# Digital Hopf Spaces and Their Duals

Authors: Dae-Woong Lee
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this article, we study the fundamental notions of digital Hopf and co-Hopf spaces based on pointed digital images. We show that a digital Hopf space, a digital associative Hopf space, a digital Hopf group, and a digital commutative Hopf space are unique up to digital homotopy type; that is, there is only one possible digital Hopf structure up to digital homotopy type on the underlying digital image. We also establish an equivalent condition for a digital image to be a digital Hopf space and investigate the difference between ordinary topological co-Hopf spaces and their digital counterparts by showing that any digital co-Hopf space is a digitally contractible space focusing on deep-learning methods in imaging science.


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## Body

## 1. Introduction

### 1.1. Hopf and Co-Hopf Spaces

In many cases in algebraic topology, it is feasible to introduce a natural group structure on some set whose elements consist of homotopy classes of the base point preserving continuous functions from a pointed topological space to another. It is well known that ifX,x0 is a group-like space or if W,w0 has a cogroup structure, then we can construct a group structure on the set W,w0,X,x0 of homotopy classes of continuous functions from W,w0 to X,x0 as inherited from various functors.A Hopf spaceX,mX is a pointed topological space X,x0 along with a base point preserving continuous function mX:X×X⟶X, called a multiplication on X, for which the constant map cx0:X⟶X is a homotopy identity.It is well known that every topological group is a Hopf space and that no even-dimensional sphere of nonnegative dimension is a Hopf space except for the zero sphereS0. What about the odd-dimensional spheres? The answer to this question is that the spheres in dimensions 1, 3, and 7 are the only spheres that are Hopf spaces. In the general case, as compared to a topological group, Hopf spaces may lack associativity and inverses.As the dual notion of a Hopf space, a pairC,φC consisting of a pointed space C=C,c0 and a continuous function φC:C⟶C∨C, called a comultiplication on C, preserving the base point is called a co-Hopf space if π1∘φC and π2∘φC are homotopic to the identity function 1C on C, where π1 and π2 are the projections C∨C⟶C onto the first and second summands, respectively; see [1–11] for further details on related topics. One of the reasons for the importance of co-Hopf spaces is that we can obtain a (not necessarily abelian) group structure on the set C,c0,Y,y0 of pointed homotopy classes just like with Hopf spaces.


### 1.2. Digital Homotopy Sets

Just like classical homotopy theory in algebraic topology, in the realm of digital geometry and computer science, we can construct a covariant functorW,w0,kW,− and a contravariant functor −,X,x0,kX from the category of pointed digital images to the category of sets (resp., groups) and functions (resp., group homomorphisms), where W,w0,kW and X,x0,kX are pointed digital images with kW-adjacency and kX-adjacency relations, respectively. We refer to the papers [12] (Theorem 4.14) and [13] as a special case of the sets of digital homotopy classes from a digital image to another.The digital fundamental groups and the digital homotopy groups allow us to have much more efficient homotopy characterization of digital images. Indeed, we can handle the fundamental properties of pointed digital images more easily by using the group structure on the setW,w0,kW,X,x0,kX of digital homotopy classes of base point preserving digital continuous functions from W,w0,kW to X,x0,kX if the pointed digital image X,x0,kX with kX-adjacency has the structure of a digital Hopf group.


### 1.3. Motivations

Historically, a Hopf space was named after Heinz Hopf [14] as the Eckmann–Hilton dual of a co-Hopf space [15]. In the early 20th century, a lot of interesting results on Lie groups as the particular case of Hopf spaces have been widely developed by suitable methods for CW-complexes from the homology and cohomology viewpoints in algebraic topology. The pointed Hopf spaces have been the direct outgrowth of compact Lie groups in classical homotopy theory; see [15–17]. Recently, the digital version of a Hopf space has been investigated by several authors; see [18–21].The aforementioned statements serve as some motivation for research on this topic. Therefore, we need to develop and investigate the basic properties of digital Hopf spaces, digital associative Hopf spaces, digital Hopf groups, digital commutative Hopf spaces, and their duals from the digital homotopy points of view as an application to computer science, focusing on deep-learning methods in imaging science. It is shown in [22] that the definition of a path in algebraic topology is coherent with respect to the one used for digital images with an adjacency relation.In the same vein, digital counterparts of geometry or topology deal with discrete sets, which are recognized to be digitized images of then-dimensional Euclidean space. For application of the powerful digital homotopy-theoretical tools, it is favorable to reformulate the digital counterparts of Hopf spaces and co-Hopf spaces in the digital homotopy categories as seen in the realms of digital geometry and computer science, based on the deep-learning methods in imaging science.


### 1.4. Organization of the Paper

This article is organized as follows. In Section2, we introduce and explain the fundamental notions of digital topology or digital images. In Section 3, we show that a digital Hopf space, a digital associative Hopf space, a digital Hopf group, and a digital commutative Hopf space are unique up to digital homotopy type; that is, there is only one possible digital Hopf structure up to digital homotopy type on the underlying digital image. We also find an equivalent condition for a digital image to be a digital Hopf space. In Section 4, we investigate the differences between ordinary topological co-Hopf spaces and digital co-Hopf spaces by showing that any digital co-Hopf space is digitally contractible. We also show that any strictly digital co-Hopf space is a set having only one element as a base point in which the result is exactly the same as the Eckmann–Hilton dual of a topological group. A summary and further research direction will be given at the end of this paper.


## 1.1. Hopf and Co-Hopf Spaces

In many cases in algebraic topology, it is feasible to introduce a natural group structure on some set whose elements consist of homotopy classes of the base point preserving continuous functions from a pointed topological space to another. It is well known that ifX,x0 is a group-like space or if W,w0 has a cogroup structure, then we can construct a group structure on the set W,w0,X,x0 of homotopy classes of continuous functions from W,w0 to X,x0 as inherited from various functors.A Hopf spaceX,mX is a pointed topological space X,x0 along with a base point preserving continuous function mX:X×X⟶X, called a multiplication on X, for which the constant map cx0:X⟶X is a homotopy identity.It is well known that every topological group is a Hopf space and that no even-dimensional sphere of nonnegative dimension is a Hopf space except for the zero sphereS0. What about the odd-dimensional spheres? The answer to this question is that the spheres in dimensions 1, 3, and 7 are the only spheres that are Hopf spaces. In the general case, as compared to a topological group, Hopf spaces may lack associativity and inverses.As the dual notion of a Hopf space, a pairC,φC consisting of a pointed space C=C,c0 and a continuous function φC:C⟶C∨C, called a comultiplication on C, preserving the base point is called a co-Hopf space if π1∘φC and π2∘φC are homotopic to the identity function 1C on C, where π1 and π2 are the projections C∨C⟶C onto the first and second summands, respectively; see [1–11] for further details on related topics. One of the reasons for the importance of co-Hopf spaces is that we can obtain a (not necessarily abelian) group structure on the set C,c0,Y,y0 of pointed homotopy classes just like with Hopf spaces.


