Normal functors in the coarse category

We define the canonical coarse structure on the spaces of the form FX, where F is a finitary normal functor of finite degree and show that every finitary (i.e., preserving the class of finite spaces) normal functor of finite degree in Comp has its counterpart in the coarse category.

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Дата:	2005
Автор:	Frider, V.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2005
Назва видання:	Algebra and Discrete Mathematics
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Цитувати:	Normal functors in the coarse category / V. Frider // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 4. — С. 16–27. — Бібліогр.: 8 назв. — англ.

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spelling	irk-123456789-1573392019-06-21T01:26:19Z Normal functors in the coarse category Frider, V. We define the canonical coarse structure on the spaces of the form FX, where F is a finitary normal functor of finite degree and show that every finitary (i.e., preserving the class of finite spaces) normal functor of finite degree in Comp has its counterpart in the coarse category. 2005 Article Normal functors in the coarse category / V. Frider // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 4. — С. 16–27. — Бібліогр.: 8 назв. — англ. 1726-3255 http://dspace.nbuv.gov.ua/handle/123456789/157339 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We define the canonical coarse structure on the spaces of the form FX, where F is a finitary normal functor of finite degree and show that every finitary (i.e., preserving the class of finite spaces) normal functor of finite degree in Comp has its counterpart in the coarse category.
format	Article
author	Frider, V.
spellingShingle	Frider, V. Normal functors in the coarse category Algebra and Discrete Mathematics
author_facet	Frider, V.
author_sort	Frider, V.
title	Normal functors in the coarse category
title_short	Normal functors in the coarse category
title_full	Normal functors in the coarse category
title_fullStr	Normal functors in the coarse category
title_full_unstemmed	Normal functors in the coarse category
title_sort	normal functors in the coarse category
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2005
url	http://dspace.nbuv.gov.ua/handle/123456789/157339
citation_txt	Normal functors in the coarse category / V. Frider // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 4. — С. 16–27. — Бібліогр.: 8 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT friderv normalfunctorsinthecoarsecategory
first_indexed	2025-07-14T09:47:03Z
last_indexed	2025-07-14T09:47:03Z
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 4. (2005). pp. 16 – 27 c© Journal “Algebra and Discrete Mathematics” Normal functors in the coarse category V. Frider Communicated by V. M. Usenko Abstract. We define the canonical coarse structure on the spaces of the form FX, where F is a finitary normal functor of finite degree and show that every finitary (i.e., preserving the class of finite spaces) normal functor of finite degree in Comp has its counterpart in the coarse category. 