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Topology Proceedings 38 (2011) Pp. 253-278: Inverse Hypersystems

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Topology Proceedings 38 (2011) Pp. 253-278: Inverse Hypersystems Volume 38, 2011 Pages 253–278 http://topology.auburn.edu/tp/ Inverse Hypersystems by Nikica Ugleˇsić Electronically published on October 22, 2010 Topology Proceedings Web: http://topology.auburn.edu/tp/ Mail: Topology Proceedings Department of Mathematics & Statistics Auburn University, Alabama 36849, USA E-mail: [email protected] ISSN: 0146-4124 COPYRIGHT ⃝c by Topology Proceedings. All rights reserved. http://topology.auburn.edu/tp/ TOPOLOGY PROCEEDINGS Volume 38 (2011) Pages 253-278 E-Published on October 22, 2010 INVERSE HYPERSYSTEMS NIKICA UGLESIˇ C´ Abstract. The notion of a (generalized) inverse hypersys- tem in a category C, that generalizes the known notion of a generalized inverse system, is introduced via a functor of a cofinally small weakly cofiltered category to C. The appropri- ate morphisms are also defined such that they generalize the morphisms of generalized inverse systems. The correspond- ing category P RO-C is constructed such that pro-C and P ro- C are subcategories of it. In comparison to the relationship between pro-C and P ro-C, the essential benefit is that there exist inverse hypersystems which are not isomorphic to any generalized inverse system. The notion of a cofinite inverse hypersystem is also introduced, and it is proven that every generalized inverse hypersystem is isomorphic to a cofinite inverse hypersystem. At the end, it is shown by example how an inverse hypersystem could occur. 1. Introduction Since 1960, when Alexander Grothendieck introduced the no- tion of a pro-category and the appropriate technique (see [7]), pro- categories have had a very wide range of applications, especially in geometric and algebraic topology (see [1], [2], [6], [11], [3]). How- ever, it has been noticed that in some considerations, the notion of an inverse system is too restrictive. Namely, there are specific circumstances in which more than one morphism between a pair 2010 Mathematics Subject Classification. Primary 18A05, Secondary 18B35. Key words and phrases. cofiltered category, generalized inverse system, in- verse system, pro-category. ⃝c 2010 Topology Proceedings. 253 254 N. UGLESIˇ C´ of terms of an inverse system has occurred. To consider such a case, Sibe Mardeˇsi´cand Jack Segal [11] introduced the notion of a generalized inverse system. This requires generalizing and extend- ing the pro-category pro-C to a larger one, denoted by P ro-C [11]. Although very useful as tools, the generalized inverse systems and \pro-category" P ro-C cannot yield any essentially new result com- pared to the inverse systems and pro-category pro-C. Namely, every generalized inverse system X admits an (ordinary) inverse system X0 which are isomorphic objects of P ro-C [11, Theorem I.1.4]. In other words, pro-C ⊆ P ro-C is a skeletal subcategory. Therefore, a new extension is needed. The presented one is based on the following replacement: Instead of the requirement that for every pair pu; pv : X휆0 ⇉ X휆, there ex- ists a pu0 : X휆00 ! X휆0 satisfying pupu0 = pvpu0 , we put a weaker condition: for every pair pu; pv : X휆0 ⇉ X휆, there exists a pair pu0 ; pv0 : X휆00 ⇉ X휆0 satisfying pupu0 = pvpv0 . The idea came from studying S-equivalence and the corresponding sequence of the Sn- equivalences (see [10], [14], [4]), where such families of morphisms between the terms of inverse sequences naturally occurred. Ac- cording to that weaker condition, the notion of a weakly cofiltered category is introduced. Consequently, in the usual way, the notion of a generalized inverse system is generalized to so-called (gener- alized) inverse