## 1.2. Digital Homotopy Sets

Just like classical homotopy theory in algebraic topology, in the realm of digital geometry and computer science, we can construct a covariant functorW,w0,kW,− and a contravariant functor −,X,x0,kX from the category of pointed digital images to the category of sets (resp., groups) and functions (resp., group homomorphisms), where W,w0,kW and X,x0,kX are pointed digital images with kW-adjacency and kX-adjacency relations, respectively. We refer to the papers [12] (Theorem 4.14) and [13] as a special case of the sets of digital homotopy classes from a digital image to another.The digital fundamental groups and the digital homotopy groups allow us to have much more efficient homotopy characterization of digital images. Indeed, we can handle the fundamental properties of pointed digital images more easily by using the group structure on the setW,w0,kW,X,x0,kX of digital homotopy classes of base point preserving digital continuous functions from W,w0,kW to X,x0,kX if the pointed digital image X,x0,kX with kX-adjacency has the structure of a digital Hopf group.


## 1.3. Motivations

Historically, a Hopf space was named after Heinz Hopf [14] as the Eckmann–Hilton dual of a co-Hopf space [15]. In the early 20th century, a lot of interesting results on Lie groups as the particular case of Hopf spaces have been widely developed by suitable methods for CW-complexes from the homology and cohomology viewpoints in algebraic topology. The pointed Hopf spaces have been the direct outgrowth of compact Lie groups in classical homotopy theory; see [15–17]. Recently, the digital version of a Hopf space has been investigated by several authors; see [18–21].The aforementioned statements serve as some motivation for research on this topic. Therefore, we need to develop and investigate the basic properties of digital Hopf spaces, digital associative Hopf spaces, digital Hopf groups, digital commutative Hopf spaces, and their duals from the digital homotopy points of view as an application to computer science, focusing on deep-learning methods in imaging science. It is shown in [22] that the definition of a path in algebraic topology is coherent with respect to the one used for digital images with an adjacency relation.In the same vein, digital counterparts of geometry or topology deal with discrete sets, which are recognized to be digitized images of then-dimensional Euclidean space. For application of the powerful digital homotopy-theoretical tools, it is favorable to reformulate the digital counterparts of Hopf spaces and co-Hopf spaces in the digital homotopy categories as seen in the realms of digital geometry and computer science, based on the deep-learning methods in imaging science.


## 1.4. Organization of the Paper

This article is organized as follows. In Section2, we introduce and explain the fundamental notions of digital topology or digital images. In Section 3, we show that a digital Hopf space, a digital associative Hopf space, a digital Hopf group, and a digital commutative Hopf space are unique up to digital homotopy type; that is, there is only one possible digital Hopf structure up to digital homotopy type on the underlying digital image. We also find an equivalent condition for a digital image to be a digital Hopf space. In Section 4, we investigate the differences between ordinary topological co-Hopf spaces and digital co-Hopf spaces by showing that any digital co-Hopf space is digitally contractible. We also show that any strictly digital co-Hopf space is a set having only one element as a base point in which the result is exactly the same as the Eckmann–Hilton dual of a topological group. A summary and further research direction will be given at the end of this paper.


## 2. Preliminaries

Letℤ be the set (or a topological space or the ring) of integers, and let X be a finite subset of ℤn for n≥1. A binary digital image (or digital image for short) is a pair X,kX, where kX indicates some adjacency relation. For a positive integer u with 1≤u≤n, an adjacency relation of a digital image in ℤn is defined as follows.Definition 1 (see [23]).
Letu and n be positive integers with u≤n. Then, two distinct points a=a1,a2,…,an and b=b1,b2,…,bn in ℤn are lu-adjacent if(i)
there exist at mostu distinct indices i such that ai−bi=1;(ii)
for all indicesj, if aj−bj≠1, then aj=bj.
The numberlu is the cardinal number kX=ku,n,1≤u≤n, of the set of elements b of ℤn which is adjacent to a given element a of X⊊ℤn, and is called an adjacency relation kX defined on ℤn. A digital image X⊊ℤn with an adjacency relation kX on ℤn is sometimes denoted as X,kX to designate the adjacency relation. For example,(i)
l1=2=k1,1, in ℤ1;(ii)
l1=4=k1,2, l2=8=k2,2 in ℤ2;(iii)
l1=6=k1,3, l2=18=k2,3, l3=26=k3,3 in ℤ3;(iv)
l1=8=k1,4, l2=32=k2,4, l3=64=k3,4, l4=80=k4,4 in ℤ4, and so on.Definition 2 (see [24]).
A digital imageX⊂ℤn with an adjacency relation kX is called kX-connected if for any pair of distinct elements a,b of X, there is a set A=a0,a1,…,as⊂X of s+1 distinct points satisfying that a=a0,as=b, and ai and ai+1 are kX-adjacent for i=0,1,2,…,s−1 for s≥1.Definition 3 (see [12, 24]).
LetX⊂ℤn1 and Y⊂ℤn2 be the digital images with kX-adjacency and kY-adjacency relations, respectively. A map f:X⟶Y between digital images is said to be kX,kY-continuous if, for any kX-connected subset A of X, the image fA is also a kY-connected subset of Y.Definition 4 (see [25]).
For the nonnegative integersc and d with c<d, a digital interval is a set of the type(1)c,dℤ=z∈ℤ|c≤z≤d,with the 2-adjacency relation in the set of all nonnegative integers.Definition 5 (see [12, 26]).
LetX:=X,kX be a digital image along with the kX-adjacency relation. Then, a (2 kX)-continuous function f:0,mℤ⟶X is called a digital kX-path in X.Definition 6 (see [12, 23]).
LetX:=X,kX and Y:=Y,kY be digital images, and let f,g:X⟶Y be kX,kY-continuous functions. Then, f is said to be digital kX,kY-homotopic to g if there exist an integer m and a function H:X×0,mℤ⟶Y satisfying that(i)
Hx,0=fx and Hx,m=gx for all x∈X;(ii)
the functionHx:0,mℤ⟶Y, x∈X, which is given by Hxt=Hx,t is 2,kY-continuous for all t∈0,mℤ;(iii)
the functionHt:X⟶Y, t∈0,mℤ, which is given by Htx=Hx,t is kX,kY-continuous for all x∈X.
In this case, the functionH is called a digital kX,kY-homotopy between f and g, denoted by(2)H:f≃kX,kYg.
It is well known that the relation≃kX,kY is an equivalence relation on the set of all kX,kY-continuous functions from X to Y. For a kX,kY-continuous function f:X⟶Y, we denote f by(3)f=g:X⟶Y|f≃kX,kYg,which is called the digital homotopy class represented by f.Definition 7 (see [12, 26]).
A pointed digital image withkX-adjacency is a triple X:=X,x0,kX. In this situation, the element x0 of X is called the base point of X,x0,kX. A base point preserving digital continuous function f:X,x0,kX⟶Y,y0,kY is a kX,kY-continuous function f:X,kX⟶Y,kY with fx0=y0.Definition 8 (see [25]).
A digital imageX:=X,x0,kX is said to be digitally contractible if the identity function 1X:X⟶X on X is digitally null-homotopic; that is, it is digital kX,kX-homotopic to the constant function cx0:X⟶X at x0.Definition 9 (see [26]).
LetX:=X,kX and Y:=Y,kY be digital images. Then, a kX,kY-continuous function f:X⟶Y is said to be a kX,kY-homotopy equivalence if there exists a kX,kY-continuous function g:Y⟶X such that(4)g∘f≃kX,kX1X,f∘g≃kY,kY1Y.
In this case, we say thatX and Y have the same digital homotopy type which is denoted by(5)X≃kX,kYY.
Convention: In this paper, we work on the pointed digital categoryD∗ of pointed digital images and base point preserving digital continuous functions. Thus, any digital image in this paper has a base point, and all maps (resp., homotopies) are base point preserving digital continuous functions (resp., pointed digital homotopies) between pointed digital images.