1. Introduction The coarse category first introduced by J. Roe [1] turned out to be an appropriate universe for different areas of modern mathematics (see, e.g. [2], [3]). In [4] the authors considered the hyperspace functor in the coarse category and established some properties of the monad generated by this functor. In particular, it was proved in [4] that the G-symmetric power functor can be extended onto the Kleisli category of this monad. Both the hyperspace functor and the G-symmetric power functor have their counterparts in the category Comp of compact Hausdorff spaces. These functors in Comp are examples of normal functors in the sense of Shchepin [5]. The theory of normal functors was developed by different authors [6]; in [5] this theory was applied to the topology of nonmetriz- able compact Hausdorff spaces and in [7] to the topology of infinite- dimensional manifolds. The aim of this paper is to show that every finitary (i.e., preserving the class of finite spaces) normal functor of finite degree in Comp has its counterpart in the coarse category. These functors are completely determined by their restrictions onto the category of spaces of cardinality ≤ n, where n denotes the degree of the functor. We show, in particular, Key words and phrases: coarse space, coarse map, normal functor. Jo u rn al A lg eb ra D is cr et e M at h .V. Frider 17 that our construction coincides with the Vietoris coarse structure on the hypersymmetric power functors (defined in [4]). 2. Preliminaries 2.1. Coarse structures For the convenience of reader we recall some definitions of the coarse topology; see, e.g. [1], [8] for details. Let X be a set and M,N ⊂ X ×X. The composition of M and N is the set MN = {(x, y) ∈ X ×X \| there exists z ∈ X such that (x, z) ∈M, (z, y) ∈ N}, the inverse of M is the set M−1 = {(x, y) ∈ X ×X \| (y, x) ∈M}. A coarse stucture on a set X is a family E of subsets, which are called the entourages, in the product X ×X that satisfies the following properties: 1. any finite union of entourages is contained in an entourage; 2. for every entourageM , its inverseM−1 is contained in an entourage; 3. for every entourages M , N their composition MN is contained in an entourage; 4. ∪E = X ×X. A coarse space is a pair (X, E), where E is a coarse structure on a set X. A coarse structure on X is called unital, if the diagonal ∆X is con- tained in an entourage. A coarse structure on X is called anti-discrete, if X ×X is an entourage. If E1, E2 are coarse structures on X, then E1 ≤ E2 means that for every M ∈ E1 there is N ∈ E2 such that M ⊂ N . Two coarse structures E1 and E2 are said to be equivalent, if E1 ≤ E2 and E2 ≤ E1. We usually identify coarse spaces with equivalent coarse structures. The weight of the coarse structure is the cardinal number w(E) = min{\|E ′\| \| E and E ′ are equivalent }. If E is a coarse structure on X, then, obviously, the coarse structure E1 = {M ∪M−1 \| M ∈ E} is equivalent to E and is symmetric in the sense that N−1 ∈ E1 for every N ∈ E1. Jo u rn al A lg eb ra D is cr et e M at h .18 Normal functors in the coarse category Given M ∈ E and A ⊂ X we define the M -neighborhood M(A) of A as follows: M(A) = {x ∈ X \| (a, x) ∈ M for sone a ∈ A}. We use the notation M({a}) instead of M(a). A set A ⊂ X is bounded if there exists x ∈ X such that A ⊂M(x). Let (Xi, Ei), i = 1, 2, be coarse spaces. A map f : X1 → X2 is called coarse, if the following two conditions hold: 1. for every M ∈ E1 there exists N ∈ E2 such that (f × f)(M) ⊂ N ; 2. for any bounded subset A of X2 the set f−1(A) is bounded. Let f, g : X1 → X2 be coarse maps. If there exists U ∈ E2 (here E2 is the coarse structure on X2) such that (f(x), g(x)) ∈ U for every x ∈ X1 then the maps f, g are said to be U -close. Let (X, d) be a metric space. The family Ed = {{(x, y) ∈ X ×X \| d(x, y) < n} \| n ∈ N} forms a metric coarse structure on X. It is easy to see that the coarse spaces and coarse maps form a cat- egory. We denote it by CS. Let U : CS → Set denote the forgetful functor into the category Set of sets. The product of coarse spaces (X1, E1), (X2, E2) is the coarse space (X1 ×X2, E1 × E2), where E1 × E2 = {U1 × U2 \| U1 ∈ E1, U2 ∈ E2}. By induction, we define the product of coarse spaces (Xi, Ei), i = 1, . . . , n. We denote the n-th power of a coarse space (X, E) by (Xn, En). Note that Xn can be naturally identified with C(n,X), the set of all maps from n to X; we say that f, g ∈ C(n,X) are U -close whenever (f, g) ∈ Un, where U ∈ E . 