MENTAL vs. CATEGORY THEORY

"Category theory is the most general and abstract branch of pure mathematics" (C.A.R. Hoare).

"The golden rule of modern mathematics is that life takes place within −and between− categories." (John Baez)

"Category theory can bring a better and more natural understanding of mathematical objects than set theory" (Tatsuya Hagino).

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Category Theory
The central idea of category theory (CT) is that, while in a set there are vertical relationships of membership of each element in the set, in a category there are horizontal relationships between the elements of the set. These relationships form a structure. CT is a general mathematical theory that deals with structures, systems of structures and how structures of different domains are related.

CT was born in algebraic topology because of the need for a formalism to describe the transformation of one type of structure into another and to assign algebraic invariants to topological structures.

The formal notion of category was introduced in 1945 with the publication of the paper "General Theory of Natural Equivalences", by Samuel Eilenberg and Saunders Mac Lane. It dealt with classes of natural transformations in algebraic topology. They formalized the concept of homology (an intuitive geometric concept) and created a homological algebra or homology theory (of axiomatic type) [Eilenberg & Mac Lane, 1945].

Eilenberg and Mac Lane found that all structures shared a number of common features:

• They refer to classes of sets.
• There are defined internal functions, which relate some elements to others.
• Functions can be composed associatively.
• There is always at least one identity function.

From all this they deduced that there was no need to mention sets explicitly and that mathematics could be based solely on the concept of function and composition of functions, instead of the classical concepts of set and membership. In this way a more generic framework was created in which sets and structures would be particular cases of categories.


Formal definition of category

A category C (which is usually represented in bold) consists of two parts:

1. A class or collection of objects. Objects are usually represented by capital letters.
   
   
2. A binary relationships, called "morphisms" or "arrows", between the objects in the class. They are usually represented by lowercase letters.

It is usually common to represent the objects of a category as points and the morphisms as arrows connecting the points (hence the name "arrows" given to the morphisms).

For any pair of objects, A and B (the same or different), of the class, there exists a set of morphisms (correspondences) Mor(A, B) (it can be the empty set), such that if f belongs to Mor(A, B), then f establishes a correspondence between object A (origin or source) and object B (destination). It is represented as f:A→B.

The composition of the morphisms f:A→B and g: B→C is another morphism g○f:A→C. Therefore, if f belongs to Mor(A, B) and g belongs to Mor (B, C), then g○f belongs to Mor(A, C).


The notion of composition of morphisms allows us to generalize fundamental concepts of mathematics, such as product, addition and exponentiation, as well as the basic mechanisms of logic.

A category C satisfies the following two axioms:

1. Associative composition of morphisms: h○(g○f) = (h○g)○f
   
   
2. Morphism identity. For every object X of category C, there exists an identity morphism that makes it correspond to itself, represented as id_x(X) o bien 1_x. Therefore, if we have the morphism f: A→B, then 1_B○f = f○1_A = f.

An object A of a category C can be considered as a morphism identity 1_A. Therefore the morphism f:A→B can be represented as composition of the morphism A and the morphism f: f○A = B.

The idea is to approach everything in terms of morphisms and their composition, not to mention objects. A category is thus characterized by its morphisms and not by its objects.


Domain and codomain of a morphism

• The domain of a morphism f of a category C is the set of source objects or origin of the morphism f.
  
  
• The codomain of a morphism f of a category C is the set of target objects of the morphism f.

There can be morphisms with the same name, the same source object and different target objects. Sometimes the functional notation f(A)=B is used, but it is only applicable when there is only one arrow from A to B.



Types of morphisms

• The inverse morphism of f:A→B is f⁻¹:B→A.
  
  
• A morphism f is an isomorphism if combining the direct morphisms (f: A→B) and inverse (f⁻¹:B→A), the properties: f○f⁻¹ = 1_B y f⁻¹○f = 1_A. The arrow is called "isomorphic" or "invertible".
  
  
• A morphism f is a monomorphism if it satisfies: if f○g = f○h, then g=h. The arrow is called "monomorphic" or "monic".
  
  
• A morphism f is an epimorphism if it satisfies: if g○f = h○f, then g=h. The arrow is called "epimorphic" or "epic".
  
  
• A morphism f is an endomorphism if the domain and codomain of f are the same.

Examples of categories

1. Set.
   Objects: All sets.
   Arrows: Functions between sets. The category Set is the paradigmatic example of category.
   
   
2. Grp.
   Objects: All groups.
   Arrows: Homomorphisms between groups.
   
   
3. Vec.
   Objects: All vector spaces.
   Arrows: Linear transformations.
   
   
4. Top.
   Objects: All topological spaces.
   Arrows: Continuous functions.
   
   
5. Pos. Sets with a partial order (poset, partial order set). A partial order relation on a set C is a relation such that for any pair of elements a and b of C it is satisfied that a≤b or b≤a (x≤y indicates "x precedes y").
   
   Objects: The elements of a set with a partial order.
   Arrows: The relationships ≤ between elements. The composition of x≤y and y≤z is x≤z.
   
   Examples: the real numbers and the relation "less than or equal to"; the set of subsets of a given set C, P(C), and the relation &rise; (contained or equal).
   
   
6. The sets with an equality relation between their elements. Objects: The elements of the set.
   Arrows: The equality relationships (=) between the elements. All elements are equal to themselves and pairs of equal elements can additionally be defined. An arrow is an element X=Y.