## 3. Digital Hopf Spaces

From now on, the base points and the adjacency relations of digital images will sometimes be omitted for our notational convenience unless we specifically state otherwise.Definition 10 (see [27]).
LetX:=X,x0,kX and Y:=Y,y0,kY be the pointed digital images. Then, the normal product adjacency NPkX,kY on the Cartesian product,(6)X×Y:=X×Y,x0×y0,NPkX,kY,is defined as follows: if x,x′∈X and y,y′∈Y, then the two elements x,y and x′,y′ of X×Y are NPkX,kY-adjacent in X×Y if and only if(i)
x=x′, and y and y′ are kY-adjacent;(ii)
x and x′ are kX-adjacent, and y=y′; or(iii)
x and x′ are kX-adjacent, and y and y′ are kY-adjacent.
Note that the normal product adjacency relation makes the Cartesian productX×Y into a digital image with the adjacency relation NPkX,kY whose base point is x0×y0.
Lets and t be the integers with 1<s≤t, and let Xi,kXi|i=1,2,…,t be a family of digital images Xi,kXi for i=1,2,…,t. Then, as a generalization of the normal product adjacency, we define the following.Definition 11 (see [28, 29]).
Letxi,xi′∈Xi for i=1,2,…,t. Then, the two elements x=x1,x2,…,xt and x′=x1′,x2′,…,xt′ of ∏i=1tXi are NPskX1,kX2,…,kXt-adjacent if and only if(i)
for at least 1 and at mosts indices i, xi, and xi′ are kXi-adjacent;(ii)
for all other indicesj, xj=xj′.
The adjacency relationNPskX1,kX2,…,kXt is said to be the generalized normal product adjacency on the Cartesian product ∏i=1tXi.