2.2. Normal functors We briefly recall some notions from the theory of normal functors in the category Comp of compact Hausdorff spaces; see, e.g., [5] for details. An endofunctor F in Comp is called normal if it is continuous , monomor- phic, epimorphic, preserves weight of infinite compacta, intersections, preimages, singletons and empty set. Suppose that F is a monomorphic functor that preserves the inter- sections, and a ∈ FX. The support of a is the set suppF,X(a) = ∩ { A ⊂ X \| A is closed and a ∈ FA } . In the obvious situation the notation suppF,X(a) is abbreviated to suppF (a) or even to supp(a). We say that a functor is finitary if it preserves the finite spaces. Jo u rn al A lg eb ra D is cr et e M at h .V. Frider 19 Examples of normal finitary functors are: 1) The hypersymmetric power functor expn. Here expnX = {A ⊂ X \| 1 ≤ A ≤ n}, expn f = f(A). 2) The G-symmetric power functor SPn G, where G is a subgroup of the symmetric group Sn. Define an equivalence relation ∼G on Xn by the conditions: (x1, . . . , xn) ∼G (y1, . . . , yn) if and only if there exists σ ∈ G such that xi = yσ(i) for all i = 1, . . . , n. We denote by [x1, . . . , xn]G the equivalence class that contains (x1, . . . , xn). By the definition, the G-symmetric power of X is SPn GX = Xn/∼G . Given a map f : X → Y , we define a map SPn Gf : SPn GX → SPn GY by the formula: SPn Gf([x1, . . . , xn]G) = [f(x1), . . . , f(xn)]G. 3) The subfunctor Pn,k of the functor of probability measures, k ∈ N. Let Pn,k = { n∑ i=1 αiδxi \| x1, . . . , xn ∈ X, αi = pi k , pi ∈ {0, 1, . . . , }, n∑ i=1 αi = 1 } . Let F be a normal functor. The degree of a point a ∈ FX is the cardinality of supp(a), whenever supp(a) is finite and ∞, otherwise. The degree of a is denoted by deg(a). The maximal possible value of deg(a) is called the degree of the functor F and denoted by deg(F ). If F is not a functor of degree n for any n ∈ N then is said to be a functor of infinite degree (this is denoted by degF = ∞). Let F be a finite normal functor of degree n in the category Comp. For every normal functor F : Comp → Comp construct the functorFβ : Tych → Tych by the following manner (Tych denotes the category of Tychonov spaces). For every X ∈ \|Tych\| let FβX = {a ∈ FβX \| supp(a) ⊂ X} ⊂ FβX (here βX denotes the Čheh-Stone compactification of the space X). Given a map f : X → Y of Tychonov spaces, we have F (βf)(FβX) ⊂ FβY . Let Fβf = Fβ\|FβX : FβX → FβY , which is the restriction F (βf) on FβX. Obviously, Fβ is an endofunctor in Tych. By definition, Fβ is an extension of F onto Tych. We keep the notion F for its canonical extension onto Tych. For every a ∈ Fn, the subfunctor Fa of F is defined as follows: Fa = ∩{F ′ \| F ′ is a normal subfunctor ofF such that a ∈ F ′n}. Jo u rn al A lg eb ra D is cr et e M at h .20 Normal functors in the coarse category Every functor of the form Fa is a quotient functor of the power functor (−)deg(a) in the sense that there exists a natural transforma- tion πa : (−)deg(a) → Fa such that all the maps πaX : Xdeg(a) → Fa are onto. Indeed, define πaX as follows: πaX(x1, . . . , xn) = Ff(a), where f : {1, . . . , n} → X is the map sending i to xi. Note, that from the normality of F , the preimages of the map πaX are finite. 