1. Introduction

Digital Topology is a branch of mathematics where the image processing and digital image processing is studied. Many Mathematicians, for example Rosenfeld, Kopperman, Han, Kong, Malgouyres, Boxer, Ayala, Karaca and others have contributed this area with their research. The notion of digital image, digital continuous map and digital homotopy studied in[3, 4, 6, 7, 12, 16]. Their recognition and efficient computation became a useful material for our study.

Then we carry this notion to category theory and we construct some fundamental category models in digital topology.

In section two we recall some definitions and properties from Boxer[5] in section three we introduce the ‘Digital Category’ and give basic examples of digital categories in order to construct a tool for category theory researchers.

2. Preliminaries

In this paper we denote the set of integers by . Then represents the set of lattice points in Euclidean dimensional spaces. A finite subset of is called to be digital image.

We will use a variety of adjacency relations it the digital image research. The following[7] are commonly used.

Two points and in are if they are distinct and differ by at most in each coordinate; and in are if they are and differ in exactly one coordinate. Two points and in are if they are distinct and differ by at most in each coordinate; they

are if they are and differ in at most two coordinates; they are if they are and differ in exactly one coordinate. For , a of a lattice point is a point that is to .

We generalize in and in by taking are if and and differ by in one coordinate and by in all other coordinates.

More extensive adjacency relations are investigated in[5]. In the following, if is an adjacency relation defined for an integer on as one of the discussed above, that is, if

or .

We assume as , as ,etc.

Suppose that be an relation defined on . A digital image is [5] if and only if for every pair of points there is a set such that and and are .

Definition 2.1. Let and are digital images such that, . Then the digital function is a function which is defined between digital images.

Definition 2.2. ([3]; see also[15]) Let and are digital images such that, . Assume that be a function. Let be an relation defined on , . is called to be if the image under of every subset of is .

A function satisfying Definition 2.1 is referred to be digitally continuous. A consequence of this definition is given below.

Definition 2.3. ([3]; see also[15]) Let and are digital images. Then the function is said to be if and only if for every such that and are , either or and are .

Definition 2.4. ([2]) Let , . A digital interval is a set of the form

in which is assumed.

For example, if is an relation on a digital image , then is if and only if for every , either or and are .

Definition 2.5. ([3]; see also[6]) suppose that and be digital images. Let be functions. Assume there is a positive integer and a function such that

i) For all and ;

ii) For all , the induced function defined by for all is.

iii) For all , the induced function defined by for all is .

Then is called to be a digital between and , and and are said to be digitally in .

We use the notation to denote and are digitally in .

Definition 2.6.[4]A digital in a digital image is a function . Also if , we say that is a digital , and the point is the base point of the loop . If is a constant function, then it is called a trivial loop.

If and are digital in such that starts where ends, the product of and , written , is intuitively, the obtained by following by . Formally, and , then is defined by

Definition 2.7. Let and be functions. Two paths and , mapping the digital interval , are said to be digital path homotopic if they have the same initial point and the same final point , and there is a map such that

for each and each . We call to be a digital path homotopy between and , and we write .

3. Digital Categories

Definition 3.1. A digital category is a quintuple where

(i) is a class whose members are called object. object are digital images.

(ii) is a class whose members are called morphisms. morphisms are digital functions which is defined between digital sets.

(iii) and are digital functions from to ( is called the domain of and is called the codomain of )

(iv) is a function from

into , called the composition law of such that the following conditions are satisfied:

(1) Matching Condition: If is defined, then and ;

(2) Associativity Condition: If and are defined, then ;

(3) Identity Existence Condition: For each object there exist morphism such that and

(a) whenever is defined, and

(b) whenever is defined;

(4) Smallness of Morphism Class Condition: For any pair of objects, the class

is a set.

Let and be digital images. We will use the notation to denote the composition .

Thus the statement that the triangle

is equivalent to the statement that . When morphisms and exist such that the above triangle commutes, we say that factor through . Similarly the statement that the square

commutes means that .

Proposition 3.1. Let be a digital category and is a object such that is a digital image. Then there exist exactly one morphismsatisfiying the properties 3(a) and 3(b) of Definition 3.1; i.e. such that

(a) whenever is defined, and

(b) whenever is defined;

Proof: Suppose that each of and is such a morphism. Then by (a) and by (b) ; hence, .

Definition 3.2. For each object of the digital category , the unique morphism satishfiying (a) and (b) above is denoted by and is called the identity of .

Definition 3.3. A digital category is said to be:

(1) Small provided that is a digital set;

(2) Discrete provided that all of its morphisms are identities;

(3) Connected that for each pair of objects, .

and

can be considered to be digital categories, but neither

nor

can be digital categories.

Definition 3.4. For each natural number , the set supplied with the usual order can be considered to be digital category , thus we have the special small digital categories:

The empty category

Boxer defined the path homotopy in [5]. We can consider equivalence classes of digital paths (path homotopy classes) because digital path homotopy relation is an equivalence relation.

Example 3.2. Given a digital image and points a digital path from to is a continuous mapping from some digital set to with and If

is a digital path from to and is a digital path from to , there is a path defined by

from to .This makes into a category, the digital path category of .

Now given digital paths

, both from to .

There is a digital continuous mapping in a digital image in such that

It easy to see that this is an equivalence relation. The quotient of the digital path category by this congruence relation is a category called the digital category of digital homotopy classes of digital paths in .

Definition 3.5. Let be a digital category and digital digital sets. A digital morphism in said to be monic if it is left cancelable.

Theorem 3.1. Suppose thatbe digital sets. Consider the following ‘cube’ of digital objects and digital morphisms in a given digital category :

(i) Suppose that all faces except the top face are given to be commutative. If is monic, then the top face is also commutative.

(ii) Suppose that all faces except the bottom face are given to be commutative. If is epic, then the bottom face is also commutative.

Proof: Let denote the digital morphism from to . We have to prove that . Now is given to be monic and (writing as juxtaposition)

Then the result follows by left cancellation of.

ii) We have to prove that [1,5] is given to be epic and

Then the result follows by right cancellation of

Example 3.3.Given a digital image and points a digital path from to is a continuous mapping from some digital set to with and . If is a digital path from to and is a digital path from to there is a path defined by

from to . This makes into a category the digital path category of. Now given digital paths , both from to , one can define if there is a continuous map

in such that

4. Conclusions

In this paper we construct the digital category model for digital images and gave some conditions made the diagram commutative.