### 3.1. Notation

We now fix the notations in this paper:(i)
cx0:X⟶X and cy0:Y⟶Y are the constant functions whose values are x0 and y0, respectively.(ii)
kX×Y is the normal product adjacency on X×Y; that is, kX×Y:=NPkX,kY, and in particular kX×X:=NPkX,kX.(iii)
kX×Y×Z is the generalized normal product adjacency on X×Y×Z; that is, kX×Y×Z:=NP3kX,kY,kZ, and in particular kX×X×X:=NP3kX,kX,kX.(iv)
TX×Y:X×Y⟶Y×X is the switching function sending x,y to y,x.(v)
ΔX:X⟶X×X is the diagonal function sending x to x×x as a kX,kX×X-continuous function.We now consider the digital versions of the Hopf space, homotopy associativity, homotopy commutativity, and Hopf group in algebraic topology as follows—see [15, 30, 31] for the terminologies in classical homotopy theory.Definition 12 (see [18]).
A digital imageX:=X,x0,kX with a kX-adjacency relation is called a digital Hopf space if there is a kX×X,kX-continuous function,(7)mX:X×X⟶X,
such that the diagram in Figure1 is digital homotopy commutative; that is,(8)mX∘cx0×1X∘ΔX≃kX,kX1X≃kX,kXmX∘1X×cx0∘ΔX.
In this case, thekX×X,kX-continuous function(9)mX:X×X⟶Xis called a digital multiplication on X,x0,kX, and the constant function cx0 is said to be a digital homotopy identity.Figure 1
A commutative diagram.Definition 13 (see [21]).
LetX:=X,x0,kX and Y:=Y,y0,kY be pointed digital Hopf spaces. Then, a kX,kY-continuous function f:X⟶Y is said to be a digital Hopf function if(10)f∘mX≃kX×X,kYmY∘f×f.Definition 14 (see [21]).
A digital multiplicationmX:X×X⟶X on a pointed digital Hopf space X:=X,x0,kX is said to be digital homotopy associative if the diagram in Figure 2 commutes up to digital homotopy. A digital Hopf space X=X,x0,kX with a digital homotopy associative multiplication mX:X×X⟶X is called a digital associative Hopf space.Figure 2
A commutative diagram.Definition 15 (see [21]).
LetX:=X,x0,kX be a digital Hopf space with a digital multiplication mX:X×X⟶X. A kX,kX-continuous function(11)νX:X⟶Xis called a digital homotopy inverse if the diagram in Figure 3 commutes up to digital homotopy.Figure 3
A commutative diagram.Remark 1.
We should be careful with the term ‘inverse’ at this moment because an inverse is usually understood to be an element of a group or more generally of a set with a binary operation. The inverse in Definition15 is a function that produces the homotopy inverse elements.
We note that, in general, there are many kinds of digital homotopy inverses and that one of the digital homotopy inversesνX:X⟶X can be constructed as a kX,kX-continuous function so that the triangles in Definition 15 are digital homotopy commutative. More precisely, we let f and g be digital homotopy classes represented by kW,kX-continuous functions f,g:W⟶X, respectively. If W:=W,w0,kW is a pointed digital image, and if X:=X,x0,kX is a pointed digital Hopf space with a digital multiplication mX:X×X⟶X, then we define a binary operation “⊞” on the set W,w0,X,x0 of all digital homotopy classes of base point preserving digital continuous functions from W to X by the digital homotopy class of the composite ([21], Definition 3.7):(12)f⊞g:W⟶ΔWW×W⟶f×gX×X⟶mXX.
LetνX:X⟶X be a digital homotopy inverse on a pointed digital Hopf space X:=X,x0,kX. Then, every element of the set W,w0,X,x0 with the binary operation “⊞” is invertible, and the inverse f−1 of f is given by(13)f−1=νX∘f,where f:W⟶X is any digital kW,kX-continuous function; see [21] (Theorem 3.24) for more details.Definition 16 (see [21]).
A digital multiplicationmX:X×X⟶X on a pointed digital Hopf space X:=X,x0,kX is said to be digital homotopy commutative if the diagram in Figure 4 is digital homotopy commutative. Here,(14)TX×X:X×X⟶X×Xis the switching function. A digital Hopf space X=X,x0,kX with a digital homotopy commutative multiplication mX:X×X⟶X is called a digital commutative Hopf space.
A topological Hopf group [30] is a pointed homotopy associative Hopf space with a homotopy inverse satisfying the group axioms up to homotopy. We remark that any topological group is a topological Hopf group. We consider a digital version of a topological Hopf group as follows.Figure 4
A commutative diagram.Definition 17.
A digital Hopf group is a digital Hopf spaceX,x0,kX furnished with an associative multiplication mX:X×X⟶X and an inverse νX:X⟶X up to digital homotopy.Example 1.
Lete=1,0,a=0,1,b=−1,0, and c=0,−1 be the elements of ℤ2, and let X:=e,a,b,c⊊ℤ2 be a digital image with the l1-adjacency relation defined on ℤ2. If we define a digital multiplication(15)mX=⊙:X×X⟶X,by the rule of Table 1, then the pointed digital image X,e,l1 becomes a group as well as a digital Hopf group; see also [20] (Example 3.8).
The following results show that a digital Hopf space, a digital associative Hopf space, a digital Hopf group, and a digital commutative Hopf space are unique up to the digital homotopy type; that is, there is only one possible digital Hopf structure up to digital homotopy on the underlying digital image.Table 1
A digital multiplicationmX=⊙:X×X⟶X.
Binary operationeabceeabcaabcebbceacceabTheorem 1.
LetX:=X,x0,kX be a digital Hopf group with a multiplication mX:X×X⟶X and an inverse νX:X⟶X up to digital homotopy, and let Y:=Y,y0,kY be any digital image. If f:X⟶Y is a kX,kY-homotopy equivalence, then Y,y0,kY becomes a digital Hopf group.Proof.
Sincef:X⟶Y is a kX,kY-homotopy equivalence, there exists a kY,kX-continuous function g:Y⟶X such that(16)g∘f≃kX,kX1X,f∘g≃kY,kY1Y.
If we define a function(17)mY:Y×Y⟶Yto be the composite(18)mY:Y×Y⟶g×gX×X⟶mXX⟶fY
of digital functions, thenmY:Y×Y⟶Y is a digital multiplication on Y:=Y,y0,kY. Indeed, since the diagram in Figure 5 is digital homotopy commutative, we obtain(19)mY∘cy0×1Y∘ΔY≃kY,kYf∘mX∘g×g∘cy0×1Y∘ΔY=f∘mX∘cx0×1X∘ΔX∘g≃kY,kYf∘1X∘g≃kY,kYf∘g≃kY,kY1Y,and similarly(20)mY∘1Y×cy0∘ΔY≃kY,kY1Y.
Therefore,Y,y0,kY is a digital Hopf space.
Secondly, we show that the diagram in Figure6 is digital homotopy commutative. Indeed, we obtain(21)mY∘f×f≃kX×X,kYf∘mX∘g×g∘f×f=f∘mX∘g∘f×g∘f≃kX×X,kYf∘mX∘1X×1X=f∘mX,and similarly(22)mX∘g×g≃kY×Y,kXg∘mY.
Thus,f:X⟶Y and g:Y⟶X are digital Hopf functions.
Thirdly, we note that five faces out of the six faces of the hexahedron in Figure7 are digital homotopy commutative, with the exception of the front face colored in blue. Indeed, using the homotopy relation (20), we can show that the left-hand face of the hexahedron is digital homotopy commutative as follows.(23)1X×mX∘g×g×g=g×mX∘g×g≃kY×Y×Y,kX×Xg×g∘mY=g×g1Y×mY,and the same applies for the other four faces. For the back face to be homotopy commutative, we need to be associative. The reader should be reminded that this is from Definition 17. For the right face, refer to the homotopy at (21). The top face is from the previous page and the homotopy between f∘g and 1Y. In addition, the bottom face is from the definition of mY.
