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.
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 if is a group-like space or if has a cogroup structure, then we can construct a group structure on the set of homotopy classes of continuous functions from to as inherited from various functors.
A Hopf space is a pointed topological space along with a base point preserving continuous function , called a multiplication on , for which the constant map 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 sphere . 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 pair consisting of a pointed space and a continuous function , called a comultiplication on , preserving the base point is called a co-Hopf space if and are homotopic to the identity function on , where and are the projections 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 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 functor and a contravariant functor from the category of pointed digital images to the category of sets (resp., groups) and functions (resp., group homomorphisms), where and are pointed digital images with -adjacency and -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 set of digital homotopy classes of base point preserving digital continuous functions from to if the pointed digital image with -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 the -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 Section 2, 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
Let be the set (or a topological space or the ring) of integers, and let be a finite subset of for . A binary digital image (or digital image for short) is a pair , where indicates some adjacency relation. For a positive integer with , an adjacency relation of a digital image in is defined as follows.
Definition 1 (see [23]). Let and be positive integers with . Then, two distinct points and in are -adjacent if(i)there exist at most distinct indices such that ;(ii)for all indices , if , then .The number is the cardinal number , of the set of elements of which is adjacent to a given element of , and is called an adjacency relation defined on . A digital image with an adjacency relation on is sometimes denoted as to designate the adjacency relation. For example,(i) in ;(ii), in ;(iii), , in ;(iv), , , in , and so on.
Definition 2 (see [24]). A digital image with an adjacency relation is called -connected if for any pair of distinct elements of , there is a set of distinct points satisfying that , and and are -adjacent for for .
Definition 3 (see [12, 24]). Let and be the digital images with -adjacency and -adjacency relations, respectively. A map between digital images is said to be -continuous if, for any -connected subset of , the image is also a -connected subset of .
Definition 4 (see [25]). For the nonnegative integers and with , a digital interval is a set of the typewith the 2-adjacency relation in the set of all nonnegative integers.
Definition 5 (see [12, 26]). Let be a digital image along with the -adjacency relation. Then, a (2 )-continuous function is called a digital -path in .
Definition 6 (see [12, 23]). Let and be digital images, and let be -continuous functions. Then, is said to be digital -homotopic to if there exist an integer and a function satisfying that(i) and for all ;(ii)the function , , which is given by is -continuous for all ;(iii)the function , , which is given by is -continuous for all .In this case, the function is called a digital -homotopy between and , denoted byIt is well known that the relation is an equivalence relation on the set of all -continuous functions from to . For a -continuous function , we denote bywhich is called the digital homotopy class represented by .
Definition 7 (see [12, 26]). A pointed digital image with -adjacency is a triple . In this situation, the element of is called the base point of . A base point preserving digital continuous function is a -continuous function with
Definition 8 (see [25]). A digital image is said to be digitally contractible if the identity function on is digitally null-homotopic; that is, it is digital -homotopic to the constant function at .
Definition 9 (see [26]). Let and be digital images. Then, a -continuous function is said to be a -homotopy equivalence if there exists a -continuous function such thatIn this case, we say that and have the same digital homotopy type which is denoted byConvention: In this paper, we work on the pointed digital category 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]). Let and be the pointed digital images. Then, the normal product adjacency on the Cartesian product,is defined as follows: if and , then the two elements and of are -adjacent in if and only if(i), and and are -adjacent;(ii) and are -adjacent, and ; or(iii) and are -adjacent, and and are -adjacent.Note that the normal product adjacency relation makes the Cartesian product into a digital image with the adjacency relation whose base point is .
Let and be the integers with , and let be a family of digital images for . Then, as a generalization of the normal product adjacency, we define the following.
Definition 11 (see [28, 29]). Let for . Then, the two elements and of are -adjacent if and only if(i)for at least 1 and at most indices , , and are -adjacent;(ii)for all other indices , .The adjacency relation is said to be the generalized normal product adjacency on the Cartesian product .
