@ Appl. Gen. Topol. 22, no. 1 (2021), 183-192doi:10.4995/agt.2021.14542 © AGT, UPV, 2021 Digital homotopic distance between digital functions Ayşe Borat Bursa Technical University, Faculty of Engineering and Natural Scieces, Department of Mathe- matics, Bursa, Turkey (ayse.borat@btu.edu.tr) Communicated by V. Gregori Abstract In this paper, we define digital homotopic distance and give its relation with LS category of a digital function and of a digital image. Moreover, we introduce some properties of digital homotopic distance such as being digitally homotopy invariance. 2010 MSC: 55M30; 68U10. Keywords: homotopic distance; Lusternik Schnirelmann category; digital topology. 1. Introduction Macias-Virgos and Mosquera-Lois introduced homotopic distance between maps in [16]. One of the benefits of homotopic distance is to cover the concepts of Lusternik-Schnirelmann category (denoted by cat, see [8]) and topological complexity (denoted by TC, see [9]). If one computes the homotopic distance between some specific maps, then they end up with cat or TC of the domain of these maps. So there is a well-defined relation between homotopic distance, TC, cat and even the sectional category (secat) of some specific fibrations. We investigate an analog of this relationship in the digital setting. After constructing the digital homotopic distance between digital functions, one of our aims is to show the relation between digital homotopic distance and digital Received 27 October 2020 – Accepted 14 December 2020 http://dx.doi.org/10.4995/agt.2021.14542 A. Borat LS category of a digital image as defined in [2] and digital LS cat of a digital function as defined in [18]. Our another aim is to investigate how the adjacency relation effects the digital homotopic distance; see Theorem 3.5 and Theorem 3.6. 2. Background In this section, we recall some definitions and theorems from digital topology. A digital image X is a subset of Zn with an adjacency relation which is defined as follows. Definition 2.1 ([6]). Let p = (p1, . . . , pn) and q = (q1, . . . , qn) in Zn. Then for 1 ≤ ℓ ≤ n, p and q are said to be cℓ-adjacent if (i) there are at most ℓ indices i which satisfies |pi − qi| = 1 We would like to call attention to that digital LS category is also introduced with a different point of view using subdivisions by Lupton, Oprea, and Scoville in [14] and [15]. (ii) pj = qj for all other indices j satisfying |pi − qi| 6= 1 Here cℓ indicates the number of adjacent points in Zn. For example, c1 = 2 in Z; c1 = 4 and c2 = 8 in Z2. Also notice that adjacency relations are often denoted by Greek letters. Definition 2.2 ([4]). A digital interval which is a subset of Z can be defined as follows [a, b]Z = {n ∈ Z|a ≤ n ≤ b} where 2-adjacency is assumed. Definition 2.3 ([5]). Let (X, κ) and (Y, λ) be digital images. Given a function f : X → Y , if f(x) and f(y) are λ-adjacent or f(x) = f(y) in Y whenever x and y are κ-adjacent in X, then f is called (κ, λ)-continuous. Definition 2.4 ([5, 13]). Let f, g : X → Y be (κ, λ)-continuous functions. If there exist m ∈ Z+ and a function F : X × [0, m]Z → Y with the following