We now prove that the front face is also commutative up to digital homotopy as follows:(24)mY∘mY×1Y≃f∘mX∘g×g∘f∘mX∘g×g×1Y≃f∘mX∘g×g∘f∘mX∘g×g×f∘g=f∘mX∘g×g∘f×f∘mX×1X∘g×g×g=f∘mX∘g×g∘f×f∘mX×1X∘g×g×g=f∘mX∘g∘f×g∘f°mX×1X∘g×g×g≃f∘mX∘1X×1X∘mX×1X∘g×g×g=f∘mX∘mX×1X∘g×g×g≃f∘mX∘1X×mX∘g×g×g=f∘mX∘1X×1X∘1X×mX∘g×g×g≃f∘mX∘g∘f×g∘f∘1X×mX∘g×g×g=f∘mX∘g×g∘f×f∘1X×mX∘g×g×g=f∘mX∘g×g∘f×f∘1X×mX∘g×g×g=f∘mX∘g×g∘f∘g×f∘mX∘g×g≃f∘mX∘g×g∘1Y×f∘mX∘g×g≃mY∘1Y×mY,where ≃ means the digital kY×Y×Y,kY-homotopy. Thus, mY:Y×Y⟶Y is an associative multiplication up to digital homotopy.
We finally define akY,kY-continuous function(25)νY:Y⟶Y,by the digital homotopy class of the compositions(26)νY:Y⟶gX⟶νXX⟶fY,
so that the diagram in Figure8 commutes up to digital homotopy. Indeed, since the diagram in Figure 9 is strictly commutative, by using the digital homotopy inverse νX, we have(27)mY∘1Y×νY∘ΔY≃f∘mX∘g×g∘1Y×f∘νX∘g∘ΔY=f∘mX∘g×g∘f∘νX∘g∘ΔY≃f∘mX∘g×1X∘νX∘g∘ΔY=f∘mX∘g×νX∘g∘ΔY=f∘mX∘1X×νX∘g×g∘ΔY=f∘mX∘1X×νX∘ΔX∘g≃f∘cx0∘g=cy0,and similarly(28)mY∘νY×1Y∘ΔY≃cy0,where ≃ means the digital kY,kY-homotopy. Therefore,(29)νY:Y⟶Yis a digital homotopy inverse.
Using the relations from (19) through (28), Y,y0,kY is a digital Hopf group, as required.⃞Figure 5
A commutative diagram.Figure 6
A commutative diagram.Figure 7
A commutative diagram.Figure 8
A commutative diagram.Figure 9
A commutative diagram.Example 2.
LetX be the digital Hopf group in Example 1 and let(30)Y,e,l1≔X∪2,0∈ℤ2,e,l1.
Then, we can show that(31)X≃kX,kYYand that the digital image Y,e,l1 becomes a digital Hopf group, where l1=4. Indeed, let f:X⟶Y be an inclusion and let g:Y⟶X be a function given by(32)gy=y,ify∈X,e,ify=2,0.
Then, it can be seen thatf and g are kX,kY- and kY,kX-continuous functions, respectively, and that(33)g∘f=1X:X⟶X.
LetH:Y×0,1ℤ⟶Y be the function given by(34)Hy,t=y,ift=0,f∘gy,ift=1,for all y∈Y. Then, we see that H is a kY×0,1ℤ,kY-continuous function and that(35)H:f∘g≃kY×0,1ℤ,kY1Y,where(36)kY×0,1ℤ=NP2kY,l1with l1=2. By (33) and (35), we have the result.
A binary operation is said to be commutative if changing the order of the operands does not change the result. It is one of the fundamental properties of many binary operations in mathematics, and many mathematical proofs depend on it. We now show that digital homotopy commutativity is digital homotopy invariant.Theorem 2.
LetX:=X,x0,kX be a digital commutative Hopf space with a multiplication mX:X×X⟶X up to digital homotopy, and let Y:=Y,y0,kY be any digital image. If f:X⟶Y is a kX,kY-homotopy equivalence, then Y becomes a digital commutative Hopf space.Proof.
We need to show that the diagram in Figure10 is digital homotopy commutative. Since the diagram in Figure 11 is strictly commutative, using the digital homotopy commutativity, we have(37)mY∘TY×Y≃f∘mX∘g×g∘TY×Y=f∘mX∘TX×X∘g×g≃f∘mX∘g×g≃mY,where ≃ means the digital kY×Y,kY-homotopy. Thus, Y:=Y,y0,kY is a digital commutative Hopf space.
As usual, we define the digital projections as follows.Figure 10
A commutative diagram.Figure 11
A commutative diagram.Definition 18 (see [32]).
The first and second digital projections(38)p1:X×Y⟶X,p2:X×Y⟶Yare defined by(39)p1x,y=x,p2x,y=yas the kX×Y,kX-continuous and kX×Y,kY-continuous functions, respectively.
We note that the digital products have the following property.Remark 2.
LetX:=X,x0,kX and Y:=Y,y0,kY be digital images. Then, for every digital image Z:=Z,z0,kZ with digital continuous functions(40)q1:Z⟶X,q2:Z⟶Y,
there exists a uniquekZ,kX×Y-continuous function φ:Z⟶X×Y given by(41)φz=q1z,q2z,such that the diagram in Figure 12 is commutative. The kZ,kX×Y-continuous function φ:Z⟶X×Y is denoted by q1,q2. In particular, if X=Y=Z, and q1 and q2 are the identity function 1X on X, then(42)φ=1X,1X:X⟶X×Xis the digital diagonal function(43)ΔX:X⟶X×X,sending x to x×x.Figure 12
A commutative diagram.Definition 19 (see [30, 33]).
LetX:=X,x0 and Y:=Y,y0 be the digital images with ku,n-adjacency relations in ℤn. The digital wedge product X∨Y is given by(44)X∨Y=X×y0∪x0×Y⊂X×Y,with the ku,2n-adjacency relation whose base point is x0,y0 in ℤ2n. Here, the normal product adjacency in Definition 10 (or the Cartesian product in graph theory [34]) is assumed.
In particular, ifp1:X×X⟶X and p2:X×X⟶X are the digital projections in Definition 18, then we can consider the projections p⌣1:X∨X⟶X and p⌣2:X∨X⟶X defined by the restrictions p⌣1=p1|X∨X and p⌣2=p2|X∨X; that is,(45)p⌣1x,x0=x=p⌣2x0,x,p⌣1x0,x=x0=p⌣2x,x0,for all x,x0 and x0,x in the digital wedge product X∨X.Lemma 1.
For every digital imageX:=X,x0, the function(46)p⌣1,p⌣2:X∨X⟶X×X,is equal to the inclusion function(47)i:X∨X⟶X×X,of X∨X into X×X.Proof.
Since both digital continuous functionsp⌣1,p⌣2 and i make the two diagrams in Figures 13 and 14 commute, by uniqueness, we have(48)p⌣1,p⌣2=i:X∨X⟶X×X,as required.Figure 13
A commutative diagram.Figure 14
A commutative diagram.Definition 20.
The digital folding map(49)∇X:X∨X⟶X,is a digital kX∨X,kX-continuous function defined by(50)∇Xx,x0=x=∇Xx0,x,for all x,x0 and x0,x in X∨X.
We now find an equivalent condition for a digital image to be a digital Hopf space as follows.Lemma 2.
A digital imageX:=X,x0,kX is a digital Hopf space with a digital multiplication(51)mX:X×X⟶X,
if and only if the diagram in Figure15 is commutative up to digital homotopy.Figure 15
A commutative diagram.Proof.
Let(52)j1,j2:X⟶X∨X,be the digital continuous functions given by(53)j1x=x,x0,j2x=x0,x,respectively. Then, it can be shown that the diagram in Figure 16 is a commutative diagram up to digital homotopy. Indeed, we have(54)mX∘i∘j1−=mX∘i−,x0=mX−,x0=mX∘1X×cx0−,−=mX∘1X×cx0∘ΔX−≃kX,kX1X−,that is,(55)mX∘i−,x0≃kX,kX1X−=∇X−,x0,and similarly(56)mX∘ix0,−≃kX,kX1X−=∇Xx0,−.
Conversely, we have(57)1X=∇X∘j1≃kX,kXmX∘i∘j1=mX∘1X×cx0∘ΔX,1X=∇X∘j2≃kX,kXmX∘i∘j2=mX∘cx0×1X∘ΔX,
that is,(58)mX:X×X⟶X,is a digital multiplication as required.□
LetA:=A,a0,kA,B:=B,b0,kB, and X:=X,x0,kX be the digital images, and let f:A⟶X and g:B⟶X be kA,kX-continuous and kB,kX-continuous functions, respectively. Then, we have the kA∨B,kX∨X-continuous function(59)f∨g:A∨B⟶X∨X,defined by(60)f∨ga,b=fa,gb,where each ordered pair should contain at least one entry from the base points a0, b0, and x0 of A, B, and X, respectively.Figure 16
A commutative diagram.Theorem 3.
The digital imageX:=X,x0,kX is a digital Hopf space if and only if for any kA,kX-continuous function f:A⟶X and any kB,kX-continuous function g:B⟶X, there exists a kA×B,kX-continuous function s:A×B⟶X such that(61)∇X∘f∨g≃kA∨B,kXs∘j,where j:A∨B⟶A×B is the inclusion function of A∨B into A×B.Proof.
LetmX:X×X⟶X be the digital multiplication on X:=X,x0,kX. Then, we can define a kA×B,kX-continuous function(62)s:A×B⟶X,by(63)s=mX∘f×g:A×B⟶f×gX×X⟶mXX.
We thus have(64)s∘j−,b0≃kA∨B,kXmX∘f×g∘j−,b0=mX∘f×g−,b0=mX∘f−,x0=mX∘1X×cx0f−,f−=mX∘1X×cx0∘ΔXf−≃kX,kX1Xf−=f−=∇X∘f−,x0=∇X∘f∨g−,b0,for all −,b0 in A∨B, and similarly(65)s∘ja0,−≃kA∨B,kX∇X∘f∨ga0,−,for all a0,− in A∨B.
Conversely, for a digital folding map(66)∇X:X∨X⟶X,
there is akX×X,kX-continuous function mX:X×X⟶X such that the diagram in Figure 17 is digital homotopy commutative. By Lemma 2, X,x0,kX becomes a digital Hopf space furnished with the digital multiplication mX.Figure 17
A commutative diagram.