3.1. Notation
We now fix the notations in this paper:(i) and are the constant functions whose values are and , respectively.(ii) is the normal product adjacency on ; that is, , and in particular .(iii) is the generalized normal product adjacency on ; that is, , and in particular .(iv) is the switching function sending to .(v) is the diagonal function sending to as a -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 image with a -adjacency relation is called a digital Hopf space if there is a -continuous function,such that the diagram in Figure 1 is digital homotopy commutative; that is,In this case, the -continuous functionis called a digital multiplication on , and the constant function is said to be a digital homotopy identity.
Definition 13 (see [21]). Let and be pointed digital Hopf spaces. Then, a -continuous function is said to be a digital Hopf function if
Definition 14 (see [21]). A digital multiplication on a pointed digital Hopf space is said to be digital homotopy associative if the diagram in Figure 2 commutes up to digital homotopy. A digital Hopf space with a digital homotopy associative multiplication is called a digital associative Hopf space.
Definition 15 (see [21]). Let be a digital Hopf space with a digital multiplication . A -continuous functionis called a digital homotopy inverse if the diagram in Figure 3 commutes up to digital homotopy.
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 Definition 15 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 inverses can be constructed as a -continuous function so that the triangles in Definition 15 are digital homotopy commutative. More precisely, we let and be digital homotopy classes represented by -continuous functions , respectively. If is a pointed digital image, and if is a pointed digital Hopf space with a digital multiplication , then we define a binary operation “” on the set of all digital homotopy classes of base point preserving digital continuous functions from to by the digital homotopy class of the composite ([21], Definition 3.7):Let be a digital homotopy inverse on a pointed digital Hopf space . Then, every element of the set with the binary operation “” is invertible, and the inverse of is given bywhere is any digital -continuous function; see [21] (Theorem 3.24) for more details.
Definition 16 (see [21]). A digital multiplication on a pointed digital Hopf space is said to be digital homotopy commutative if the diagram in Figure 4 is digital homotopy commutative. Here,is the switching function. A digital Hopf space with a digital homotopy commutative multiplication 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.
Definition 17. A digital Hopf group is a digital Hopf space furnished with an associative multiplication and an inverse up to digital homotopy.
Example 1. Let , and be the elements of , and let be a digital image with the -adjacency relation defined on . If we define a digital multiplicationby the rule of Table 1, then the pointed digital image 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.
Theorem 1. Let be a digital Hopf group with a multiplication and an inverse up to digital homotopy, and let be any digital image. If is a -homotopy equivalence, then becomes a digital Hopf group.
Proof. Since is a -homotopy equivalence, there exists a -continuous function such thatIf we define a functionto be the compositeof digital functions, then is a digital multiplication on . Indeed, since the diagram in Figure 5 is digital homotopy commutative, we obtainand similarlyTherefore, is a digital Hopf space.
Secondly, we show that the diagram in Figure 6 is digital homotopy commutative. Indeed, we obtainand similarlyThus, and are digital Hopf functions.
Thirdly, we note that five faces out of the six faces of the hexahedron in Figure 7 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.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 and . In addition, the bottom face is from the definition of .
We now prove that the front face is also commutative up to digital homotopy as follows:where means the digital -homotopy. Thus, is an associative multiplication up to digital homotopy.
We finally define a -continuous functionby the digital homotopy class of the compositionsso that the diagram in Figure 8 commutes up to digital homotopy. Indeed, since the diagram in Figure 9 is strictly commutative, by using the digital homotopy inverse , we haveand similarlywhere means the digital -homotopy. Therefore,is a digital homotopy inverse.
Using the relations from (19) through (28), is a digital Hopf group, as required.⃞
Example 2. Let be the digital Hopf group in Example 1 and letThen, we can show thatand that the digital image becomes a digital Hopf group, where . Indeed, let be an inclusion and let be a function given byThen, it can be seen that and are - and -continuous functions, respectively, and thatLet be the function given byfor all . Then, we see that is a -continuous function and thatwherewith . 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. Let be a digital commutative Hopf space with a multiplication up to digital homotopy, and let be any digital image. If is a -homotopy equivalence, then becomes a digital commutative Hopf space.
