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This document is draft.

Table of Contents

• Degrees of endomorphisms
• Definition
  • Terminological notes
• Groupoid of endomorphisms
• Category of elements
  • Projection of a morphism
  • Category of elements
    • Category of elements of endomorphisms
  • Image of an object or a set
• Square limiting and ancestry
  • Square limiting
  • Endomorphism series
  • Ancestry
  • Ancestry morphisms and ancestry categories

I suggest to add this (or at least a part of this) to programs of math faculties of Universities as a sub-course of Category Theory course. Probably, this is especially important for students specializing in Math Logic of Computer Science, it is also important for algebraists. This article quite suits to be used as a studybook.

Category of pseudomorphisms is used in my algebraic theory of formulas (expressions) which is developed by me for purposes of math logic and computer science.

This is a generalization of the concept of dependencies and categories of dependencies as defined in my earlier articles on this topic.

This variant of the article is shortened compared with old variant. In this version I have also defined the concept of ancestry for categories whose projection is a complete lattice instead for these categories which are complete lattices.

Note that in this article I use the standard definition of category (with single source and destination), not my alternative definition with multiple sources and destinations. However I do not require the sets of objects and morphisms to be disjoint.

Degrees of endomorphisms

Below we will need the concept of degrees of an endomorphism.

Let $U$ is an endomorphism of an object $A$ of some category. Then $U^i$ is defined for any $i=1,2,3,\dots$ and $U^i$ is also an endomorphism of the object $A$.

Additionally we can define: $U^0=1_A$.

Obvious  $U^j\circ U^i=U^{i+j}$ for any $i,j=0,1,2,\dots$

Definition

Let we have some category (original category) and let the set of its morphisms is partially ordered (by relation $\subseteq$). Then we can define a new category which I call category of pseudomorphisms™. (This can be generalized for the case of $\subseteq$ being arbitrary reflexive and transitive relation of morphisms.)

Below we will also consider the case when $\subseteq$ is the equality relation ($=$). In this case the category of pseudomorphisms will be instead called category of intermorphisms.

I recall that an endomorphism is such morphism whose source and destination objects is the same. So I will speak about object of an endomorphism, instead of its source and destination.

Let $U$ and $V$ are endomorphisms of the original category. By definition $f\in Pseud\left(U,V\right)$ if and only if $f$ is a morphism from the object of $U$ to the object of $V$ and $f\circ U\subseteq V\circ f$.

The category of pseudomorphisms is defined as follows:

• Objects of this category are endomorphisms of the original category, and morphisms (pseudomorphisms) from $U$ to $V$ are triples $\left(U,V,f\right)$ where $f\in Pseud\left(U,V\right)$.
• Composition of morphisms (pseudomorphisms) of the category of pseudomorphisms is naturally induced by composition of morphisms in the original category.
• Let $U$ is an object of the category of pseudomorphisms (that is an endomorphism of the original category). The identity morphism of $U$ (in the category of pseudomorphisms) is defined as $\left(U,U,U^0\right)$.

To prove correctness of the above definition, we need to prove that:

1. Composition of pseudomorphisms is a pseudomorphism.
2. Composition of pseudomorphisms is associative.
3. Identity morphisms of the category of pseudomorphisms are really identity morphisms.

◄

Composition of pseudomorphisms is a pseudomorphism.

Let $\left(U,V,f\right)$ and $\left(V,W,g\right)$ are pseudomorphisms. Then their composition is $\left(U,W,g\circ f\right)$.

We have:

• $f\circ U\subseteq V\circ f$;
• $g\circ V\subseteq W\circ g$;

$$g\circ f\circ U\subseteq g\circ V\circ f\subseteq W\circ g\circ f\text{.}$$

So $\left(U,W,g\circ f\right)$ is a pseudomorphism.

Associativity.

Directly follows from associativity of composition of morphisms in the original category.

Identity morphisms are really such.

Let $U$ is an endomorphism. Then its identity pseudomorphism is $\left(U,U,U^0\right)$.

That its composition with any applicable pseudomorphism is equal to that pseudomorphism is obvious.

This means that it is really an identity morphism of the category of pseudomorphisms.

►

Remark  $Pseud\left(U,V\right)$ is not the set of pseudomorphisms from $U$ to $V$.

An equivalent definition of the category of pseudomorphisms using the concept of inverse image of a category by a function: the category of pseudomorphisms for a given category $C$ is the subcategory of the category ${\left(\mathrm{Dst}\cap \mathrm{Src}\right)}^{-1}C$ for whose morphisms $\left(f,U,V\right)$ holds $f\in Pseud\left(U,V\right)$.