## 3.1. Notation

We now fix the notations in this paper:(i)
cx0:X⟶X and cy0:Y⟶Y are the constant functions whose values are x0 and y0, respectively.(ii)
kX×Y is the normal product adjacency on X×Y; that is, kX×Y:=NPkX,kY, and in particular kX×X:=NPkX,kX.(iii)
kX×Y×Z is the generalized normal product adjacency on X×Y×Z; that is, kX×Y×Z:=NP3kX,kY,kZ, and in particular kX×X×X:=NP3kX,kX,kX.(iv)
TX×Y:X×Y⟶Y×X is the switching function sending x,y to y,x.(v)
ΔX:X⟶X×X is the diagonal function sending x to x×x as a kX,kX×X-continuous function.We now consider the digital versions of the Hopf space, homotopy associativity, homotopy commutativity, and Hopf group in algebraic topology as follows—see [15, 30, 31] for the terminologies in classical homotopy theory.Definition 12 (see [18]).
A digital imageX:=X,x0,kX with a kX-adjacency relation is called a digital Hopf space if there is a kX×X,kX-continuous function,(7)mX:X×X⟶X,
such that the diagram in Figure1 is digital homotopy commutative; that is,(8)mX∘cx0×1X∘ΔX≃kX,kX1X≃kX,kXmX∘1X×cx0∘ΔX.
In this case, thekX×X,kX-continuous function(9)mX:X×X⟶Xis called a digital multiplication on X,x0,kX, and the constant function cx0 is said to be a digital homotopy identity.Figure 1
A commutative diagram.Definition 13 (see [21]).
LetX:=X,x0,kX and Y:=Y,y0,kY be pointed digital Hopf spaces. Then, a kX,kY-continuous function f:X⟶Y is said to be a digital Hopf function if(10)f∘mX≃kX×X,kYmY∘f×f.Definition 14 (see [21]).
A digital multiplicationmX:X×X⟶X on a pointed digital Hopf space X:=X,x0,kX is said to be digital homotopy associative if the diagram in Figure 2 commutes up to digital homotopy. A digital Hopf space X=X,x0,kX with a digital homotopy associative multiplication mX:X×X⟶X is called a digital associative Hopf space.Figure 2
A commutative diagram.Definition 15 (see [21]).
LetX:=X,x0,kX be a digital Hopf space with a digital multiplication mX:X×X⟶X. A kX,kX-continuous function(11)νX:X⟶Xis called a digital homotopy inverse if the diagram in Figure 3 commutes up to digital homotopy.Figure 3
A commutative diagram.Remark 1.
We should be careful with the term ‘inverse’ at this moment because an inverse is usually understood to be an element of a group or more generally of a set with a binary operation. The inverse in Definition15 is a function that produces the homotopy inverse elements.
We note that, in general, there are many kinds of digital homotopy inverses and that one of the digital homotopy inversesνX:X⟶X can be constructed as a kX,kX-continuous function so that the triangles in Definition 15 are digital homotopy commutative. More precisely, we let f and g be digital homotopy classes represented by kW,kX-continuous functions f,g:W⟶X, respectively. If W:=W,w0,kW is a pointed digital image, and if X:=X,x0,kX is a pointed digital Hopf space with a digital multiplication mX:X×X⟶X, then we define a binary operation “⊞” on the set W,w0,X,x0 of all digital homotopy classes of base point preserving digital continuous functions from W to X by the digital homotopy class of the composite ([21], Definition 3.7):(12)f⊞g:W⟶ΔWW×W⟶f×gX×X⟶mXX.
LetνX:X⟶X be a digital homotopy inverse on a pointed digital Hopf space X:=X,x0,kX. Then, every element of the set W,w0,X,x0 with the binary operation “⊞” is invertible, and the inverse f−1 of f is given by(13)f−1=νX∘f,where f:W⟶X is any digital kW,kX-continuous function; see [21] (Theorem 3.24) for more details.Definition 16 (see [21]).
A digital multiplicationmX:X×X⟶X on a pointed digital Hopf space X:=X,x0,kX is said to be digital homotopy commutative if the diagram in Figure 4 is digital homotopy commutative. Here,(14)TX×X:X×X⟶X×Xis the switching function. A digital Hopf space X=X,x0,kX with a digital homotopy commutative multiplication mX:X×X⟶X is called a digital commutative Hopf space.
A topological Hopf group [30] is a pointed homotopy associative Hopf space with a homotopy inverse satisfying the group axioms up to homotopy. We remark that any topological group is a topological Hopf group. We consider a digital version of a topological Hopf group as follows.Figure 4
A commutative diagram.Definition 17.
A digital Hopf group is a digital Hopf spaceX,x0,kX furnished with an associative multiplication mX:X×X⟶X and an inverse νX:X⟶X up to digital homotopy.Example 1.
Lete=1,0,a=0,1,b=−1,0, and c=0,−1 be the elements of ℤ2, and let X:=e,a,b,c⊊ℤ2 be a digital image with the l1-adjacency relation defined on ℤ2. If we define a digital multiplication(15)mX=⊙:X×X⟶X,by the rule of Table 1, then the pointed digital image X,e,l1 becomes a group as well as a digital Hopf group; see also [20] (Example 3.8).
The following results show that a digital Hopf space, a digital associative Hopf space, a digital Hopf group, and a digital commutative Hopf space are unique up to the digital homotopy type; that is, there is only one possible digital Hopf structure up to digital homotopy on the underlying digital image.Table 1
A digital multiplicationmX=⊙:X×X⟶X.
Binary operationeabceeabcaabcebbceacceabTheorem 1.
LetX:=X,x0,kX be a digital Hopf group with a multiplication mX:X×X⟶X and an inverse νX:X⟶X up to digital homotopy, and let Y:=Y,y0,kY be any digital image. If f:X⟶Y is a kX,kY-homotopy equivalence, then Y,y0,kY becomes a digital Hopf group.Proof.
Sincef:X⟶Y is a kX,kY-homotopy equivalence, there exists a kY,kX-continuous function g:Y⟶X such that(16)g∘f≃kX,kX1X,f∘g≃kY,kY1Y.
If we define a function(17)mY:Y×Y⟶Yto be the composite(18)mY:Y×Y⟶g×gX×X⟶mXX⟶fY
of digital functions, thenmY:Y×Y⟶Y is a digital multiplication on Y:=Y,y0,kY. Indeed, since the diagram in Figure 5 is digital homotopy commutative, we obtain(19)mY∘cy0×1Y∘ΔY≃kY,kYf∘mX∘g×g∘cy0×1Y∘ΔY=f∘mX∘cx0×1X∘ΔX∘g≃kY,kYf∘1X∘g≃kY,kYf∘g≃kY,kY1Y,and similarly(20)mY∘1Y×cy0∘ΔY≃kY,kY1Y.
Therefore,Y,y0,kY is a digital Hopf space.
Secondly, we show that the diagram in Figure6 is digital homotopy commutative. Indeed, we obtain(21)mY∘f×f≃kX×X,kYf∘mX∘g×g∘f×f=f∘mX∘g∘f×g∘f≃kX×X,kYf∘mX∘1X×1X=f∘mX,and similarly(22)mX∘g×g≃kY×Y,kXg∘mY.
Thus,f:X⟶Y and g:Y⟶X are digital Hopf functions.
Thirdly, we note that five faces out of the six faces of the hexahedron in Figure7 are digital homotopy commutative, with the exception of the front face colored in blue. Indeed, using the homotopy relation (20), we can show that the left-hand face of the hexahedron is digital homotopy commutative as follows.(23)1X×mX∘g×g×g=g×mX∘g×g≃kY×Y×Y,kX×Xg×g∘mY=g×g1Y×mY,and the same applies for the other four faces. For the back face to be homotopy commutative, we need to be associative. The reader should be reminded that this is from Definition 17. For the right face, refer to the homotopy at (21). The top face is from the previous page and the homotopy between f∘g and 1Y. In addition, the bottom face is from the definition of mY.