Special objects of a category

• Objeto inicial. An object I of a category C is initial if for each object X of C there is only one morphism I→X.
  
  
  
• Objeto terminal. An object T of a category C is terminal if for each object X of C there is only one morphism X→T.

Subobjects

The concept of subobject is a generalization of the notion of subset in set theory.

The formal definition of subobject, and the relations between subobjects, is grounded in morphisms, rather than referring to objects:

• A subobject S of an object A is a monomorphism from S to A.
  
  
  
• If A is an object of a category, and we have two morphisms u:S→A and v: T→A, then u≤v is said to exist if w: S→T such that u = v○w. And u≡v if and only if u≤v and v≤u.

In the category Sets, a subobject of an object (set) A is a subset of A. In the category Grp, a subobject is a subset.

This definition of subobject makes the collection of subobjects of an object A a partial order. This binary relationship between each object and its subobjects is an equivalence relationship. The equivalence classes are the subobjects of A. The dual concept of subobject is the quotient object.

An equivalence relation on a set is a relation between the elements of the set that share the same property. This allows the elements of the set to be classified into equivalence classes. That set of equivalence classes is the quotient set.

Subobject classifier

A subobject classifier is special object Ω of a category. It is a generalization of the truth value set {0, 1} of Boolean logic. Intuitively, the subobjects of an object X correspond to the morphisms of X to Ω. For example, in the category Sets, if Ω={0,1}, each element of a subset S of X corresponds to 1 if it belongs to X and 0 if it does not belong. This is what is called the "characteristic function" of S in X.


Product of two objects of a category

The product of objects is a notion that attempts to generalize many operations of this type, such as Cartesian product of sets, direct product of groups, direct product of rings, product of topological spaces, and so on.

The product of two objects A₁ and A₂ of category C is another object P = A₁×A₂, so that two morphisms p₁: P→A₁ y p₂: P→A₂, for every object X, and for every pair of morphisms f₁: X→A₁ y f₂: X→A₂, there is only one morphism f:X→P that satisfies.

p₁○f = f₁   p₂○f = f₂

That is, f is uniquely determined by f₁ and f ₂ and is represented as f = ⟨f₁, f₂⟩. Therefore,

p₁○⟨f₁,f₂⟩ = f₁   p₂○⟨f₁,f₂⟩ = f₂

The morphisms f₁ and f₂ are the projections of f with respect to p₁ and p₂.

This is usually represented like this:

X→A1×A2

X→A1,   X→A2

meaning: X→A₁×A₂ naturally corresponds to morphism pairs X→A₁ y X→A₂.

The product of objects satisfies the commutative and associative properties.

The identity element is object 1: 1×A = A. It is shown that if A×A = 1, then A = 1.

In the case of the category Set, the product of objects (sets) corresponds to the Cartesian product.


Sum of two objects of a category

The sum −also called coproduct or dual product− of two objects A₁ and A₂ of category C is another object S = A₁+A₂, such that two morphisms j₁: A₁→S y j₂: A₂→S such that for every object X and for every pair of morphisms f₁: A₁→X y f₂: A₂→X there is only one morphism f:S→X that satisfies.

f○j₁ = f₁   f○j₂ = f₂

Note that the direction of the arrows is the opposite of that of the product. That is, f is uniquely determined by f₁ and f₂ and is represented as.

f=	{	f1
		f2

The morphisms j₁ and j₂ are the sum injections of A₁ and A₂.

This is usually represented like this:

A1+A2→X

A1→X,   A2→X

meaning: A₁+A₂→X naturally corresponds to morphism pairs A₁→X y A₂→X.

The sum of objects satisfies the commutative and associative properties.

The identity element is the object 0: 0+A = A. A negative object of A is another B, such that A+B = 0.

In the case of the category Set, the sum of objects corresponds to disjoint union (union of sets without common elements).


Exponentiation

If C is a category and if Y and Z are objects of C, Y^Z is another −also called "power object" (power object)− which is the set of all morphisms of Z in Y.

The laws of exponentiation are:

1^Z ≡ 1
Y¹ ≡ Y
Y⁰ ≡ 1
(Y^Z)^S ≡ Y^Z+S
(Y₁+Y₂)^Z ≡ Y₁^Z +Y₂^Z
Y^Z₁+Z₂ ≡ Y^Z₁×Y^Z₂

The laws for exponentials are generic. Their application to arithmetic is only a special case.

Exponentiation is usually represented like this:

X→YZ

Z×X→Y

meaning: the morphism X→Y^Z naturally corresponds to the morphism Z×X→Y.

In the category Set, the exponential object Y^Z is the set of functions between Z and Y.


Category types

• A discrete category is a category in which all morphisms are identities. That is, there are no connections between the objects in the category.
  
  
• A connected category is a category in which for every pair of objects there is at least one morphism between them.
  
  
• A subcategory is a selection of objects in a category together with all morphisms between them.
  
  
• The dual (or opposite) category C' of a category C is the same category C but with the arrows in the opposite direction (i.e., with the reverse morphisms).
  
  
• A distributive category is the one that satisfies, for any three objects, A₁, A₂ and A₃:
  
  
  A₁×A₂ + A₁×A₃ ≡ A₁×(A₂+A₃)
  
  
• A linear category is the one which, for any pair of objects A and B, A×B is isomorphic to A+B.
  
  
• Quotient category. If we have an object X of a category C, C/X is another category, called "quotient category", in which each object is a morphism of C with codomain X, i.e., it is the set of arrows of C that have destination X.
  