In the case when $\subseteq$ is the equality relation of morphisms of the original category, I will call pseudomorphisms intermorphisms™ and the category of pseudomorphisms category of intermorphisms. (Formerly I was instead calling these homomorphisms but it was a terminological glitch because in category theory the term homomorphism is used as a synonym of morphism.)

Remark  See also math encyclopedias about intertwiners and intertwining if you are curious about terminology.

I will call the natural functor from a category of pseudomorphisms (do not mess with unrelated term natural transformation) the functor which maps every endomorphism to its object and every pseudomorphisms $\left(U,V,f\right)$ to the morphism $f$.

It is really a functor because it obviously maps identity morphisms of the category of pseudomorphisms to identity morphisms of the original category and preserves composition.

This functor is trivial and has no immediate interesting properties.

Theorem  Let $U$ and $V$ are endomorphisms. Then $f\in Pseud\left(U,V\right)$ if and only if $f\in Pseud\left(U^n,V^n\right)$.

◄

The reverse implication is obvious.

Let now $f\in Pseud\left(U,V\right)$. That $f\in Pseud\left(U^0,V^0\right)$ is obvious.

We will prove $f\in Pseud\left(U^n,V^n\right)$ for $n>0$ by induction. For $n=1$ it is given to be true. Let it is true for $n=k$, that is $f\circ U^k\subseteq V^k\circ f$. Then $$f\circ U^{k+1}=f\circ U^k\circ U\subseteq V^k\circ f\circ U\subseteq V^k\circ V\circ f=V^{k+1}\circ f\text{.}$$

►

Terminological notes

The full names for the introduced categories (for relations $\subseteq$ and $=$ correspondingly) are:

• category of endomorphisms and pseudomorphisms™;
• category of endomorphisms and intermorphisms™.

In practice it is more convenient to use shorter names:

• category of pseudomorphisms™;
• category of intermorphisms™.

Groupoid of endomorphisms

Theorem

The following statements are equivalent:

1. $\left(U,V,f\right)$ is an an isomorphism in the category of pseudomorphisms;
2. $\left(U,V,f\right)$ is an an isomorphism in the category of intermorphisms;
3. $\left(U,V,f\right)$ is an intermorphism and $f$ is an isomorphism (in the original category).

◄

$\text{(2)}\Rightarrow \text{(1)}$

Obvious.

$\text{(2)}\Rightarrow \text{(3)}$

Let $\left(U,V,f\right)$ is an isomorphism in category of intermorphisms. The above mentioned functor maps $\left(U,V,f\right)$ to $f$. So $f$ is an isomorphism, because any functor maps isomorphisms to isomorphisms.

$\text{(3)}\Rightarrow \text{(2)}$

Let $\left(U,V,f\right)$ is an intermorphism ($f\circ U=V\circ f$) and $f$ is an isomorphism of the original category. Then reverse isomorphism $f^{-1}$ exists. So $f^{-1}\circ f$ and $f\circ f^{-1}$ are identity morphisms.

Multiplying the formula above with $f^{-1}$ at both left and right sides we get $U\circ f^{-1}=f^{-1}\circ V$. So $\left(V,U,f^{-1}\right)$ is an intermorphism. But it is the reverse of $\left(U,V,f\right)$.

So $\left(U,V,f\right)$ has the reverse, that is it is an isomorphism.

$\text{(1)}\Rightarrow \text{(2)}$

Let $\left(U,V,f\right)$ is an isomorphisms in the category of pseudomorphisms ($f\circ U\subseteq V\circ f$). Then there exists the reverse pseudomorphism $\left(V,U,f^{-1}\right)$ ($f^{-1}$ can be only the reverse of the morphism $f$ because compositions of $f$ and $f^{-1}$ are identity morphisms). So $f^{-1}\circ V\subseteq U\circ f^{-1}$. Multiplying this formula with $f$ at both left and right sides we get $V\circ f\subseteq f\circ U$. Comparing this formula with the formula above, we get $f\circ U=V\circ f$, that is $\left(U,V,f\right)$ is an intermorphism. Multiplying the last formula with $f^{-1}$ at both left and right sides we get $U\circ f^{-1}=f^{-1}\circ V$, so $\left(V,U,f^{-1}\right)$ is an intermorphism. It is the reverse intermorphism of $\left(U,V,f\right)$, so it is an isomorphism in the category of intermorphisms.

►

We can call triples conforming to the equivalent statements of the above theorem isomorphisms between endomorphisms (or more specifically e.g. isomorphisms between systems of formulas, when an endomorphism will represent a system of formulas).

BTW, it defines the groupoid of endomorphisms for any given category. (There are no requirement for morphisms to be partially ordered.)