conditions, then F is called a (κ, λ)-homotopy, and f and g are called (κ, λ)-homotopic in Y (denoted by f ≃κ,λ g). (i) For all x ∈ X, F(x, 0) = f(x) and F(x, m) = g(x). (ii) For all x ∈ X, the induced function Fx : [0, m]Z → Y , Fx(t) = F(x, t) is (2, λ)-continuous. (iii) For all t ∈ [0, m]Z, the induced function Ft : X → Y , Ft(x) = F(x, t) is (κ, λ)-continuous. Proposition 2.5 ([5]). If f : X → Y and g : Y → Z are (κ, λ)-continuous and (λ, γ)-continuous functions, respectively, then g ◦ f : X → Z is (κ, γ)- continuous. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 184 Digital homotopic distance between digital functions Definition 2.6 ([7]). If κ and λ are two adjacency relations on X, then we say κ dominates λ (denoted by κ ≥d λ) if for x, y ∈ X and if x, y are κ-adjacent imply x, y are λ-adjacent. Proposition 2.7 ([7]). Let κ, κ1, κ2 be adjacency relations on X and λ, λ1, λ2 be adjacency relations on Y . (a) If f : X → Y is (κ, λ1)-continuous and λ1 ≥d λ2, then f is (κ, λ2)- continuous. (b) If f : X → Y is (κ1, λ)-continuous and κ2 ≥d κ1, then f is (κ2, λ)- continuous. Definition 2.8 ([1, 17]). Let (X, κ) and (Y, λ) be digital images. Two elements (x1, y1), (x2, y2) ∈ X × Y are called NP(κ, λ)-adjacent if either (i) x1 = x2 and y1, y2 are λ-adjacent or (ii) y1 = y2 and x1, x2 are κ-adjacent or (iii) x1, x2 are κ-adjacent and y1, y2 are λ-adjacent. Definition 2.9. If a (κ, λ)-continuous function f : X → Y is (κ, λ)-homotopic to a constant map c : X → Y , c(x) = y0, then f is said to be (κ, λ)- nullhomotopic. If f : X → X is (κ, κ)-nullhomotopic, then we omit one of the adjacency relations and simply write “κ-nullhomotopic”. 3. Digital Homotopic Distance Recall that a covering of a space X is a collection of subsets of X whose union is X. Definition 3.1. Let f, g : X → Y be (κ, λ)-continuous functions. The (κ, λ) homotopic distance (so-called digital homotopic distance) between f and g is the least non-negative integer n such that there exists a covering U0, U1, . . . , Un of the digital image X with the property f|Ui ≃κ,λ g|Ui for each i. It is denoted by Dκ,λ(f, g). If there is no such covering, we define Dκ,λ(f, g) = ∞. Proposition 3.2. Let f : (X, κ) → (Y, λ) be continuous. If X is finite and κ-connected, then Dκ,λ(f, g) < ∞. Proof. Let Ux = {x}. Then {Ux |x ∈ X} is a finite covering of X. Since X is κ- connected, for each x ∈ X there is a (c1, κ)-continuous fx : [0, mx]Z → X such that fx(0) = x and fx(mx) ∈ f−1(g(x)). Then the function H : Ux×[0, mx]Z → Y defined by H(x, t) = f(fx(t)) is a homotopy between f|Ux and g|Ux. It follows that Dκ,λ(f, g) < ∞. � Proposition 3.3. Let f, g : X → Y be (κ, λ)-continuous. The following prop- erties hold. (a) Dκ,λ(f, g) = Dκ,λ(g, f). (b) Dκ,λ(f, g) = 0 iff f ≃κ,λ g. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 185 A. Borat Proposition 3.4. Let (X, κ) and (Y, λ) be digital images. Let f, f′, g, g′ : X → Y be (κ, λ)-continuous functions. If f ≃κ,λ f′ and g ≃κ,λ g′ then Dκ,λ(f, g) = Dκ,λ(f ′, g′). Proof. Suppose Dκ,λ(f′, g′) = n. Then there exist subsets U0, U1, . . . , Un cov- ering (X, κ) such that f′|Ui ≃κ,λ g ′|Ui for all i. From the assumption f ≃κ,λ f′ and g ≃κ,λ g′, we have f|Ui ≃κ,λ f ′|Ui and g|Ui ≃κ,λ g ′|Ui for all i. Therefore, for all i, we have f|Ui ≃κ,λ f ′|Ui ≃κ,λ g ′|Ui ≃κ,λ g|Ui. Hence Dκ,λ(f, g) ≤ n. The other way around can be proved similarly. Thus we conclude that Dκ,λ(f ′, g′) = Dκ,λ(f, g). � The following theorems state how the adjacency relations in the domain and in the image affect the digital homotopic distance. Theorem 3.5. Let (X, κ), (Y, λ) and (Y, λ′) be digital images. Let f, f′ : X → Y be (κ, λ)-continuous and g, g′ : X → Y (κ, λ′)-continuous functions. If f ≃κ,λ f ′, g ≃κ,λ′ g′ and λ′ ≥d λ, then Dκ,λ(f, g) ≤ Dκ,λ′(f′, g′). Proof. Suppose Dκ,λ′(f′, g′) = n. Then there exist subsets U0, U1, . . . , Un cov- ering (X, κ) such that f′|Ui ≃κ,λ′ g ′|Ui for all i. From the assumption f ≃κ,λ f′ and g ≃κ,λ′ g′, we have f|Ui ≃κ,λ f ′|Ui and g|Ui ≃κ,λ′ g ′|Ui for all i. On the other hand, since λ′ ≥d λ and g, g′ are (κ, λ′)-continuous, by Propo- sition 2.7(a) g, g′ are (κ, λ)-continuous. Moreover, (κ, λ′)-homotopies are also (κ, λ)-homotopies. Hence, since g|Ui ≃κ,λ′ g ′|Ui and f ′|Ui ≃κ,λ′ g ′|Ui, thus g|Ui ≃κ,λ g ′|Ui and f ′|Ui ≃κ,λ g ′|Ui for all i. Therefore, for all i, we have f|Ui ≃κ,λ f ′|Ui ≃κ,λ g ′|Ui ≃κ,λ g|Ui. It follows that f|Ui ≃κ,λ g|Ui for all i. So we have Dκ,λ(f, g) ≤ n. Thus we conclude that Dκ,λ(f, g) ≤ Dκ,λ′(f′, g′). � Theorem 3.6. Let (X, κ), (X, κ′) and (Y, λ) be digital images. Let f, f′ : X → Y be (κ, λ)-continuous and g, g′ : X → Y (κ′, λ)-continuous functions. If f ≃κ,λ f ′, g ≃κ′,λ g′ and κ′ ≥d κ, then Dκ′,λ(f′, g′) ≤ Dκ,λ(f, g). Proof. Suppose Dκ,λ(f, g) = n. Then there exist subsets U0, U1, . . . , Un cover- ing (X, κ) such that f|Ui ≃κ,λ g|Ui for all i. From the assumption f ≃κ,λ f′ and g ≃κ′,λ g′, we have f|Ui ≃κ,λ f ′|Ui and g|Ui ≃κ′,λ g ′|Ui for all i. On the other hand, since κ′ ≥d κ and f, f′ are (κ, λ)-continuous, by Propo- sition 2.7(b) f, f′ are (κ′, λ)-continuous. Moreover, (κ, λ)-homotopies are also (κ′, λ)-homotopies. Hence, since f|Ui ≃κ,λ f ′|Ui and f|Ui ≃κ,λ g|Ui, it follows that f|Ui ≃κ′,λ f ′|Ui and f|Ui ≃κ′,λ g|Ui for all i. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 186 Digital homotopic distance between digital functions Therefore, for all i, we have f ′|Ui ≃κ′,λ f|Ui ≃κ′,λ g|Ui ≃κ′,λ g ′|Ui. It follows that f′|Ui ≃κ,λ g ′|Ui for all i. So we have Dκ′,λ(f ′, g′) ≤ n. Notice that Ui’s also admit κ′-adjacency. Thus we conclude that Dκ′,λ(f′, g′) ≤ Dκ,λ(f, g). � Proposition 3.7. If f, g : X → Y are (κ, λ)-continuous maps and if {U0, U1, . . . , Un} is a finite covering of X, then we have Dκ,λ(f, g) ≤ n ∑ i=0 Dκ,λ(f|Ui, g|Ui) + n. Proof. Suppose Dκ,λ(f|Ui, g|Ui) = mi for each i = 0, 1, . . . , n. So there exist U0i , U 1 i , . . . , U mi i covering (Ui, κ) such that f|Uj i ≃κ,λ g|Uj i for all i, j. Consider the collection U = {U00 , U 1 0 , . . . , U m0 0 , U 0 1 , U 1 1 , . . . , U m1 1 , . . . , U 0 n, U 1 n, . . . , U mn n }. Notice that ⋃ V ∈U V = X. Moreover, f|V ≃κ,λ g|V for all V ∈ U. Hence Dκ,λ(f, g) ≤ (m0 + 1) + (m1 + 1) + . . . + (mn + 1) = (m0 + m1 + . . . + mn) + n = n ∑ i=0 Dκ,λ(f|Ui, g|Ui) + n. � 