3. Normal functors in the coarse category 3.1. Construction Let F be finitary normal functor of degree n ≥ 1, (X, E) a coarse space. For any U ∈ E define Û ={(a, b) ∈ FX × FX \| there exist W1, . . . ,Wk ∈ E , f1, . . . , f2k ∈ C(n,X), c1, . . . , ck ∈ Fn such that W1 . . .Wk ⊂ U, are f2i−1, f2i U -close, i = 1, . . . , k, i Ff1(c1) = a, Ff2k(ck) = b, Ff2j(cj) = Ff2j+1(cj+1), j = 1, . . . , k − 1}. Note that here we consider the set X as a discrete topological space, that is why it is possible to consider the discrete space FX, which is identified with the underlying set. Proposition 3.1. The family {Û \| U ∈ E} forms a coarse structure on FX. Proof. It is easy to see that Û−1 = (Û)−1. Suppose that Û , V̂ ∈ Ê . Let us show that for any W ∈ E we have Û V̂ ⊂ Ŵ . Let (a, b) ∈ Û V̂ , this means that there exists c ∈ FX such that (a, c) ∈ Û , (c, b) ∈ V̂ . There exist U1, . . . , Uk, V1, . . . , Vl ∈ E , a1, . . . , ak, b1, . . . , bl ∈ Fn, and f1, f2, . . . , f2k, g1, g2, . . . , g2l : n→ X such that the following holds: 1. U1 . . . Uk ⊂ U , V1 . . . Vl ⊂ V ; 2. Ff1(a1) = a, Ff2i(ai) = Ff2i+1(ai+1) for every i = 1, 2, . . . , k − 1, Ffk(ak) = Fg1(b1) = c, Fg2j(bj) = Fg2j+1(bj+1) for every j = 1, 2, . . . , l − 1, Fg2l(bl) = b; 3. the maps f2i−1, f2i are Ui-close, i = 1, . . . , k and the maps g2j−1, g2j are Vj-close, j = 1, . . . , l. Jo u rn al A lg eb ra D is cr et e M at h .V. Frider 21 Since U1 . . . UkV1 . . . Vl ⊂ UV ⊂W , from the definition of E it follows that (a, b) ∈ Ŵ . We are going to show that ∪Ê = FX × FX. Choose arbitrary a, b ∈ FX and x ∈ X and check that (a, ηX(x)) ∈ Û , (ηX(x), b) ∈ Û for some U ∈ E . There exists U ∈ E such that (y, x) ∈ U for every y ∈ supp(a). Let a′ ∈ Fn and f ∈ Xn be such that Ff(a′) = a. Denote by g : n→ X the constant map with value {x}. Since f and g is U -close, we see that (a, ηX(x)) ∈ Û . Similarly, one can show that (ηX(x), b) ∈ Û for some U ∈ E . In the sequel, we will denote the defined above coarse structure on FX by EF . Proposition 3.2. The coarse structure EF is unital if and only if so is E. Proof. Choose an arbitrary a ∈ FX. Since E is a unital coarse structure, there exists U ∈ E such that (xi, xi) ∈ U for every i = 1, . . . , n, where xi ∈ supp(a). Let us consider the map φ : n → X acting in the following way: φ(i) = xi and b ∈ Fn such that Fφ(b) = a. Notice that Fφ(b) = a and for every i = 1, . . . , n, (φ(i), φ(i)) ∈ U . Thus (a, a) ∈ Û . Therefore, we obtain that ∆FX ⊂ Û Lemma 3.3. Let (X, E) be a coarse space and F a finitary functor of finite degree. Let a ∈ FX, x ∈ X and a ∈ Û(ηX(x)). Then supp(a) ⊂ U(x). Proof. Since (a, ηX(x)) ∈ Û , there exist U1, . . . , Uk ∈ E , a1, . . . , ak ∈ Fn, and maps f1, f2, . . . , f2k : n→ X such that the following holds: (1) U1 . . . Uk ⊂ U ; (2) Ff1(a1) = a, Ff2k(ak) = ηX(x), and Ff2i(a2i) = Ff2i+1(a2i+1) for every i = 1, . . . , k − 1; (3) the maps f2i−1, f2i are Ui-close, i = 1, . . . , k. Let y ∈ supp(a). By induction, one can find points zi ∈ supp(ai), i = 1, . . . , k, such that f1(z1) = y, f2i(zi) = f2i+1(zi+1), i = 1, . . . , k − 1. Since (f2i−1(zi), f2i(zi)) ∈ Ui, i = 1, . . . , k, we conclude that (y, x) ∈ U1 . . . Uk ⊂ U . Let now f : (X, EX) → (Y, EY ) be a coarse map between coarse spaces. It is natural to define the map Ff : FX → FY . Proposition 3.4. The map Ff : (FX, EXF ) → (FY, EY