Category of elements

In this section we will descent from morphisms to relations of individual elements.

Projection of a morphism

Let $Pr$ (projection) is a functor from some category (original category) to the category whose objects are sets and whose morphisms are binary relations between these sets.

Proposition  $Prf\subseteq \left(PrDstf\right)\times \left(PrSrcf\right)$ for any morphism $f$.

◄  By definition of a functor.  ►

Proposition  $Pr{1}_A=I_{PrA}$ (identity relation on the set $PrA$) for any object $A$ of the original category.

◄  By definition of a functor.  ►

Normally, $Pr$ is a monotonous function (regarding $\subseteq$), but we do not need this property:

Having projection also allows to define for any morphism $f$ its set theoretic domain and image:

• $imf=imPrf$;
• $domf=domPrf$.

Remark  In category theory the words image and domain can be used in other sense, as synonyms of source and destination.

Proposition  $domf\subseteq PrSrcf$; $imf\subseteq PrDstf$ for any morphism $f$ of the original category.

◄  Follows from properties of a functor and that a relation lies completely in the Cartesian product of its image and domain.  ►

We will call a morphism monovalued if and only if its projection is monovalued.

Category of elements

I will call elements pairs $\left(A,a\right)$, where $A$ is an object of the original category, $a\in PrA$.

The category of elements is defined as follows:

• its objects are all elements;
• its morphisms (element morphisms) from element $\left(A,a\right)$ to element $\left(B,b\right)$ are such triples $\left(f,a,b\right)$, where $f$ is a morphism (of the original category) from $A$ to $B$ and $\left(a,b\right)\in Prf$;
• its composition of morphisms $\left(f,a,b\right)$ and $\left(g,b,c\right)$ is $\left(g\circ f,a,c\right)$.

To prove correctness of the above definition we need to prove that:

1. composition of two element morphisms is an element morphism;
2. there are identity morphism for every element.

◄

By items:

1. Let $\left(f,a,b\right)$ and $\left(g,b,c\right)$ are element morphisms. Their composition is $\left(g\circ f,a,c\right)$. To prove that it is an element morphism we need to prove that $\left(a,c\right)\in Pr\left(g\circ f\right)$.

   It follows from $\left(a,b\right)\in Prf$, $\left(b,c\right)\in Prg$ and $\left(Prg\right)\circ \left(Prf\right)=Pr\left(g\circ f\right)$.

2. $\left(1_A,a,a\right)$ is an endomorphism of the element $\left(A,a\right)$ because $\left(a,a\right)\in Pr{1}_A$. Obviously the endomorphism $\left(1_A,a,a\right)$ is the identity morphism for element $\left(A,a\right)$.

►

There exists a functor from the category of elements to the original category, which maps element $\left(A,a\right)$ to $A$ and $\left(f,a,b\right)$ to $f$. It is really a functor because it preserves identities and composition.

By this functor in an obvious way (as inverse image) to every subcategory of the original category corresponds some subcategory of the category of elements. So we can speak about categories of isomorphic, pseudomorphic, intermorphic, etc. elements.

When two elements are related with a morphism of such a category, I will call second of these elements isomorphic, pseudomorphic, intermorphic, etc. image of the first one.

Proposition  If two elements $\left(A,a\right)$ and $\left(B,b\right)$ are isomorphic, then objects $A$ and $B$ are also isomorphic.

◄  Because a functor maps an isomorphism to isomorphism.  ►

Remark

The above proposition reflects the fact that an element encapsulates the structure of the object (like as in object oriented programming an instance encapsulates the structure of the class), not just an element of a set. So every element contains all information about the structure of the object.

Two elements are isomorphic means that these elements belong to isomorphic objects and are in equivalent places of that objects.

Category of elements of endomorphisms

For the category of pseudomorphisms (or category of intermorphisms) we will define functor $Pr$ as follows:

• $PrU=PrSrcU=PrDstU$ for an object $U$ of the category of pseudomorphisms (that is an endomorphism of the original category).
• Projection of a pseudomorphism is equal to projection of the corresponding morphism of the original category.

Remark  When $U$ is an endomorphism (of the original category), $PrU$ is ambiguous, because $U$ can be considered either as a morphism of the original category or as an object of the category of pseudomorphisms. So I will use explicit wording to disambiguate in such cases.

We need to prove that it is really a functor.