We now prove that the front face is also commutative up to digital homotopy as follows:(24)mY∘mY×1Y≃f∘mX∘g×g∘f∘mX∘g×g×1Y≃f∘mX∘g×g∘f∘mX∘g×g×f∘g=f∘mX∘g×g∘f×f∘mX×1X∘g×g×g=f∘mX∘g×g∘f×f∘mX×1X∘g×g×g=f∘mX∘g∘f×g∘f°mX×1X∘g×g×g≃f∘mX∘1X×1X∘mX×1X∘g×g×g=f∘mX∘mX×1X∘g×g×g≃f∘mX∘1X×mX∘g×g×g=f∘mX∘1X×1X∘1X×mX∘g×g×g≃f∘mX∘g∘f×g∘f∘1X×mX∘g×g×g=f∘mX∘g×g∘f×f∘1X×mX∘g×g×g=f∘mX∘g×g∘f×f∘1X×mX∘g×g×g=f∘mX∘g×g∘f∘g×f∘mX∘g×g≃f∘mX∘g×g∘1Y×f∘mX∘g×g≃mY∘1Y×mY,where ≃ means the digital kY×Y×Y,kY-homotopy. Thus, mY:Y×Y⟶Y is an associative multiplication up to digital homotopy.
We finally define akY,kY-continuous function(25)νY:Y⟶Y,by the digital homotopy class of the compositions(26)νY:Y⟶gX⟶νXX⟶fY,
so that the diagram in Figure8 commutes up to digital homotopy. Indeed, since the diagram in Figure 9 is strictly commutative, by using the digital homotopy inverse νX, we have(27)mY∘1Y×νY∘ΔY≃f∘mX∘g×g∘1Y×f∘νX∘g∘ΔY=f∘mX∘g×g∘f∘νX∘g∘ΔY≃f∘mX∘g×1X∘νX∘g∘ΔY=f∘mX∘g×νX∘g∘ΔY=f∘mX∘1X×νX∘g×g∘ΔY=f∘mX∘1X×νX∘ΔX∘g≃f∘cx0∘g=cy0,and similarly(28)mY∘νY×1Y∘ΔY≃cy0,where ≃ means the digital kY,kY-homotopy. Therefore,(29)νY:Y⟶Yis a digital homotopy inverse.
Using the relations from (19) through (28), Y,y0,kY is a digital Hopf group, as required.⃞Figure 5
A commutative diagram.Figure 6
A commutative diagram.Figure 7
A commutative diagram.Figure 8
A commutative diagram.Figure 9
A commutative diagram.Example 2.
LetX be the digital Hopf group in Example 1 and let(30)Y,e,l1≔X∪2,0∈ℤ2,e,l1.
Then, we can show that(31)X≃kX,kYYand that the digital image Y,e,l1 becomes a digital Hopf group, where l1=4. Indeed, let f:X⟶Y be an inclusion and let g:Y⟶X be a function given by(32)gy=y,ify∈X,e,ify=2,0.
Then, it can be seen thatf and g are kX,kY- and kY,kX-continuous functions, respectively, and that(33)g∘f=1X:X⟶X.
LetH:Y×0,1ℤ⟶Y be the function given by(34)Hy,t=y,ift=0,f∘gy,ift=1,for all y∈Y. Then, we see that H is a kY×0,1ℤ,kY-continuous function and that(35)H:f∘g≃kY×0,1ℤ,kY1Y,where(36)kY×0,1ℤ=NP2kY,l1with l1=2. By (33) and (35), we have the result.
A binary operation is said to be commutative if changing the order of the operands does not change the result. It is one of the fundamental properties of many binary operations in mathematics, and many mathematical proofs depend on it. We now show that digital homotopy commutativity is digital homotopy invariant.Theorem 2.
LetX:=X,x0,kX be a digital commutative Hopf space with a multiplication mX:X×X⟶X up to digital homotopy, and let Y:=Y,y0,kY be any digital image. If f:X⟶Y is a kX,kY-homotopy equivalence, then Y becomes a digital commutative Hopf space.Proof.
We need to show that the diagram in Figure10 is digital homotopy commutative. Since the diagram in Figure 11 is strictly commutative, using the digital homotopy commutativity, we have(37)mY∘TY×Y≃f∘mX∘g×g∘TY×Y=f∘mX∘TX×X∘g×g≃f∘mX∘g×g≃mY,where ≃ means the digital kY×Y,kY-homotopy. Thus, Y:=Y,y0,kY is a digital commutative Hopf space.
As usual, we define the digital projections as follows.Figure 10
A commutative diagram.Figure 11
A commutative diagram.Definition 18 (see [32]).
The first and second digital projections(38)p1:X×Y⟶X,p2:X×Y⟶Yare defined by(39)p1x,y=x,p2x,y=yas the kX×Y,kX-continuous and kX×Y,kY-continuous functions, respectively.
We note that the digital products have the following property.Remark 2.
LetX:=X,x0,kX and Y:=Y,y0,kY be digital images. Then, for every digital image Z:=Z,z0,kZ with digital continuous functions(40)q1:Z⟶X,q2:Z⟶Y,
there exists a uniquekZ,kX×Y-continuous function φ:Z⟶X×Y given by(41)φz=q1z,q2z,such that the diagram in Figure 12 is commutative. The kZ,kX×Y-continuous function φ:Z⟶X×Y is denoted by q1,q2. In particular, if X=Y=Z, and q1 and q2 are the identity function 1X on X, then(42)φ=1X,1X:X⟶X×Xis the digital diagonal function(43)ΔX:X⟶X×X,sending x to x×x.Figure 12
A commutative diagram.Definition 19 (see [30, 33]).
LetX:=X,x0 and Y:=Y,y0 be the digital images with ku,n-adjacency relations in ℤn. The digital wedge product X∨Y is given by(44)X∨Y=X×y0∪x0×Y⊂X×Y,with the ku,2n-adjacency relation whose base point is x0,y0 in ℤ2n. Here, the normal product adjacency in Definition 10 (or the Cartesian product in graph theory [34]) is assumed.
In particular, ifp1:X×X⟶X and p2:X×X⟶X are the digital projections in Definition 18, then we can consider the projections p⌣1:X∨X⟶X and p⌣2:X∨X⟶X defined by the restrictions p⌣1=p1|X∨X and p⌣2=p2|X∨X; that is,(45)p⌣1x,x0=x=p⌣2x0,x,p⌣1x0,x=x0=p⌣2x,x0,for all x,x0 and x0,x in the digital wedge product X∨X.Lemma 1.
For every digital imageX:=X,x0, the function(46)p⌣1,p⌣2:X∨X⟶X×X,is equal to the inclusion function(47)i:X∨X⟶X×X,of X∨X into X×X.Proof.
Since both digital continuous functionsp⌣1,p⌣2 and i make the two diagrams in Figures 13 and 14 commute, by uniqueness, we have(48)p⌣1,p⌣2=i:X∨X⟶X×X,as required.Figure 13
A commutative diagram.Figure 14
A commutative diagram.Definition 20.
The digital folding map(49)∇X:X∨X⟶X,is a digital kX∨X,kX-continuous function defined by(50)∇Xx,x0=x=∇Xx0,x,for all x,x0 and x0,x in X∨X.
We now find an equivalent condition for a digital image to be a digital Hopf space as follows.Lemma 2.
A digital imageX:=X,x0,kX is a digital Hopf space with a digital multiplication(51)mX:X×X⟶X,
if and only if the diagram in Figure15 is commutative up to digital homotopy.Figure 15
A commutative diagram.Proof.
Let(52)j1,j2:X⟶X∨X,be the digital continuous functions given by(53)j1x=x,x0,j2x=x0,x,respectively. Then, it can be shown that the diagram in Figure 16 is a commutative diagram up to digital homotopy. Indeed, we have(54)mX∘i∘j1−=mX∘i−,x0=mX−,x0=mX∘1X×cx0−,−=mX∘1X×cx0∘ΔX−≃kX,kX1X−,that is,(55)mX∘i−,x0≃kX,kX1X−=∇X−,x0,and similarly(56)mX∘ix0,−≃kX,kX1X−=∇Xx0,−.
Conversely, we have(57)1X=∇X∘j1≃kX,kXmX∘i∘j1=mX∘1X×cx0∘ΔX,1X=∇X∘j2≃kX,kXmX∘i∘j2=mX∘cx0×1X∘ΔX,
that is,(58)mX:X×X⟶X,is a digital multiplication as required.□
LetA:=A,a0,kA,B:=B,b0,kB, and X:=X,x0,kX be the digital images, and let f:A⟶X and g:B⟶X be kA,kX-continuous and kB,kX-continuous functions, respectively. Then, we have the kA∨B,kX∨X-continuous function(59)f∨g:A∨B⟶X∨X,defined by(60)f∨ga,b=fa,gb,where each ordered pair should contain at least one entry from the base points a0, b0, and x0 of A, B, and X, respectively.Figure 16
A commutative diagram.Theorem 3.
The digital imageX:=X,x0,kX is a digital Hopf space if and only if for any kA,kX-continuous function f:A⟶X and any kB,kX-continuous function g:B⟶X, there exists a kA×B,kX-continuous function s:A×B⟶X such that(61)∇X∘f∨g≃kA∨B,kXs∘j,where j:A∨B⟶A×B is the inclusion function of A∨B into A×B.Proof.
LetmX:X×X⟶X be the digital multiplication on X:=X,x0,kX. Then, we can define a kA×B,kX-continuous function(62)s:A×B⟶X,by(63)s=mX∘f×g:A×B⟶f×gX×X⟶mXX.
We thus have(64)s∘j−,b0≃kA∨B,kXmX∘f×g∘j−,b0=mX∘f×g−,b0=mX∘f−,x0=mX∘1X×cx0f−,f−=mX∘1X×cx0∘ΔXf−≃kX,kX1Xf−=f−=∇X∘f−,x0=∇X∘f∨g−,b0,for all −,b0 in A∨B, and similarly(65)s∘ja0,−≃kA∨B,kX∇X∘f∨ga0,−,for all a0,− in A∨B.
Conversely, for a digital folding map(66)∇X:X∨X⟶X,
there is akX×X,kX-continuous function mX:X×X⟶X such that the diagram in Figure 17 is digital homotopy commutative. By Lemma 2, X,x0,kX becomes a digital Hopf space furnished with the digital multiplication mX.Figure 17
A commutative diagram.