  
• Closed Cartesian category. It is a category that satisfies 3 properties:
  
  
  1. It has a terminal object.
     
     
  2. Any two objects X and Y of C, have a product X×Y in C.
     
     
  3. Any two objects Y and Z of C, have an exponent Z ^Y (the set of morphisms from Y to Z) in C.
  
  In such a category, it is satisfied that f:X×Y→Z is equivalent to λg: X→Z^Y (morphisms of Z to Y), i.e., a function of two variables can be defined as a function of one variable. This is called "currification".
  
  An example of this category is Set, the category of all sets with functions between them as mofisms. In this case, X×Y is the Cartesian product and Z^Y is the set of all functions between Y and Z.

Diagrams

A diagram is a graphical representation of the arrow structure of a category, where objects are vertices. A path in a diagram is a finite succession or chain of arrows. The basic diagrams are the triangle and the square:



The triangle is said to commute if the following is true: g○f = h.
The square is said to commute if: g○f = k○h.

In general, a diagram commutes if any two paths with the same origin and the same destination, and of length > 1, are equal, i.e., the arrow obtained by composition of the arrows of any connected path depends only on the ends of the path.


Limits and collimits of a category

A category C with limits or boundaries is one that has the following properties:

Property	Boundary	Limits
Special object	Initial object: 0	Terminal object: 1
Operation	Sum of objects	Product of objects
Null operation	0+A = A	1×A = A
Quotient set C/X (for every object X)	C/X has sums	C/X has products

Functor

A functor is a morphism between categories that preserves structure when passing from one category to another. The functors are usually represented with capital letters to differentiate it from the internal morphisms in a category. A functor associates to each object of a category an object of the other category and to each arrow of the first category an arrow of the second category. Formally, a functor F from category C to category D:

1. Associates to each object X of category C a single object F(X) = Y of category D.
   
   
2. Associates each arrow f of category C a single arrow F(f) = g of category D. It is said to be a 2-morphism (morphism of order 2). Normal morphisms of a category are 1-morphisms.

In addition, the following properties are fulfilled, which preserve the structure when passing from one category to another:

1. Si f:X→Y, entonces F(f):F(X)→F(Y).
   
   
2. For all morphisms of C, it is satisfied that F (g○f) = F(g)○F(f). The functor preserves the composition.
   
   
3. For every object X of C, it is satisfied that F(1_x) = 1_F(x). The functor preserves the identities.

The functor so defined is said to be covariant. A contravariant functor is a covariant functor between C' (the dual category) and D.

Example functor: in the category Set, the transformation P(f): P(X)→ P(Y), where P(X) is the set of subsets of X, is a functor.


Identity functor

The identity functor of a category C is a functor I_C: C → C, such that I_C(X)=X for every object X of C.


Reverse functor

If F: C→D is a functor , F is said to be an isomorphism if there exists another functor G: D→C such that F○G = I_C y G○F = I_D. The functor G is called the inverse functor of F.


Natural transformation between functors

While functors are morphisms that allow one to move from one category to another, natural transformations provide an analogous relationship between functors: morphisms between categories of functors. A natural transformation T between two functors F and G, between the categories C and D, is the one that meets the conditions reflected in the diagram. That is;

G(f)○T_X = T_Y○F(f)

Attached functors

There are several definitions of adjoint functors [Mac Lane, 1998], but the most widely used is the one using a natural isomorphism of sets of morphisms, which is as follows. Given two categories, C and D, and two functors F: C→D y G: D→C, F is attached to the left of G (and G is a functor attached to the right of F) if, for each object X of C and for each object Y of D, there are two unique morphisms, f_C y f_D tales que f_C: F(X)→Y y f_C:X→G(Y). That is, there is a natural transformation between the objects X and Y, as illustrated in the figure.


In particular it is verified that G○F=I_C y F○G=I_D. And, therefore, G○F○G=G and F○ G○F=F, siendo I_C and I_D the identity functors of the &1 categories; strong>C y D, respectivamente. That is, two adjoint functors are inverse functors when the morphisms that apply to X and to Y are 1_X and 1_Y, respectively.


"Small" categories
If the class of objects in a category is a set, the category is said to be "small".

Examples

1. A set C determined partially ordered.
   
   Objects: The elements of the set C.
   Arrows: A partial order relation.
   
   
2. A given graph.
   
   Objects: The vertices of the graph.
   Arrows: The pairs of connected vertices (the edges between vertices).
   
   
3. Monoid. A category that has only one object.
   
   Objects: Only one (X).
   Arrows: You can have many of the same arrows (from X to X) but with different names.

Product of small categories

Product of two small categories, C and D, is another category:

Objects: Cartesian product elements C×D.

Arrows: The elements of the Cartesian product F×G (F is the set of morphisms of C, and G is the set of the morphisms of D).

Higher Order Categories
There are many types. The most prominent are as follows:

n-categorie

A n-category consists of:

• 0-objects: A, B, ...
• 1-objects: morphisms (arrows) between objects.
• 2-objects: morphisms between morphisms (2-morphisms, morphisms of order 2, arrows between arrows).
• 3-objects: morphisms between morphisms of order 2.
• etc., up to n-objects.

There are also the ∞-category (or ω-category) in which the process continues indefinitely.


Multicategory

It is a category in which the arrows are of type

(A₁, A₂, ... , A_k) → B

where the domain is a sequence of objects and the codomain is a single object. Such an arrow can be imagined as a box with k inputs and one output:


An example of a multicategory is vector spaces.