F ) is coarse if so is f . Jo u rn al A lg eb ra D is cr et e M at h .22 Normal functors in the coarse category Proof. Let us verify that the map Ff is coarsely uniform. Given Û ∈ ÊXF and (a, b) ∈ Û , find U1, . . . , Uk ∈ E , a1, . . . , ak ∈ Fn, and f1, . . . , f2k : n→ X such that the following holds: (1) U1 . . . Uk ⊂ U ; (2) Ff1(a1) = a, Ff2k(ak) = b, and Ff2i(a2i) = Ff2i+1(a2i+1) for every i = 1, . . . , k − 1; (3) the maps f2i−1, f2i are Ui-close, i = 1, . . . , k. Since f is coarsely uniform, for every i = 1, . . . , k, there exists Vi ∈ EY such that (f × f)(Ui) ⊂ Vi. Find V ∈ EY such that V1 . . . Vk ⊂ V . Let gi = ffi : n → Y . With these g1, . . . , g2k, V1, . . . , Vk and a1, . . . , ak we see that (Ff(a), Ff(b)) ∈ V̂ . This means that (Ff × Ff)(Û) ⊂ V̂ and the coarse uniformity of Ff is proved. Let us demonstrate that the map Ff is coarsely proper. Suppose that a set A is bounded in FY , i.e. there exists V ∈ EY and y ∈ Y such that A ⊂ V̂ (ηY (y)). Since f is coarsely proper, there exists x ∈ X and U ∈ EX such that f−1(Y (y)) ⊂ U(x) . We are going to show that (Ff)−1(A) ⊂ V̂ (ηX(x)). Suppose then there exists a ∈ FX with Ff(a) = b ∈ A. Since (b, ηY (y)) ∈ V̂ , by Lemma 4.2, supp(b) ⊂ V and, therefore supp(a) ⊂ U(x). It is obvious that then (a, ηX(x)) ∈ Û . 3.2. Hypersymmetric power Let (X, E) be a coarse space. On the hypersymmetric power expn one can consider, in addition to the Vietoris coarse structure, Ẽ , the defined above structure Ê . Proposition 3.5. The coarse structures Ẽ and Ê on the set expnX are coarsely equivalent. Proof. Consider an arbitrary entourage U ∈ E and show that Ũ ⊂ Û(A,B) ∈ Û . Given (a, b) ∈ Û , find a1, . . . , ak ∈ expn n and f1, . . . , f2k : n→ X such that the following holds: (1) U1 . . . Uk ⊂ U ; (2) expn f1(a1) = a, expn f2k(ak) = b, and expn f2i(a2i) = expn f2i+1(a2i+1) for every i = 1, . . . , k − 1; (3) the maps f2i−1, f2i are Ui-close, i = 1, . . . , k. For any x ∈ a there exists c1 ∈ a1 ⊂ n with f1(c1) = x. Since expn f2i(ai) = expn f2i+1(ai+1), one can choose a sequence ci ∈ ai, for i = 2, . . . , k so that (f2i−1(ci), f2i(ci)) ∈ Ui, i = 1, . . . , k. Let f2k(c2k) = y. Then (x, y) ∈ U1 . . . Uk ⊂ U . Therefore, a ⊂ U(b). One can similarly show that b ⊂ U(a). This means that (a, b) ∈ Ũ . Jo u rn al A lg eb ra D is cr et e M at h .V. Frider 23 Let (a, b) ∈ Ũ , i.e. a ⊂ U(b) and b ⊂ U(a). For every x ∈ a, find h(x) ∈ b such that (x, h(x)) ∈ U . Similarly, for every y ∈ b, find g(y) ∈ a such that (g(y), y) ∈ U . Denote by r : a → g(b) the retraction such that r(x) = g(h(x)), for every x ∈ a \ g(b). Let f1 : n→ X be a map for which f1(n) = a, f2 = rf1, f4 : n→ X be a map for which f4(n) = b, f3 = gf4, then f2(n) = rf1(n) = r(a) = g(b) = gf4(n) = f3(n). Since, for every i ∈ n, (f1(i), h(f1(i))) ∈ U and (h(f1(i)), g(h(f1(i)))) ∈ U , we see that (f1(i), f2(i)) ∈ U2. Similarly, (f3(i), f4(i)) ∈ U , therefore the maps f1 and f4 are U3-close. Finally, Ũ ⊂ Û3. 3.3. G-symmetric power The G-symmetric power functor SPn G is endowed with the coarse struc- ture Ê defined in the following way: M̂ ∈ Ê ⇔ M̂ = {([x1, . . . , xn]G, [y1, . . . , yn]G) \| there exist σ ∈ G, such that for all i = 1, . . . , n, (xi, yσ(i)) ∈M ∈ E} (see [4] for details). Further we will denote ESP n G by Ẽ . Proposition 3.6. The coarse structures Ê and Ẽ are equivalent. Proof. Consider an arbitrary entourage M ∈ E and show that M̃ and M̂ are coarsely equivalent. Let ([x1, . . . , xn]G, [y1, . . . , yn]G) ∈ M̂ , this means that there exists σ ∈ G such that (xi, yσ(i)) ∈ M for all i = 