◄

Let $\left(U,V,f\right)$ is a pseudomorphism. Let the objects of $U$ and $V$ are $A$ and $B$ correspondingly. Then source and destination of $f$ are correspondingly $A$ and $B$. Projections of $U$ and $V$ (understood as objects of the category of pseudomorphisms) are equal to $PrA$ and $PrB$ correspondingly. Projection of $\left(U,V,f\right)$ what is the same as projection of $f$ is a morphism between $PrA$ and $PrB$. So projection of $\left(U,V,f\right)$ is a morphism from $PrA$ to $PrB$.

We have left to prove that $Pr$ preserves composition and identities.

That $Pr$ preserves composition follows from that composition of pseudomorphisms is naturally corresponding to the composition of corresponding morphisms of the original category.

Let $U$ is an endomorphism of the original category, let the object of $U$ is the object $A$ of the original category. The identity pseudomorphism of the object $U$ of the category of pseudomorphisms is $\left(U,U,U^0\right)$. The object of $U^0$ is $A$. So the projection of $\left(U,U,U^0\right)$ or what is the same the projection of $U^0$ is a morphism of $PrA$. But as projection of $U^0$ it is the identity morphism of $PrA$. So projection of any identity morphism of the category of pseudomorphisms in an identity morphism.

►

So for the category of pseudomorphisms is also defined the category of elements. I will call this category (in relation to the original category) the category of elements and pseudomorphisms (or the category of elements and intermorphisms in the case of intermorphisms instead of pseudomorphisms).

• Objects of this category (elements of endomorphisms) are pairs $\left(U,a\right)$ where $U$ is an endomorphism of the original category and $$a\in PrSrcU=PrDstU\text{.}$$

• Morphisms of this category are triples $\left(f,a,b\right)$ where $f$ is a pseudomorphism and

  • $a\in SrcPrf=PrSrcf$;
  • $b\in DstPrf=PrDstf$.

  That is morphisms of this category are $\left(\left(U,V,f\right),a,b\right)$ where $U$ and $V$ are endomorphisms, $f$ is a morphism (of the original category) from $U$ to $V$ ($f\in Pseud\left(U,V\right)$),

  • $a\in SrcPrf=PrSrcf=PrSrcU=PrDstU$;
  • $b\in DstPrf=PrDstf=PrSrcV=PrDstV$.

So we can speak about categories of isomorphic, pseudomorphic, intermorphic, etc. elements of endomorphisms. When we will come to study of endomorphisms representing systems of expressions, we will so speak about isomorphic, pseudomorphic, intermorphic, etc. formulas.

Image of an object or a set

A relation $f$ applied to a set $A$ is by definition the set $$f\left(A\right)=\left\{b|\exists a:\left(a,b\right)\in f\right\}\text{.}$$

By definition a morphism $f$ applied to set $A$ is $f\left(A\right)=\left(Prf\right)\left(A\right)$

Remark  We can also define morphism $f$ applied to an object $A$: $f\left(A\right)=f\left(PrA\right)=\left(Prf\right)\left(PrA\right)$, but do not need this.

Obvious  (For our original category) image of an object by composition of morphisms is the image of the image of this object, that is $\left(g\circ f\right)\left(A\right)=g\left(f\left(A\right)\right)$.

Obvious  $imf=f\left(domf\right)=f\left(Srcf\right)$ for a morphism $f$ of our original category.

Obvious  $f\left(A\cup B\right)=f\left(A\right)\cup f\left(B\right)$ for sets $A$ and $B$.

Square limiting and ancestry

Square limiting

Let in addition to the requirements of the previous section for any morphism (of original category) $U$ and a set $A$ is defined an endomorphism (of original category) $U\square A$ ($U$ square limited to $A$) such that $$PrSrc\left(U\square A\right)=PrDst\left(U\square A\right)=A\text{.}$$

We can also define square limiting for right argument $A$ being an object of the original category (instead of a set) by the formula $U\square A=U\square \left(PrA\right)$.

Remark  It is enough for our purposes if square limiting is defined only for endomorphisms (instead of all morphisms of the original category).

Example  Square limiting sometimes may be defined as $f\square A=f\cap \left(A\times A\right)$.

Normally $\square$ follow certain axioms, which we however will not need in this section:

• $\square$ is monotonous by both left and right arguments.
• $\left(f\square A\right)\subseteq f$.

Moreover normally $\square$ is (infinitely) distributive by left argument regarding operation $\cup$.

Endomorphism series

For any endomorphism $U$ of a category whose morphisms form a complete lattice (the most important example is the category of binary relations) I will define:

• $S_1\left(U\right)=U\cup U^2\cup U^3\cup \dots$;
• $S\left(U\right)=U^0\cup S_1\left(U\right)=U^0\cup U^1\cup U^2\cup \dots$.