Multicategories can be generalized to other types of objects, such as tree of objects, array of objects, and so on.

An operand is a multicategory with a single object.

Multicategories are to operands as categories are to monoids.


Categories of categories

They are categories whose objects are categories and whose arrows are functors. For example, the category formed by:

Objects: All small categories.
Arrows: All the functors between them.

Topos Theory
The topos theory −Greek, "place"; in poetry it means "common place"− is a branch of CT. The word "topos" is singular, the plural being "topoi" or "toposes".

The concept of topos was elaborated by Alexander Grothendieck, around 1960, as a generalization of the concept of sheaf of algebraic geometry. William Lawvere, together with Myles Tierney [Lawvere, 1963] generalized this concept by means of elementary (first order) axioms leading to its present notion: the elementary topos, which is a generalization of set theory, an elementary set category theory, a generalization of the notion of topological space, and a generalization of Boolean logic. In short, an autonomous universe for mathematical discourse.

For Lawvere, a topos can be considered as a category of variable sets over a general topological space. And the usual category of sets as composed of fixed (or constant) sets.

A topos (or elementary topos) is a category with:

1. Limits and colimits.
   
   
2. Products. For every pair of objects (X, Y) of the category, there exists the object product X×Y.
   
   
3. Exponentials. For every pair of objects (X, Y) of the category, there is an object Y ^X (exponential) representing all morphisms from X to Y.
   
   
4. A subobject classifier Ω. For any object X, the X→Ω morphisms are equivalent to the subobjects of X.

A topos is a closed Cartesian category with a subobject classifier.

An example of a topos is the category Set of finite sets and functions between them and where the subobject classifier is the set {V, F}.


Categorical Logic
Categorical logic is a branch of CT, which is based on a categorical approach to logic.

The origin of categorial logic was William Lawvere's paper "Functorial Semantics of Algebraic Theories" [1963]. Lawvere rethought logic (syntax and semantics) from the categorial point of view. He conceived an algebraic logical theory as a special kind of category, which is known as "functorial semantics", which formalizes semantics by means of functors. Syntax and semantics are represented by a category. An interpretation of a theory is realized by a functor between the two (the theory and its interpretation).

The notion of elementary topos provides a unified treatment of the syntax and semantics of higher-order predicate logic. Lawvere wanted to construct a higher-order predicate logic in terms of CT. The result (categorical logic) was an intuitionistic type of logic. Lawvere thought that since a topos possesses a classifier of subobjects , which can be truth values {V, F}, then the logic could be built on a topos. But Lawvere considered that the logic might not be Boolean, and constructed it by means of a Heyting algebra. Heyting algebras are a generalization of Boolean algebra that models intuitionistic propositional logic. The internal logic of an elementary topos is based on the Heyting algebra of subobjects of the terminal object 1, ordered by inclusion, which are equivalent to the morphisms of 1 to the subobject classifier Ω.

Algebraic systems in logic were not new. They had already been established with Boole and Tarski. It was an algebrist, Joachim Lambek, who first discovered analogies between CT and logic, namely between the axioms of the categories and deductive systems such as Gentzen's natural deduction. Lawvere introduced algebraic theories in place of equational (or substitution) theories. Existential and universal quantification are formalized as adjunct functors −left and right, respectively− of the logical operation of substitution. If A and B are two sets, the quantifiers are defined as morphisms between P(A) and P(B), the respective power sets.


Category Theory Valuation
A generalization process

CT is the culmination of a generalization process by substituting the constant for the variable:

• Elementary algebra is a generalization of arithmetic and results from the substitution of constants (numbers) for variables, keeping the operations fixed.
  
  
• Abstract algebra allows operations to vary to form fixed and determinate mathematical structures (group, ring, etc.).
  
  
• CT allows even the form of structures to vary, to give rise to a general theory of structures and forms.

In CT even the semantics varies, depending on the framework under consideration. Indeed:

• The traditional conception of mathematics is based on the fixed universe of sets and, consequently, with a fixed and determined semantics. For example, the concept of group in set theory is a set equipped with an internal operation satisfying certain axioms expressed in terms of the elements of the set. So the interpretation of group is always referred to the same framework: the universe of sets.
  
  
• In CT, on the contrary, mathematical concepts possess a meaning relative to the framework (category) under consideration. In the case of the group, its elements are replaced by arrows, thus making it interpretable not only in the universe (category) of sets, but in any category or frame considered. The category and the corresponding semantics may vary. CT has ambiguity of reference, as each frame has its own concept of object and morphism.

Arrow semantics

The concept of arrow or morphism is ambiguous. It is also said to be "multifaceted". It only states that it is an entity f that matches an object A an object B, but does not specify the concrete meaning of "match". The semantics is open. In fact there may be different interpretations of an arrow f:A→B, depending on the frame under consideration. For example:

• Function f of argument A, which produces the result B: f(A)=B.
  
  
• Monadic operator f of argument A and yielding B.
  
  
• Process f that brings the system from state A to state B.
  
  
• Link between objects A and B of name f.
  
  
• Sequence formed by objects A and B of name or attribute f.
  
  
• Information flow between A and B through the f communication channel.
  
  
• Function f that assigns to the object A the type or class B.
  
  
• f as a logical implication (if A then B) or as a deduction of a logical system (starting from A, one deduces B).
  
  
• Rule f which transforms A into B.
  
  
• Button f of an automaton that makes change from state A to state B.
  