1, . . . , n. Taking the maps f1 : n → X sending i to xi and f2 : n → X sending i to yσ(i) we obtain that SPn Gf1([1, . . . , n]G) = [x1, . . . , xn]G and SPn Gf2([1, . . . , n]G) = [y1, . . . , yn]G. Note that f1 and f2 are M -close. Hence, ([x1, . . . , xn]G, [y1, . . . , yn]G) ∈ M̃ . To show that every ([x1, . . . , xn]G, [y1, . . . , yn]G) ∈ M̃ is containing in M̂ we choose entourages U1, . . . , Uk ∈ E , points a1, . . . , ak ∈ SPn Gn and maps f1, . . . , f2k ∈ C(n,X) for which SPn Gf1(a1) = [x1, . . . , xn]G, SPn Gf2k(ak) = [y1, . . . , yn]G, SPn Gf2i(ai) = SPn Gf2i+1(ai+1) for i = 1, . . . , n and f2i−1, f2i are Ui-close for every i = 1, . . . , k , where U1 · . . . ·Uk ⊂M . We have (xi, f2(i)) ∈ U1, (f3(i), f4(i)) ∈ U2, . . . , (f2k(i), yi) ∈ Uk and [f2i(1), . . . , f2i(n)]G = SPn Gf2i(ai) = SPn Gf2i+1(ai+1) = = [f2i+1(1), . . . , f2i+1(n)]G. Jo u rn al A lg eb ra D is cr et e M at h .24 Normal functors in the coarse category Therefore, we conclude that f2i(j) = f2i+1(j) for every i = 1, . . . , k and j = 1, . . . , n and that implies (xi, yi) ∈ U1 · . . . · Uk ⊂M . Now taking σ = e, where e is the identity permutation we obtain that [x1, . . . , xn]G, [y1, . . . , yn]G) ∈ M̂ . 4. Normality properties In this section we define counterparts of the properties from the definition of the normal functor for the functors in the asymptotic category. Definition 4.1. Recall a map f : (X, EX) → (Y, EY ) between coarse spaces to be monomorphic, if there exists a map g : f(X) → Y such that the following holds: there exists UX ∈ EX , UY ∈ EY , such that for every x ∈ X, y ∈ f(X) we have (x, gf(x)) ∈ UX , (y, fg(y)) ∈ UY . Proposition 4.2. The functor F : CS → CS defined above preserves the class of monomorphic maps. Proof. Let f , g and Ui, i = 1, 2 be as above. The map Ff : FX → FY is monomorphic, this follows from the commutative diagram: FX Ff �� Ff // idFX !!{ w s o k g c _ [ W S O K G C Ff(FX) Fg // FX Ff �� Ff(FX) Fg // idFf(FX) == G K O S W [ _ c g k o s w FX Ff OO Ff // Ff(FX) Definition 4.3. Recall a map f : (X, EX) → (Y, EY ) between coarse spaces to be epimorphic, if the image f(X) is coarsely dense in Y . This means that exists UY ∈ EY such that UY (f(X)) ⊃ Y . Proposition 4.4. The functor F : CS → CS preserves the class of epi- morphic maps. Proof. Let f : X → Y be an epimorphic map. Show that Ff : FX → FY is also epimorphic. Fix an arbitrary b ∈ FY . Since supp(b) = y1, . . . , yn, there exists b∗ ∈ Fn such that Fφ(b∗) = b, where φ : n → Y acts in the following way: φ(i) = yi. Jo u rn al A lg eb ra D is cr et e M at h .V. Frider 25 The map f is epimorphic, this means that there exists UY ∈ EY for which Y ⊂ UY (f(X)). Hence, for every i there exists y∗I with yi ∈ UY (y∗i ). Consider the map ϕ : n→ X, ϕ(i) = xi, xi ∈ X and f(xi) = y∗i . Let a = Fϕ(b∗). Finally, we have to show that (Ff(a), b) ∈ ÛY . With this aim we notice that: 1) Fψ(b∗) = a, fϕ(b∗) = b; 2) ϕ and ψ are UY -close, because ϕ(i) = yi, ψ(i) = xi and f(xi) = y∗i . We have (yi, y ∗ i ) ∈ UY , so we are done. Definition 4.5. Let f : (X, EX) → (Y, EY ) be a coarse map and B ⊂ Y . A coarse preimage of the set B is the set A together with two maps g : A→ B, j : A→ X for which the following holds: if there are two coarse maps α : Z → X, β : Z → B such that fα ∼ iβ, then exists a map h : Z → A such that gh ∼ β, jh ∼ α. For the convenience of the reader let us introduce this definition by the diagram: Z α ## h ��@ @ @ @ β �� A j // g �� X f �� B i // Y Proposition 4.6. Any two coarse preimages of B are coarsely equivalent. Proof. Let (A, g, j), (A′, g′, j′) be different coarse preimages of the set B. For the coarse preimage A put Z = A′, α = j′, β = g′. We obtain that fα = fj′, iβ = ig′. Since A′ is also a coarse preimage, fj′ ∼ ig′ implies fα ∼ iβ. That is why we can find the map h : A′ → A such that gh ∼ g′ and jh ∼ j′. Similarly, one can show that for A′ there exists a map h′ : A → A′ for which g′h′ ∼ g and j′h′ ∼ j. Therefore, g′ ∼ gh ∼ g′h′h⇒ h′h ∼ idA′ j ∼ j′h′ ∼ jhh′ ⇒ hh′ ∼ idA This completes the proof. Proposition 4.7. The functor F preserves coarse preimages. Jo u rn al A lg eb ra D is cr et e M at h .26 Normal functors in the coarse category Proof. Let f : X → Y be a coarse map and A be a coarse preimage for B. Γα π2 ''PPPPPPPPPPPPPP β A A A A A j // g �� X f �� B ⊂ Y We build the set Γα = {(z, x)\|x ∈ suppα(z)} ⊂ Z × X. Note that f(π2(Γα)) ⊂ B, where π2 is a projection map on the second factor of the product. Since A is a coarse preimage of B, there exists β : Z → A such that jβ ∼ π2. Moreover, Ff(α(Z)) ⊂ FB, whence supp(Ff ◦α(Z)) ⊂ B and ⇒ f(suppα(Z)) ⊂ B. Put ξ(z) = Fβ ◦ α(z). This map is sending any point x ∈ Z to ξ(z) ∈ FA i.e. ξ : Z → FA. It is easy to see that the diagram Z α ((RRRRRRRRRRRRRRRR ξ !!C CC CC CC C FA Fj // Fg �� FX Ff �� FB ⊂ FY is commutative, thus FA ∼= (Ff)−1(FB). This ends the proof. Definition 4.8. A functor F : CS → CS is normal in CS if: 1) F preserves weight; 2) F is monomorphic; 3) F is epimorphic; 4) F preserves preimages; 5) F preserves ∅ (i.e. bounded coarse spaces). Remark 4.9. There is now sense to consider properties such as preserv- ing intersections and continuity in coarse category. For example, simple intersection between sets of odd natural numbers and even natural num- bers is empty set, but in coarse category their intersection is equivalent to all natural row. Proposition 4.10. If F is a finitary normal functor of finite degree n ≥ 1, then F is normal in CS. Proof. The preservation of weights is in a follows from the definition of the coarse structure on FX (see definition). All the other properties follow from propositions 4.7, 4.9 and 4.12. Jo u rn al A lg eb ra D is cr et e M at h .V. Frider 27 References [1] J. Roe, Index theory, coarse geometry, and the topology of manifolds, Regional Conference Series on Mathematics, vol. 90, CBMS Conference Proceedings, Amer- ican Mathematical Society, 1996. [2] N. Higson and J. Roe, Analytic K-homology, Oxford University Press, Oxford, 2000. [3] N. Higson, E.K.Pedersen, and J. Roe, C*-algebras and controlled topology, Kthe- ory, 11(1997), 209-239. [4] Frider V., Zarichnyi M., Hyperspace functor in the coarse category, Visn. L’viv. un-tu. Ser. Mech.-Mat. (2003), № 61, 78-86. [5] Shchepin Ye. V., Functors and ucountable degrees of compacta, Uspekhi Mat. Nauk, (1981), V. 36, №3(219) 4-61. [6] Telejko A., Zarichnyj M., Categorical topology of compact Hausdorff spaces.- Lviv:VNTL,1999. [7] Fedorchuk V. V., Covariant functors in category of compact Hausdorff spaces, absolute retracts and Q-manifolds, Uspekhi Mat. Nauk, (1981), V. 36, №3(219) 177-195. [8] P. D. Mitchener, Addendum to ”Coarse homology theories”, Algebraic and Geo- metric Topology, 1 (2001), 271-297. P. D. Mitchener, Addendum to ”Coarse ho- mology theories”, Algebraic and Geometric Topology, 3 (2003), 1089-1101. Contact information V. Frider Ivan Franko National University of Lviv, 1 Universitetska Str. 79000 Lviv, Ukraine Received by the editors: 19.09.2005 and final form in 15.12.2005.

Normal functors in the coarse category

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