4. Relations Between the Digital Analogs of Homotopic Distance and cat In this section we introduce the relation between digital homotopic distance and digital LS category both of a digital image and a digital function. Let us first recall the definition of digital LS categories. Definition 4.1 ([2]). The digital LS category of a digital image (X, κ) is the least non-negative integer k such that there is a covering U0, U1, · · · , Uk of X such that inclusion map ιi : Ui →֒ X is digitally κ-nullhomotopic in X for each i = 0, 1, . . . , k. It is denoted by catκ(X) = k. Definition 4.2 ([18]). Let f : X → Y be a (κ, λ)-continuous function. The digital LS category of f is the least non-negative integer k such that there is a covering {U0, . . . , Uk} of X such that f|Uj is (κ, λ)-nullhomotopic for each j = 0, 1, . . . , k. It is denoted by catκ,λ(f). By Definition 4.2, if f : X → Y is (κ, λ)-continuous and c : X → Y is a digital constant function, then catκ,λ(f) = Dκ,λ(f, c) provided X and Y are κ- and λ-connected, respectively. By Definition 4.1, catκ(X) = Dκ,κ(Id, c) where c : X → X is a digital constant map and X is κ-connected. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 187 A. Borat Example 4.3. Consider the digital image ({0, 2}, c1). catc1({0, 2}) = 1 but Dc1,c1(Id, c) = ∞. Digital Lusternik-Schnirelmann category can be written in terms of digital homotopic distance as follows. Theorem 4.4. For a fixed x0 ∈ X, let i1 : X → X × X, i1(x) = (x, x0) and i2 : X → X × X, i2(x) = (x0, x). Then catκ(X) = Dκ,NP (κ,κ)(i1, i2). Proof. Let us first show that Dκ,NP (κ,κ)(i1, i2) ≤ catκ(X). Let catκ(X) = k. Then there are U0, U1, . . . , Uk covering X such that ji : Ui →֒ X is κ- nullhomotopic for each i = 0, 1, . . . , k. Let F i : Ui × [0, mi]Z → X be (κ, κ)-homotopy such that (a1) F i(x, 0) = ji(x) = x and F i(x, mi) = c(x) = x0 where c : X → X, c(x) = x0 is a constant map. (a2) For all x ∈ Ui, the induced function F ix : [0, mi]Z → X, F i x(t) = F i(x, t) is (2, κ)-continuous. (a3) For all t ∈ [0, mi]Z, the induced function F it : Ui → X, F i t (x) = F i(x, t) is (κ, κ)-continuous. Define Hi : Ui × [0, 2mi]Z → X × X as follows. Hi(x, t) = { (F i(x, t), x0), t ∈ [0, mi]Z (x0, F i(2mi − t)), t ∈ [mi, 2mi]Z Then we have (b1) Hi(x, 0) = i1(x) and Hi(x, 2mi) = i2(x). (b2) For all x ∈ Ui, the induced function Hix : [0, mi]Z → X × X, H i x(t) = Hi(x, t) is (2, NP(κ, κ))-continuous: Suppose t1, t2 are 2-adjacent. Case I: If t1, t2 ∈ [0, mi]Z, then Hix(t1) = (F i(x, t1), x0) and Hix(t2) = (F i(x, t2), x0) are NP(κ, κ)-continuous since the first com- ponents are κ-adjacent or equal from (a2) and the second components are equal. Case II: If t1, t2 ∈ [mi, 2mi]Z, then Hix(t1) = (x0, F i(x, 2mi − t1)) and Hix(t2) = (x0, F i(x, 2mi − t2)) are NP(κ, κ)-continuous since the first components are equal and the second components are κ-adjacent or equal from (a2). Case III: If mi ∈ {t1, t2} then either Case I or Case II applies. (b3) For all t ∈ [0, mi]Z, the induced function Hit : Ui → X, H i t(x) = Hi(x, t) is (κ, NP(κ, κ))-continuous: Suppose x, y ∈ Ui are κ-adjacent. Case I: If t ∈ [0, mi]Z, then