  
• Evaluation function f where A evaluates to B.
  
  
• Communication path f between A and B.

This possibility of varying the interpretative framework is what makes CT general, but only in a formal or syntactic sense.


The importance of adjunct functors

The basic building blocks of CT are:

• Category. They are objects and morphisms between objects.
• Functors. They are morphisms between categories.
• Natural transformations. They are morphisms between functors.
• Attached functors.

Mac Lane and Eilenberg were saying that categories were defined to define functors, and functors were defined to define natural transformations. In fact, their original paper was not titled "General Theory of Categories" but "General Theory of Natural Equivalences".

The adjoint functors were not defined by Mac Lane and Eilenberg. They were defined later, by Daniel Kan in 1958, but their importance has grown over time. Currently, the concept of adjoint functors is one of the most important in CT:

• They are present in numerous fields of mathematics. For example, logical quantifiers (the universal and the existential) are formalized as adjoint functors with respect to the substitution operation.
  
  
• They unify a large number of mathematical concepts.
  
  
• They are related to the intuitive notion of optimization, since they provide the most efficient solution.
  
  
• They allow to characterize what is important and essential in mathematics.
  
  
• They clarify the fields where they apply.
  
  
• They capture important mathematical concepts that are invisible without the CT lens.
  
  
• They are very versatile. They can be looked at from several points of view. That is why they admit many definitions.

That is why it is considered mathematically elegant to define, whenever possible, mathematical notions in terms of adjoint functors.


The semantics of functor

A functor can have several interpretations, including the following:

• A functor F: C→D as a "representation" of C in D. The idea is to match objects and morphisms of an abstract category (C) to objects and morphisms of another more concrete category (D).
  
  
• The category C has the semantics D. The latter interpretation comes from Lawvere's [1963] concept of "functorial semantics".
  
  
• The category C is a theory, and the category D is a model of this theory.

Applications

Because of its abstract nature, CT is being applied to numerous areas:

• Foundations of mathematics.
  Since everything has structure, CT has been proposed as the foundation of mathematics.
  
  CT is intended to play a role analogous to that played by set theory and to provide a new foundation for mathematics, and contributing to its evolution along the path of increasing abstraction, generalization and unification. It is intended to connect, systematize and organize the different domains of mathematics in order to discover their common deep structures that serve as a foundation for mathematics.
  
  Lawvere was convinced that CT could serve as a foundation for mathematics, as an alternative to the official foundation (set theory with the ZFC aximata). To this end, he proposed topos theory as an alternative to set theory as the foundation of mathematics, being a more abstract mathematical structure with greater algebraic power. However, Lawvere's last proposal is to base mathematics on a reflexive concept: the category theory of categories, which includes CT itself, set theory and logic.
  
  
• Cognitive psychology.
  We must start from the premise that present science cannot explain, not only consciousness, mind and life, but the very nature of matter itself. A new mathematics is needed and CT has been turned to in the hope that it would shed some light on the underlying deep reality, in two possible ways: 1) Consciousness as a kind of universal category; 2) CT as a framework for the construction of a theory of consciousness and for the representation of cognition.
  
  In favor of this approach are the following aspects:
  
  
  • The morphism or arrow f:A→B is a triad formed by A, B and f. The number 3 is the number of consciousness, for it reconciles and harmonizes the opposites from a higher point of view. In this case, the opposites are A and B, which are harmonized by means of the third element (f). In this sense, f would be associated with consciousness, since it is situated at a higher level than A and B and connects them. The basic mechanism of mind and consciousness is supposed to be relationship, and morphism is the basic mechanism of relationship between two elements.
    
    
  • Two adjoint functors (F and G), over two categories C and D, are usually represented in the form F: C↔D:G, which indicate the union of opposites, an essential characteristic or manifestation of consciousness. In this sense, adjunctions can be associated with consciousness.
  
  Notable proposals in this sense are:
  
  
  • Kato &Struppa [1999] have developed a mathematical formalism based on the concepts of presheaf and sheaf in the framework of CT, to address a general theory of consciousness. The presheaf, as a functor, represents the internal dynamics of the conscious entity. The sheaf represents the manifestation in the target category. The sheafs are formed from the pre-sheafs. Consciousness (consheaf) is associated with an intermediate element between sheaf and presheaf.
    
    
  • Struppa et al. [2002] have developed a conceptual framework for the study of consciousness, which is based on a few fundamental principles of a philosophical nature. These principles are: self-organization, complementarity (or duality), complexity, causality, self-similarity, and nonlocality (or totality). The mathematical tools to formalize these principles are oriented towards CT and in the sheaf theory. CT would be the model of consciousness. And the concept of sheaf as a model for the entities of the consciousness. They also introduce the concept of "consheaf, conscious sheaf), an intermediate concept between sheaf and pre-sheaf.
    
    
  • Kafatos & Draganescu, in their article "Toward an Integrative Science" [Internet] state that an integrative approach to science is needed to understand the nature of mind, consciousness, matter and life. This approach or theory must integrate the qualitative and the quantitative, as well as the structural and the phenomenological. They propose CT as the most appropriate mathematical formalism for this purpose. And they propose two categories: the category of neural structures and the category of phenomenological senses. These two categories are connected by a pair of adjoint functors.
  
  
• Biology.
  CT has recently been proposed as a model for the study of biological processes:
  
  
  • Robert Rosen uses arrows to represent enzyme metabolism.
    
