U \land f (q) = a}

{Planes}_{q} U = {U |_{q = a} | a \in {Pr}_{q} U}

For simplicity I will write just $Planes U$ instead of ${Planes}_{Z} U$ .

I call $Planes U$ planes of a systems of constructs (that ternary relation) $U$ .

By definition for three systems of constructs (parametrized dependencies) $U$ , $V$ , and $λ$ $V \circ_{λ} U = {((X, f (X)), (Y, g (Y)), (Z, h (Z))) | f \in U \land g \in U \land h \in λ \land f (Y) = g (X) \land f (Z) = h (X) \land g (Z) = h (Y)} .$

[TODO: Draw the picture (triangle with vertices $U$ , $V$ , $λ$ and three free lines from every vertex). On the picture all three systems $U$ , $V$ , and $λ$ are symmetric. Also simplify the above formula (by introducing new concepts and symbols). Where I have seen something similar, maybe in physics, theory of elementary particles?]

Remark The definition of a dependency may be relaxed (generalized) in such a way that the operation $\circ$ would become the same as above defined $\circ_{λ}$ , by replacing operation comma with arbitrary (possibly partial or multivalued) binary operation $λ$ .

Definition of Meta-Categories

I will call a meta-category (this term is probably not the best one and in a future version of this article I may replace meta-category with some other word) a pair of a system of constructs (that is a parametrized dependency) $U$ and an associative partial binary operation $λ$ on $ind U$ .

Remark A meta-category is a pair of two systems of constructs, that is a pair of two ternary relations.

Planes of $U$ are called morphisms of the meta-category. The operation $\circ_{λ}$ is called composition of morphisms of the meta-category.

[TODO: Are all terms defined below useful?]

I will call a transitive meta-category such a meta-category that $U \circ_{λ} U = U$ .

I will call a weakly reflexive meta-category such a meta-category that ${Pr}_{Z} U \supseteq (=) |_{dom U \cup im U}$ .

Obvious For a weakly reflexive meta-category $dom U = im U = dom U \cup im U .$

I will call a weakly symmetric meta-category such a meta-category that ${Pr}_{Z} U$ is a symmetric relation.

I will call an identity morphism of a meta-category such morphism $e$ that for any any morphism $f$ of this meta-category $f \circ_{λ} e \in {\emptyset, f}$ , $e \circ_{λ} f \in {\emptyset, f}$ .

I will call identity morphism of object $a$ (regarding some meta-category) such identity morphism $1_{a}$ that $(a, a) \in Pr 1_{a}$ .

Proposition There exist no more than one identity morphism of object $a$ .

◄ Suppose that $f$ and $g$ are such identity morphisms. It is enough to prove that $g \circ_{λ} f \neq \emptyset$ as in this case $f = g \circ_{λ} f = g$ . That $g \circ_{λ} f \neq \emptyset$ follows from that $(a, a) \in Pr f$ and $(a, a) \in Pr g$ and that $λ$ is defined on $(a, a)$ . ►

I will call a meta-category with identities such a meta-category that for any $a \in dom U \cup im U$ exists identity morphism $1_{a}$ .

I will call a meta-category with self identities such a meta-category with identities that for any $1_{a} = a$ .

Theorem If $U$ is a meta-category with identities then $U \circ_{λ} U \supseteq U$ .

◄ Let $f$ is a non-empty morphism. Then $dom f$ is not empty. Let $x \in dom f$ . Then $f \circ_{λ} 1_{x} = f$ . Uniting all such $λ$ -compositions of non-empty morphism, we prove that $U \circ_{λ} U \supseteq U$ . ►

[TODO: About reindexations between meta-categories with identities and meta-categories with self identities.]

Obvious A meta-category with identities is weakly reflexive.

Definition of Category

I will define a category without identities (with multiple sources and destinations) as a system of:

a set $Ob$ (elements of which are called objects);
a set $Mor$ (elements of which are called morphisms);
a function $Field$ defined on the set $Mor$ , whose values are non-empty relations on the set $Ob$ .
an associative binary operation λ (called composition of morphisms) on the set Mor such that for any morphisms f and g
- $Field (g) \circ Field (f) = Field λ (f, g)$ . when $λ (f, g)$ is defined.
- $Field (g) \circ Field (f) = \emptyset$ (or equivalently $im Field f \cap dom Field g = \emptyset$ ) when $λ (f, g)$ is undefined.

By definition an identity morphism of a category without identities $C$ (sorry for silly sounding collocation) is such morphism $e$ of $C$ that for any morphism $f$ of $C$ :

$λ (f, e) = f$ if $λ (f, e)$ is defined.
$λ (e, f) = f$ if $λ (e, f)$ is defined.

I will call an identity morphism for an object $a$ for a category without identities $C$ such identify morphism $1_{a}$ of this category that $(a, a) \in Field 1_{a}$ .