Hit(x) = H i(x, t) = (F(x, t), x0) and Hit(y) = H i(y, t) = (F(y, t), x0). So the first components are κ- adjacent or equal due to (a3) and the second components are equal. Case II: If t ∈ [mi, 2mi]Z, then Hit(x) = H i(x, t) = (x0, F(x, 2mi−t)) and Hit(y) = H i(y, t) = (x0, F(y, 2mi − t)). So the first components © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 188 Digital homotopic distance between digital functions are equal and the second components are κ-adjacent or equal due to (a3). Hence i1|Ui ≃κ,NP (κ,κ) i2|Ui. This establishes the desired inequality. Now let us show that catκ(X) ≤ Dκ,NP (κ,κ)(i1, i2). Let Dκ,NP (κ,κ)(i1, i2) = k. Then there are U0, U1, . . . , Uk covering X × X such that i1|Uj ≃κ,NP (κ,κ) i2|Uj for each j = 0, 1, . . . , k. Let Gj : Uj × [0, mj]Z → X × X be (κ, NP(κ, κ))-homotopy such that (c1) Gj(x, 0) = i1(x) and Gj(x, mj) = i2(x). (c2) For all x ∈ Uj, the induced function Gjx : [0, mj]Z → X × X, G j x(t) = Gj(x, t) is (2, NP(κ, κ))-continuous. (c3) For all t ∈ [0, mj]Z, the induced function G j t : Uj → X × X, G j t(x) = Gj(x, t) is (κ, NP(kappa, κ))-continuous. Let ιj : Uj →֒ X be the inclusion function and c : Uj → X, c(x) = x0 be a constant function. Define Kj : Uj × [0, mj]Z → X by Kj(x, t) = pr1 ◦ G j(x, t) where pr1 : X×X → X is the projection to the first factor. Notice that pr1 is (NP(κ, κ), κ)- continuous, [10]. Then we have (d1) Kj(x, 0) = x = ιj(x) and Kj(x, mj) = x0 = c(x). (d2) For all x ∈ Uj, the induced function Kjx : [0, mj]Z → X, K j(x, t) =: Kjx(t) = pr1 ◦G j x(t) is (2, NP(κ, κ))-continuous due to Proposition 2.5, the (NP(κ, κ), κ)-continuity of pr1 and the (2, NP(κ, κ))-continuity of Gjx. (d3) For all t ∈ [0, mj]Z, the induced function K j t : Uj → X ×X, K j(x, t) =: K j t (x) = pr1 ◦ G j t(x) is (κ, κ)-continuous due to due to Proposition 2.5, the (NP(κ, κ), κ)-continuity of pr1 and the (2, NP(κ, κ))-continuity of Gjx. Thus we have ιj|Uj ≃κ,κ c for each j. � Theorem 4.5 ([18]). catκ,NP (κ,κ)(∆X) = catκ(X) where ∆X : X → X × X, ∆X(x) = (x, x) is (κ, NP(κ, κ))-continuous. Corollary 4.6. Let i1, i2 be inclusions as defined in Theorem 4.4 and ∆X : X → X × X be (κ, NP(κ, κ))-continuous digital diagonal function. Then Dκ,NP (κ,κ)(∆X, c) = Dκ(i1, i2). Proof. This follows from Theorems 4.4 and 4.5. � 5. Digitally Homotopy Invariance of Digital Homotopic Distance Theorem 5.1 which states that the digital homotopic distance is homotopy invariant, is the main theorem of this section. Before we mention the theorem, let us recall right and left digital homotopy equivalences. A (κ, λ)-continuous function f : X → Y is left digital homotopy inverse if there exists a (λ, κ)-continuous function g : Y → X such that g ◦ f ≃κ,κ IdX. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 189 A. Borat A (κ, λ)-continuous function f : X → Y is right digital homotopy inverse if there exists a (λ, κ)-continuous function h : Y → X such that f ◦ g ≃λ,λ IdY . Theorem 5.1. Let f, g : X → Y be (κ, λ)-continuous and f′, g′ : X′ → Y ′ be (κ′, λ′)-continuous functions. If h2 : X′ → X and h1 : Y → Y ′ have left and right digital homotopy equivalences respectively such that the following diagram is commutative both with respect to f and g in the following sense: h1 ◦ f ◦ h2 ≃κ′,λ′ f ′ and h1 ◦ g ◦ h2 ≃κ′,λ′ g′. Then Dκ,λ(f, g) = Dκ′,λ′(f′, g′). X Y X′ Y ′ f g h1h2 f ′ g ′ An immediate consequence of Theorem 5.1 is the following. Corollary 5.2 ([2]). Digital LS category is digitally homotopy invariant. For a proof of Theorem 5.1 we need the following lemmas. Lemma 5.3. Let f, g : X → Y be (κ, λ)-continuous and h : Y → Z be (λ, γ)- continuous functions. Then Dκ,γ(h ◦ f, h ◦ g) ≤ Dκ,λ(f, g). Proof. Let Dκ,λ(f, g) = k. Then there exists a covering U0, . . . , Uk of X such that f|Ui ≃κ,λ g|Ui for each i = 0, 1, . . . , k. Then for each i, we have (h ◦ f)|Ui = h ◦ f|Ui ≃κ,γ h ◦ g|Ui = (h ◦ g)|Ui where the (κ, γ)-homotopy follows from Proposition 2.5. Hence Dκ,γ(h ◦ f, h ◦ g) ≤ Dκ,λ(f, g). � Lemma 5.4. Let f, g : X → Y be (κ, λ)-continuous and h : Z → X be (γ, κ)- continuous functions. Then Dγ,λ(f ◦ h, g ◦ h) ≤ Dκ,λ(f, g). Proof. Let Dκ,λ(f, g) = k. Then there exists a covering U0, . . . , Uk of X such that f|Ui ≃κ,λ g|Ui for each i = 0, 1, . . . , k. Consider h−1(Uj) ⊆ Z. Notice that {h−1(Uj)}kj=0 is a covering of X and the restriction map hj : h−1(Uj) → Z can be written in terms of h as hj : h−1(Uj) h −→ Uj ι −֒→ Z, hj = ι ◦ h. Then we have (f ◦ h)|h−1(Uj) = fh−1(Uj) ◦ hj ≃γ,λ g|h−1(Uj ) ◦ hj = g|h−1(Uj ) ◦ (ι ◦ h)|h−1(Uj) = g ◦ (ι ◦ h)|h−1(Uj) = (g ◦ h)|h−1(Uj) So (f ◦ h)|h−1(Uj ) ≃γ,λ (g ◦ h)|h−1(Uj) for each i. Notice that the (γ, λ)- homotopy on above line follows from Proposition 2.5. Hence Dγ,λ(f ◦h, g◦h) ≤ k. � By using these lemmas we prove the following propositions. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 190 Digital homotopic distance between digital functions Proposition 5.5. Let f, g : X → Y be (κ, λ)-continuous and h1 : Y → Y ′ be (λ, λ′)-continuous function with a left digital homotopy inverse. Then Dκ,λ′(h1 ◦ f, h1 ◦ g) = Dκ,λ(f, g). Proof. By Proposition 3.4 and Lemma 5.3, it follows that Dκ,λ(f, g) = Dκ,λ(h◦ h1 ◦ f, h ◦ h1 ◦ g) ≤ Dκ,λ′(h1 ◦ f, h1 ◦ g) ≤ Dκ,λ(f, g). � Proposition 5.6. Let f, g : X → Y be (κ, λ)-continuous and h2 : X′ → X be (κ′, κ)-continuous function with a right digital homotopy inverse. Then Dκ′,λ(f ◦ h2, g ◦ h2) = Dκ,λ(f, g). Proof. By Proposition 3.4 and Lemma 5.4, it follows that Dκ,λ(f, g) = Dκ,λ(f ◦ h2 ◦ h, g ◦ h2 ◦ h) ≤ Dκ′,λ(f ◦ h2, g ◦ h2) ≤ Dκ,λ(f, g). � Proof of Theorem 5.1. By Proposition 5.5 and Proposition 5.6, we have D(f′, g′) = D(h1 ◦ f ◦ h2, h1 ◦ g ◦ h2) = D(f ◦ h2, g ◦ h2) = Dκ,λ(f, g). 6. Future Work There is a relation between usual homotopic distance and TC. A similar relation can be found between digital homotopic distance and digital TC (as defined in [12]). Higher digital topological complexity is studied by Is and Karaca in [11]. Motivated from the fact that if digital homotopic distance is a generalization of digital TC, it can be predicted that a higher analog of homotopic distance (defined in a similar way as in [16]; see also [3]) can be realized as a general- ization of higher digital TC. Acknowledgements. 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