    
  • A.C. Ehresmann [2007] developed a model called "Memory Evolutive Systems" for autonomous and open evolutionary self-organizing systems. The dynamics (cooperative or competitive) of these systems are based on arrows.
  
  
• Mathematics.
  Constructivist and intuitionist mathematics. Model Theory. Functional analysis. Combinatorics. Homological algebra. Algebraic geometry. Synthetic differential geometry, an alternative to standard and non-standard analysis.
  
  
• Computer science.
  Programming semantics. Algebraic specifications. Modular decomposition of programs. Software systems. Modular structures. Semantics of programming languages. Functional programming. Type theory. Automata theory. Formalization of important notions such as compositionality, abstraction, representation independence, generality, etc., which have a mathematical foundation.
  
  Tatsuya Hagino [1987] has invented a programming and specification language, with few primitives, based on CT concepts, especially the adjoint functors
  
  
• Physics.
  CT has been proposed as a new paradigm to address the problems of modern (quantum and relativistic) physics. Classical physics is considered to be associated with set theory. And that for modern physics a more abstract mathematics must be used: CT, which provides a new framework for understanding and formalizing physical theories. In this sense, CT is being applied to different fields:
  
  
  • Relativistic physics: Space-time as a category, whose basic objects are the sheets (sheets are a kind of complex planes); Supradimensional spaces, where higher order categories are used; Imaginary n-dimensional space-time, where n-categories are used; etc.
    
    
  • Quantum physics: Category of Hilbert spaces; Category of quantum states (and quantum jumps as morphisms).
  
  
• Logic.
  Categorical logic: logical systems as categories, proof of theorems by means of general CT constructions.
  
  
• Philosophy.
  CT is of great philosophical interest because: 1) it is a framework that favors the investigation of concepts such as space, time and truth; 2) it helps to clarify epistemology and ontology.

MENTAL vs. Category Theory
The comparison between MENTAL and CT can be made according to the following aspects:

• Categories.
  In philosophy, the categories are the ultimate concepts of reality, the most generic ones, those of supreme level of abstraction, the cognitive roots that ground our understanding of reality. By means of the categories, entities are recognized, differentiated and classified. Aristotle was the first to propose a list of 10 categories: substance, quantity, quality, relation, place, time, situation, condition, action and passion. The concept of category in CT does not conform to this philosophical conception nor to its usual or informal meaning. Actually, CT uses only one philosophical category: relation, conceived as morphism.
  
  The true categories, the "natural" or primary categories are the universal semantic primitives of MENTAL, which are semantic axioms, mental dimensions, archetypes of consciousness and philosophical categories. Through instances of these philosophical categories arise all possible expressions that can be constructed with language.
  
  In CT there are infinite categories, as many as types of structures. In MENTAL there are 12 categories and their opposites or duals.
  
  
• Morphism semantics.
  In CT semantics is ambiguous, it has many interpretations. This is because CT uses only one primitive concept: morphism. In MENTAL, primitives have a precise semantics.
  
  
• Semantics and functors.
  Lawvere interprets functors as semantics. But semantics, because of its deep character, cannot be formalized. One can only connect semantics and syntax by means of the primary archetypes.
  
  
• Monoparadigm theory.
  CT is a monoparadigm theory: every mathematical entity has a structure based on morphisms (the arrows), since objects are considered identity morphisms. Everything is defined from this primary concept. It is a monoparadigm philosophy, analogous to set theory.
  
  MENTAL provides a universal paradigm, the source of all particular paradigms, including the concept of morphism in whatever interpretation it is given.
  
  
• Abstraction.
  CT is a theory of supreme level of conceptual abstraction. MENTAL is based on primary conceptual abstractions, on universal concepts present in all domains.
  
  
• Set.
  CT attempts to eliminate the concept of set, masquerading it as "collection", "family", "class", etc. On the other hand, in the topos category it is contemplated.
  
  The concept of set cannot be dispensed with because it is a primary archetype and a philosophical category. Neither can one dispense with the concept of sequence. And both concepts are part of MENTAL as dual primary archetypes.
  
  
• Abstract space.
  CT considers abstract space to be the space where objects reside and where connections (arrows) between objects are established. In topos theory it is an algebraic space.
  
  MENTAL envisages an abstract space in which relationships between expressions are established. The space is created as the relations are created.
  
  
• Foundation.
  CT is only one branch of mathematics, the most abstract one, which undoubtedly allows to establish horizontal relations between different mathematical fields, such as: set theory, abstract algebra, algebraic geometry, topology, logic, etc. by discovering formal analogies between structures, which helps to organize mathematical knowledge. CT allows reasoning about structures and transformations of structures, abstracting details and posing generalizations.
  
  In this sense, CT is a structural model theory, a particular version of general model theory, being based solely on the concept of morphism. CT lacks a deep conceptual foundation. CT is heir to the axiomatic tradition, with a Hilbertian (purely formal) and not a Freguean (conceptual) orientation, and mathematics needs both visions. CT does not serve to found mathematics.
  
  MENTAL is based on primary and universal concepts, so it grounds all fields of mathematics and even transcends mathematics itself. It can be said that mathematics is a manifestation of MENTAL, as it is also a manifestation of computer science.
  
  Mathematics evolves by raising the level of abstraction, that is, by recognizing at a deep level common structures of various apparently distinct fields. In this sense, it was inevitable that mathematics would encounter philosophy and seek to ground itself in philosophical categories. But CT ultimately proposes only one category: morphism. To try to base mathematics on a particular paradigm (the concept of morphism) is an error of the same caliber as trying to base it on only one conceptual primitive, such as the set (the official theory) or logic (as Frege and Russell pretended).
  