Proposition There exist no more than one identity morphism for an object $a$ for a category without identities $C$

◄ Suppose that $f$ and $g$ are two identity morphisms for an object $a$ . Because $(a, a) \in Field f$ and $(a, a) \in Field g$ then $im Field f \cap dom Field g \neq \emptyset$ and so $λ (f, g)$ is defined. So because $f$ is an identity element $λ (f, g) = f$ , analogously $λ (f, g) = g$ ; so $f = g$ . ►

By definition a weak identity morphism morphism of a category $C$ for object $a$ is such morphism $e$ of $C$ that $(a, a) \in Field e$ and for any morphism $f$ of $C$

$a \in dom Field f \Rightarrow λ (e, f) = f$ ;
$a \in im Field f \Rightarrow λ (f, e) = f$ .

Proposition An identity morphism for an object $a$ is a weak identity morphism for an object $a$ .

◄ Let $a \in dom Field f$ . $λ (f, 1_{a})$ is defined because otherwise $Field f \circ Field 1_{a} = \emptyset$ , what is impossible because $a \in im Field 1_{a}$ . So $λ (f, 1_{a}) = f$ ; analogously $λ (1_{a}, f) = f$ . ►

By definition a category is a category without identities every object of which has an identity morphism.

Relation between Categories and Meta-Categories

Having a category (not necessarily with identities) we will define its corresponding system of constructs (ternary relation) $U$ as: $U (a, b, i)$ if and only if $i$ is a morphism from $a$ to $b$ . So to every category corresponds a meta-category with $λ$ being the composition of morphisms of the category. (I have taken in account that fields of category morphisms are non-empty, and so $ind U$ is all $Mor$ so that $λ$ is defined on $ind U$ .)

This way to every non-zero morphism $i$ of the category bijectively corresponds index value $i$ and the corresponding non-empty plane $U |_{Z = i}$ in the system of constructs of the meta-category.

Obvious $Field i = U_{Z} (i)$ for any morphism $i$ of the category.

Proposition Composition of morphisms of the meta-category which corresponds to a category is related by the above bijection with composition of morphisms of the category. (Composition of morphisms of the corresponding meta-category is empty if and only if the composition of morphisms of the category to which it corresponds is undefined).

◄ We need to prove that $U |_{Z = λ (f, g)} = (U |_{Z = g}) \circ_{λ} (U |_{Z = f})$ . It easily follows from $Field λ (f, g) = Field (g) \circ Field (f)$ . ►

Theorem

A meta-category corresponds to some category without identities if and only if $U_{Z} \circ λ = U \circ U$ where $λ$ is understood as a binary relation (whose first argument is a pair of indices). Note that the operation at the left part of this formula is called reindexation.

◄

U_{Z} \circ λ = U \circ U \Leftrightarrow \forall i, j : (U_{Z} \circ λ) (i, j) = {(U \circ U)}_{Z} (i, j) \Leftrightarrow \forall i, j : U_{Z} (λ (i, j)) = U_{Z} (i) \circ U_{Z} (j) .

Direct implication

We have $Field i = U_{Z} (i)$ . So from a formula in the definition of category $U_{Z} (λ (i, j)) = U_{Z} (i) \circ U_{Z} (j) .$

Reverse implication

Let $Ob = dom U \cup im U$ ; let $Mor = ind U$ .

Let $Field a = U_{Z} (a)$ .

$λ$ is an associative partial binary operation; let $λ$ is the composition of morphisms.

When $λ (f, g)$ is defined, $Field λ (f, g) = U_{Z} (λ (f, g)) = U_{Z} (f) \circ U_{Z} (g) = Field (g) \circ Field (f)$ .
When $λ (f, g)$ is undefined, $Field (g) \circ Field (f) = U_{Z} (i) \circ U_{Z} (j) = U_{Z} (λ (i, j)) = \emptyset .$

That this system forms a category without identities follows from the definition of category without identities.

That our meta-category corresponds to this system is obvious.

►

[TODO: Additionally it seems that meta-categories corresponding to categories are transitive or at least $U \circ_{λ} U \subseteq U$ (the semantics is that composition of two morphisms is also a morphism). How it relates with the above formula?]

Theorem A meta-category corresponding to a category without identities is a meta-category with identities if and only if the category without identities is a category (with identities).

◄

It follows from the previous big theorem, the theorem about bijective relation of composition of morphisms in a category and the composition of morphisms in the corresponding meta-category, and the definition of identity morphisms for categories and for meta-categories.

►

To every category (with multiple domains and ranges) corresponds exactly one equivalent transitive, weakly reflexive, and meta-category with identities; and vice verse.

Postface

So I have defined categories in the terms of two dependencies with parameter (two systems of constructs, two ternary relations).

It is only a just started research, and we should yet to see how dependencies theory may be used in category theory research.

However this my work has shown that the theory of dependencies may be considered as a more general theory than category theory, and category theory as a particular case of dependencies theory. Dependencies theory is more algebraic than category theory, what may simplify category theory research.

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