Proof. We need to show that the diagram in Figure 10 is digital homotopy commutative. Since the diagram in Figure 11 is strictly commutative, using the digital homotopy commutativity, we havewhere means the digital -homotopy. Thus, is a digital commutative Hopf space.
As usual, we define the digital projections as follows.
Definition 18 (see [32]). The first and second digital projectionsare defined byas the -continuous and -continuous functions, respectively.
We note that the digital products have the following property.
Remark 2. Let and be digital images. Then, for every digital image with digital continuous functionsthere exists a unique -continuous function given bysuch that the diagram in Figure 12 is commutative. The -continuous function is denoted by . In particular, if , and and are the identity function on , thenis the digital diagonal functionsending to .
Definition 19 (see [30, 33]). Let and be the digital images with -adjacency relations in . The digital wedge product is given bywith the -adjacency relation whose base point is in . Here, the normal product adjacency in Definition 10 (or the Cartesian product in graph theory [34]) is assumed.
In particular, if and are the digital projections in Definition 18, then we can consider the projections and defined by the restrictions and ; that is,for all and in the digital wedge product .
Lemma 1. For every digital image , the functionis equal to the inclusion functionof into .
Proof. Since both digital continuous functions and make the two diagrams in Figures 13 and 14 commute, by uniqueness, we haveas required.
Definition 20. The digital folding mapis a digital -continuous function defined byfor all and in .
We now find an equivalent condition for a digital image to be a digital Hopf space as follows.
Lemma 2. A digital image is a digital Hopf space with a digital multiplicationif and only if the diagram in Figure 15 is commutative up to digital homotopy.
Proof. Letbe the digital continuous functions given byrespectively. Then, it can be shown that the diagram in Figure 16 is a commutative diagram up to digital homotopy. Indeed, we havethat is,and similarlyConversely, we havethat is,is a digital multiplication as required.□
Let , and be the digital images, and let and be -continuous and -continuous functions, respectively. Then, we have the -continuous functiondefined bywhere each ordered pair should contain at least one entry from the base points a0, b0, and x0 of A, B, and X, respectively.
Theorem 3. The digital image is a digital Hopf space if and only if for any -continuous function and any -continuous function , there exists a -continuous function such thatwhere is the inclusion function of into .
Proof. Let be the digital multiplication on . Then, we can define a -continuous functionbyWe thus havefor all in , and similarlyfor all in .
Conversely, for a digital folding mapthere is a -continuous function such that the diagram in Figure 17 is digital homotopy commutative. By Lemma 2, becomes a digital Hopf space furnished with the digital multiplication .
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 image with a -adjacency relation is called a digital co-Hopf space if there is a -continuous functionsuch that the two triangles in Figure 18 are digital homotopy commutative, where is the identity function on , andare the first projection and the second projection, respectively. In this case, the -continuous function above is called a digital comultiplication on .
Let and be digital homotopy classes represented by -continuous functions , respectively. If is a pointed digital co-Hopf space with a digital comultiplication and if is a digital image, then we also define a binary operation “” on the set of all digital homotopy classes of base point preserving digital continuous functions from to by the digital homotopy class of the compositethat is,where is the digital folding map. We note that the binary operation “” makes the set 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.
Theorem 4. Any digital co-Hopf space is digitally contractible.
Proof. Let be a digital co-Hopf space together with a digital comultiplication.If is any element of , thenwhere either or is the base point of . We thus havein the digital co-Hopf space . If , thenwhere is the constant function at . Similarly, if , thenThus, is a contractible digital image.□
Theorem 4 asserts that the set of all digital homotopy classes of digital comultiplications is the set consisting of a single element; that is, .
A topological group is a group together with a topology on such that the binary operation and the inverse function are continuous functions with respect to the topology on . 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 “” in Definition 21 by the strict equality “” as follows.
Definition 22. A digital image along with -adjacency relation is called a strictly digital co-Hopf space if there is a -continuous function such that and .
Remark 3. By Theorem 4 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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2018R1A2B6004407).