  Nor does it make sense to ground mathematics in topos theory, since it is a very complex and abstruse theory, imposes restrictions on CT itself, and is not intuitive. Mathematics must have a simple foundation, based on degrees of freedom, on primary archetypes. MENTAL is the proposal in this sense, a simple and clear proposal.
  
  
• Duality.
  Although there are dual definitions in CT (sum and product of objects of a category, functor and adjoint functor, etc.), the basic duality consists in the inversion of the arrows. This duality is essentially formal, not conceptual.
  
  MENTAL also has a dual character, but at the conceptual level. Each primitive has its opposite or dual. And it unites the opposites or duals in all aspects. In particular it unites ontology (the categorization of what exists) and epistemology (the categorization of what we know).
  
  MENTAL is the integral union of opposites. It unites semantics and syntax, theory and practice, ontology and epistemology, mathematics and metamathematics, the descriptive and the operative, and so on. This union of opposites or duals constitutes the key to consciousness.
  
  
• Relations.
  CT confuses "structure" with a collection of morphisms. But structure is something else: it is a set of relations that go beyond morphisms.
  
  In MENTAL, relations are determined by universal semantic primitives. It contemplates relations of all kinds: causal, decisional, modal, etc. and also contemplates linked, virtual, imaginary expressions, etc.
  
  
• Consciousness.
  Consciousness cannot be modeled or formalized with CT because consciousness cannot be explained. It can only be said to be based on primary archetypes, which unite pairs of opposites or duals, especially the internal and the external. And morphism is not an archetype. An archetype must unite semantics and syntax, and semantics is undefined, ambiguous. Therefore, CT has no relation to consciousness, nor can it serve as a model of consciousness.
  
  MENTAL is a language of consciousness because it precisely unites pairs of opposites.
  
  
• Language.
  CT is purely descriptive, definitional, without a formal language. It only provides a notation for the definitions of concepts and requires natural language as support. There are no constructive procedures. The emphasis is on the "what" (the descriptive) and not on the "how" (the constructive). For example, there are no explicit mechanisms for constructing object structures such as sequences, trees, matrices, etc.
  
  Instead, MENTAL is a formal language based on simple concepts from which expressions of any degree of complexity can be created.
  
  
• Limitations.
  Many mathematical notions can be expressed with CT. But not all of them. In addition, there are combinatorial restrictions. For example, you cannot define sets of any class (objects, arrows, functors), you cannot define parameterized elements, infinite categories, virtual categories, etc.
  
  All these limitations disappear with MENTAL, because of its supreme level of conceptual abstraction. Combinatorics is based on primitives and there are no restrictions. We can specify all kinds of mathematical entities, including those contemplated in CT. MENTAL is at a deeper and more fundamental level than CT.
  
  
• Applicability.
  CT, being based only on the concept of morphism, a concept that is too ambiguous, makes it difficult to apply to concrete, low-level issues.
  
  MENTAL, on the other hand, unites high and low level theory and practice, since primitives are present at all levels. It is applied in all domains where a formal language is needed.
  
  
• Natural numbers.
  CT allows us to formalize the natural numbers by means of a categorial axiomatics, which is different from Peano's axiomatics, but more complex and less intuitive.
  
  In MENTAL, number is a primary archetype or semantic axiom.
  
  
• Problem solving. Some difficult problems in some mathematical areas can be solved more easily from the CT point of view. One result demonstrated in CT generates many results in different fields of mathematics.
  
  With MENTAL, all problems are solved, simplified or clarified.
  
  
• Logic.
  In CT, logic is a structure, a derived concept. Logic is of an algebraic type. There is no logic associated with the decision.
  
  CT −in particular, topos−theory attempts to generalize logic because classical (Aristotelian) logic is too superficial and cannot be applied to complex domains such as topology, algebraic geometry, etc. Therefore the theory needs a deeper logic: a logic of intuitionistic character.
  
  In MENTAL, logic is based on a single primitive: "Condition", which is a primary archetype and a philosophical category already conceived by Aristotle. This primitive has a precise semantics and solves the problem of implication of propositional logic when combined with another primary archetype, "Generalization". Language, and not only logic, is intuitionistic in the sense that it is based on primary archetypes.
  
  Heyting algebra is a generalization of Boolean algebra, but it is not the natural, simplest and most intuitive generalization.
  
  
• Complexity.
  Despite the simplicity of the idea of morphism in CT, its development leads to a very complex and unintuitive abstract theory. Simple concepts such as natural number, addition, inclusion, etc. become tremendously complicated. The complexity derives from the fact that it tries to formalize some primary concepts, which does not make sense.
  
  
• Philosophy.
  CT has contributed to bring mathematics and philosophy closer together, by raising many epistemological and ontological issues, as well as the problem of universals. It has also forced to question the very distinction between mathematics and metamathematics, as it has had an impact on the very foundation of mathematics.
  
  In MENTAL the distinction between mathematics and metamathematics is diluted because the foundations (the primitives) are philosophical categories.
  
  
• Integration of schools of mathematical foundations.
  According to the mathematician Joachim Lambek [1989], CT makes it possible to reconcile the four great foundationalist schools of mathematics: platonism, logicism, formalism and intuitionism.
  