If $Pr$ is a functor from some category $C$ (original category) to a category whose morphisms form a complete lattice, then for every endomorphism $U$ of $C$ we can also define $S_1\left(U\right)$ and $S\left(U\right)$:

• $S_1\left(U\right)=S_1\left(PrU\right)$;
• $S\left(U\right)=S\left(PrU\right)$.

Ancestry

By definition exclusive ancestry™ and ancestry™ of a set $A$ by an (endo)morphism $U$ (of the original category) are correspondingly the endomorphisms

• ${Fam}_1\left(U,A\right)=U\square \left(S_1\left(U\right)\right)\left(A\right)$;
• $Fam\left(U,A\right)=U\square \left(S\left(U\right)\right)\left(A\right)$.

Remark  $Fam$ is an abbreviation of the English word family.

Ancestry and exclusive ancestry of an element of a pseudomorphism are defined correspondingly as:

• ${Fam}_1\left(U,a\right)={Fam}_1\left(U,\left\{a\right\}\right)$;
• $Fam\left(U,a\right)=Fam\left(U,\left\{a\right\}\right)$.

We can also define (exclusive) ancestries for pairs of an endomorphism and an object of original category:

• ${Fam}_1\left(U,A\right)={Fam}_1\left(U,PrA\right)$;
• $Fam\left(U,A\right)=Fam\left(U,PrA\right)$.

Example

Oriented graphs are morphisms of a category whose objects are sets of points, composition of two orgraphs is simply their set theoretic union, identity morphism of a set of points is a discrete graph (graph without vertices).

Ancestry of a point of an (oriented) graph is the subgraph which can be reached from this point. (Inclusive ancestry will contain this point, and exclusive ancestry does not necessarily contain this point.)

Proposition  ${\left(Fam\left(U,a\right)\right)}^n\left(x\right)=U^n\left(x\right)$ for any $x\in S\left(U\right)\left(a\right)$

◄  $U^n\left(x\right)\in S\left(U\right)\left(a\right)$  ►

Consequence  ${\left(Fam\left(U,a\right)\right)}^n\left(a\right)=U^n\left(a\right)$.

Let $\left(U,a\right)$ is an element of an endomorphism. I will call the corresponding detached element the pair $\left(Fam\left(U,a\right),a\right)$.

I will call a head element such element of an endomorphism which does not change being detached. Equivalently: a head element of an endomorphism $U$ is such element of this endomorphism whose ancestry is $U$.

Remark  When a system of an expression and all its subexpressions is represented as an endomorphism, head element corresponds to the main expression (as opposed to subexpressions).

BTW, now we can define two way connected endomorphism: an endomorphism is two way connected if and only if each its element is head.

Ancestry morphisms and ancestry categories

Remark  The terminology defined in this subsection is yet unstable and may vary in future works.

Remark  Ancestry morphisms (see below) are important for the theory of formulas where e.g. two formulas (expressions) being ancestry isomorphic means that they have the same structure of subformulas (subexpressions).

I will call the ancestry category™ of isomorphisms (pseudomorphisms, intermorphisms, etc.) the following category:

• Objects are head elements of endomorphisms.
• Morphisms from an object $\left(U,a\right)$ to an object $\left(V,b\right)$ are are are triples $\left(f,a,b\right)$ where $f$ is an isomorphism (pseudomorphism, intermorphism, etc.) between $U$ and $V$.
• Composition of morphisms is naturally induced by composition of pseudomorphisms.

I leave to the reader the trivial problem to prove that it is really a category.

This category can be intersected with the category of elements of endomorphisms. I call the resulting category strict ancestry category of isomorphisms (pseudomorphisms, intermorphism, etc.)

So we can speak about (strictly) ancestry isomorphic, pseudomorphic, intermorphic, etc. head elements of endomorphisms.

This can be generalized for any elements of endomorphisms calling them (strictly) ancestry isomorphic (pseudomorphic, intermorphic, etc.) if and only if the corresponding detached elements are (strictly) ancestry isomorphic (pseudomorphic, intermorphic, etc.) In this version of the article I leave a problem to the reader to define the corresponding categories.

Related Links

• Category theory pages on this site.
• Theory of Dependencies (on this site).
• Definition of category with multiple domains and ranges.
• www.mathematics21.org
• Theory of Formulas (on this site).
• Journal of post-Axiomatic Mathematics and Logic.

License

See here about licensing of this text etc.

Related Web Categories

See also my suggestions for new mathematics classification categories.

• /Science/Math/Algebra/Category_Theory/
• /Science/Math/Publications/Online_Texts/
• http://www.3apes.com/directory/index.cgi?page=category_theory

Copyright © 2005 Victor Porton

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