## 4. A Difference between Ordinary Co-Hopf Spaces and Digital Co-Hopf Spaces

We now consider the digital version of a co-Hopf space in algebraic topology as follows; see [15, 30, 31] for the terminology in classical homotopy theory.Definition 21.
A digital imageC:=C,c0,kC with a kC-adjacency relation is called a digital co-Hopf space if there is a kC,kC∨C-continuous function(67)φC:C⟶C∨C,
such that the two triangles in Figure18 are digital homotopy commutative, where 1C is the identity function on C, and(68)π1,π2:C∨C⟶C,are the first projection and the second projection, respectively. In this case, the kC,kC∨C-continuous function φC:C⟶C∨C above is called a digital comultiplication on C,c0,kC.
Letf and g be digital homotopy classes represented by kC,kY-continuous functions f,g:C⟶Y, respectively. If C:=C,c0,kC is a pointed digital co-Hopf space with a digital comultiplication φC:C⟶C∨C, and if Y:=Y,y0,kY is a digital image, then we also define a binary operation “⊛” on the set C,c0,Y,y0 of all digital homotopy classes of base point preserving digital continuous functions from C to Y by the digital homotopy class of the composite(69)f⊛g:C⟶φCC∨C⟶f∨gY∨Y⟶∇YY,that is,(70)f⊛g=f⊛g=∇Y∘f∨g∘φC,where ∇Y:Y∨Y⟶Y is the digital folding map. We note that the binary operation “⊛” makes the set C,c0,Y,y0 into an algebraic set just like the aforementioned binary operation “⊞” in Section 3. Note that there are many kinds of homotopy comultiplications with many distinctive properties in classical homotopy theory; see [2], and [3] (Definitions 3.1, 3.5, and 3.12) for further details.
We note that a topological co-Hopf space needs not be contractible even though it is a finite dimensional space. Take the spheres and the finite wedge products of spheres as examples.
The following theorem shows that there is a disparity or a difference between ordinary topological co-Hopf spaces and digital co-Hopf spaces.Figure 18
A commutative diagram.Theorem 4.
Any digital co-Hopf space is digitally contractible.Proof.
LetC:=C,c0,kC be a digital co-Hopf space together with a digital comultiplication.(71)φC:C⟶C∨C.
Ifx is any element of C, then(72)φCx=c,d,where either c or d is the base point c0 of C. We thus have(73)1C−≃kC,kCπ1∘φC−=π1c,d=c,1C−≃kC,kCπ2∘φC−=π2c,d=d,in the digital co-Hopf space C:=C,c0,kC. If c=c0, then(74)1C−≃kC,kCcc0−=c0,where cc0:C⟶C is the constant function at c0. Similarly, if d=c0, then(75)1C−≃kC,kCcc0−.
Thus,C is a contractible digital image.□
Theorem4 asserts that the set S of all digital homotopy classes of digital comultiplications φC:C⟶C∨C is the set consisting of a single element; that is, S=1.
A topological group is a groupG together with a topology on G such that the binary operation and the inverse function are continuous functions with respect to the topology on G. A Hopf group is a generalization of the topological group which is obtained by replacing the strict equality by the classical homotopy relation.
We now replace the digital homotopy “≃kC,kC” in Definition 21 by the strict equality “=” as follows.Definition 22.
A digital imageC:=C,c0,kC along with kC-adjacency relation is called a strictly digital co-Hopf space if there is a kC,kC∨C-continuous function φC:C⟶C∨C such that π1°φC=1C and π2°φC=1C.Remark 3.
By Theorem4 and by restricting the digital homotopy to the strict equality, we can see that any strictly digital co-Hopf space is a set consisting of a single element as the base point; see also [35] (Lemma 3.8).


## 5. Summary

We have investigated the fundamental notions of digital Hopf spaces, digital Hopf groups, digital co-Hopf spaces, and digital co-Hopf groups based on digital images. We have proven that digital Hopf spaces, digital associative Hopf spaces, digital Hopf groups, and digital commutative Hopf spaces are unique up to the digital homotopy type; that is, there is only one possible digital Hopf structure up to digital homotopy on the underlying digital image.We have established a necessary and sufficient condition for a digital image to be a digital Hopf space and investigated the relationship or the difference between ordinary topological co-Hopf spaces and digital co-Hopf spaces by showing that any digital co-Hopf space is digitally contractible. We have also noticed that any strictly digital co-Hopf space is a set having only one element as a base point in which the result is exactly the same as the Eckmann–Hilton dual of a topological group as an application in computer science focusing on the deep-learning methods in imaging science.



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