  MENTAL integrates and harmonizes the platonist (primitives are of platonic type), formalist (primitives are semantic axioms and there are formal axioms relating primitives) and intuitionist (primitives are of intuitive type) schools.
  
  
• Economy and abstraction.
  CT offers economy of thought and expression, dispensing with detail. But it dispenses with too much detail.
  
  MENTAL is a language that follows Occam's razor principle ("Entities should not be multiplied without necessity") and Einstein's razor principle ("Everything should be made as simple as possible, but not simpler"). As simplicity and abstraction are in direct relation, Einstein's razor can be expressed as "Everything should be made as abstract as possible, but not more abstract". This is exactly what happens with CT, which takes abstraction beyond what is reasonable, that is, beyond the primary concepts (archetypes). Abstraction must stop at the archetypes, which connect the internal with the external. In the 1940s, when CT was developed, there was talk of "general abstract nonsense" to refer to the disproportionate level of CT abstraction with respect to other fields of mathematics, and which makes it impractical.
  
  
• Relationships between fields.
  CT has helped to relate many domains of mathematics and also other disciplines such as logic, computer science, physics, biology, general systems theory, etc. And that general results obtained in CT are transferred as particular results to those domains. But it has not succeeded in blurring the boundaries between the different fields.
  
  With MENTAL the boundaries between the fields are blurred. Everything is the same thing: in all fields the same primary archetypes appear.
  
  
• Higher order categories.
  There are numerous ways of conceiving of higher order categories. One should also consider the categories themselves as particular cases of a higher category. In this second case, what kind of category is this? Since the category of categories is also a category, it would have to include itself, so that we would end up in Russell's paradox, since there is a self-referent concept.
  
  To solve the problem of categories of categories, three solutions have been proposed:
  
  
  1. Consider only "small" categories, i.e., categories that are sets. The category of small categories generalizes the notion of class of all sets, but does not include the category of sets nor the category of structures.
     
     
  2. Add a new axiom to set theory so that hierarchies of classes (classes of classes, etc.) can be considered. In this way it is possible to obtain categories whose objects are classes and hierarchies of classes, but without arriving at the category of all categories. This solution was proposed by Grothendieck.
     
     
  3. Axiomatize CT itself, as was done with set theory. The axiomatic system of set theory would be a particular case of the axiomatic system of CT when considering discrete categories (categories whose functions are only identity functions). This solution was proposed by Lawvere in 1996.
  
  With MENTAL this problem disappears, since the most that can happen is that descriptive expressions of fractal type appear.
  
  
• Universal category.
  In CT there is neither a universal category nor the null category. In MENTAL there is the universal expression (Ω), the existential expression (α) and the null expression (θ).

Conclusions

The fundamentals of CT are not yet clear. The theory is not conceptually consolidated and is currently still evolving. Even the very definition of category has changed over time, depending on the objectives or the needs

• The categorical logic is not sufficiently elaborated and the developments made are very complex.
  
  
• There are many ways of conceiving higher-order categories.
  
  
• There are "enriched" categories, categories that try to overcome the limitations of CT. For example, a metric space can be considered an enriched category. In addition, there are several ways to enrich categories.

MENTAL goes in the absolute opposite direction to CT (including topos theory), for it is based on supreme conceptual simplicity, not on supreme formal abstraction. The true categories are the universal semantic primitives. MENTAL is a universal paradigm, a theory of everything.

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Addenda
Topology and topological spaces

Topology is the science that studies topological spaces. A topological space is a set X together with a collection τ of open subsets of X. This τ collection is called "a topology on X".

An open set is a set in which each and every one of its elements is surrounded by elements that also belong to the set. Open sets have the property that the union and intersection of two open sets is an open set. The empty set is, by definition, an open set.

Examples: 1) X={1, 2, 3}, τ={{1}, {2}, {1,2}} is a topological space; 2) X={1, 2, 3}, τ={{1}, {3}, {1,2}} is not a topological space; 3) The power set of a set (set of all subsets) is a topological space.

The concept of topological space is very general and has its application in virtually all branches of modern mathematics, since it allows the formal definition of many concepts such as continuity, connectivity and convergence.


Bourbaki and category theory

The Bourbaki group rejected CT for several reasons: 1) because they considered it a "competition" to their structuralist method based on set theory; 2) because there were problems in reconciling CT and structure theory; 3) because they did not want to lose all the work developed in structure theory; 4) because of philosophical problems; 5) because of the difficulty of giving a conjunctive foundation to CT.


Presheaf and sheaf

A presheaf is a functor between a topological space (considered as a category) and another target category (usually, the category Set). This concept captures the idea of the association of local information to a topological space.

A sheaf H on a topological space X maps each open set U of X to a set H(U). A sheaf is a tool for the global study of local entities (the open sets).

The sheaf theory was developed by Jean Leray in the 1940s and 1950s for the treatment of some fundamental problems in the theory of differential equations. This theory was quickly extended to algebraic geometry, differential geometry and topology. Today the sheaf has become a fundamental generic concept in mathematics.

Sheafs obey an intrinsic logic of their own. The categories of sheafs constitute alternative universes that enrich classical mathematics. The logic of sheafs coincides with Heyting's intuitionistic logic. The logic of classical sets is reduced to the special case of fixed (non-variable) sets. The logic of sheafs on topological spaces is a logic that is halfway between classical logic and intuitionistic logic, whose laws reflect its geometrical properties. Intuitionistic logic allows to generalize even more abstract structures than sheafs.


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