hypersystem. More precisely, a generalized inverse hypersystem X ≡ (X휆; pu; Λ) in a category C is a (covariant) func- tor X :Λ !C of any cofinally small weakly cofiltered category Λ to the category C. Further, the notion of a map of generalized inverse systems is generalized to a map of (generalized) inverse hypersys- tems, X ! Y = (Y휇; qv;M), such that, for every 휇 2 Ob(M), a unique f휇 : Xf(휇) ! Y휇 is replaced by a set F휇 of morphisms sub- jected to certain conditions. Finally, the morphisms of generalized inverse hypersystems are defined to be the equivalence classes of the corresponding maps by an appropriate equivalence relation. The obtained category is denoted by P RO-C. Its subcategory P RO1-C, determined by all the morphisms having the representatives with all F휇 singletons, is also considered. By construction, pro-C ⊆ P ro-C ⊆ P RO1-C ⊆ P RO-C holds, and P ro-C is not a skeleton of P RO1-C (Theorem 4.6). INVERSE HYPERSYSTEMS 255 The main results of the paper are as follows: 1. There exist categories C (for example, C 2 fSet; Top; H(Top)g) and there exist generalized inverse hypersystems in C which are not isomorphic in P RO-C to any generalized inverse system in C (Theorem 4.6 and Corollary 4.8). 2. Every generalized inverse hypersystem X = (X휆; pu; Λ) in a category C is isomorphic to a \small subhypersystem" X0 of the 0 0 0 0 0 0 kind (X휆0 ; pu; (Λ ; ≤)), where 휆1 ≤ 휆2 if and only if Λ (휆2; 휆1) 6= ? (Theorem 4.11). By that fact, it makes sense (and, above all, is very useful) to con- sider inverse hypersystems (Definition 4.12) which are those gen- eralized inverse hypersystems X = (X휆; pu; Λ) having a directed preorder ≤ on the set Ob(Λ) such that (8휆, 휆0) 2 Ob(Λ)휆 ≤ 휆0 , Λ(휆0; 휆) 6= ?: Such an inverse hypersystem X is denoted by (X휆; pu; (Λ; ≤)). Fur- ther, the notion of a cofinite inverse hypersystem in a category C is introduced in the most natural way. The main fact is as follows. 3. Every generalized inverse hypersystem X = (X휆; pu; Λ) in a category C is isomorphic to an inverse hypersystem Y = (Y휇; qv; (M; ≤)) with M cofinite and ordered such that every Y휇 is an X휆(휇) and fqv j qv : Y휇0 ! Y휇g = fpu j pu : X휆(휇0) ! X휆(휇)g (Theorem 5.3). 2. A weakly cofiltered category Recall that a category Λ is called cofiltered (dual to filtered, see [1], [2], [6], [11], [3]; in some places the words “filtering” and “cofil- tering" are used), if the following two conditions are fulfilled. (i) (8휆j 2 Ob(Λ), j = 1; 2)(9uj 2 Λ(휆, 휆j), j = 1; 2), i.e., every pair 휆1 and 휆2 of objects admits a diagram 휆1 - u1 휆 ; . u2 휆2 256 N. UGLESIˇ C´ (ii) (8u; v 2 Λ(휆0; 휆)(9w 2 Λ(휆00; 휆0)) uw = vw, i.e., the follow- ing diagram commutes u 휆 휆0 w 휆00: v Example 2.1 ([8, VI. 16]). Every category satisfying condition (i) and having equalizers is cofiltered. A category Λ is said to be cofinally (or essentially) small, pro- vided there exists a small subcategory Λ0 ⊆ Λ which is cofinal in Λ; i.e., for every object 휆 of Λ, there exist an object 휆0 of Λ0 and a morphism u : 휆0 ! 휆. The simplest example of a (cofinally) small cofiltered category is a directed preordered set (Λ; ≤). In that case, for every pair 휆, 휆0 2 Λ, card(Λ(휆0; 휆)) ≤ 1, and Λ(휆0; 휆) 6= ? if and only if 휆 ≤ 휆0. Example 2.2. Let A be an infinite set, and let B = fbi j i 2 Ng ⊆ A be a countable subset such that for i 6= j, bi 6= bj. Put 휆1 = A, and by induction, 휆i+1 = 휆i n fbig, i 2 N. Further, for every i 2 N, put ui : 휆i+1 ! 휆i to be the inclusion function, and let vi : 휆i+1 ! 휆i be the function defined by { a, a 6= bi+1 vi(a) = bi+2, a = bi+1 : Let us define a category Λ by putting Ob(Λ) = f휆i j i 2 Ng, 0 Λ(휆 ; 휆 ) = f1 g, Λ(휆 0 ; 휆 ) = ? whenever i < i, and let Λ(휆 0 ; 휆 ) i i 휆i i i i i be the set of all possible compositions of the above defined functions whenever i0 > i. Then Λ is a small cofiltered category. Indeed, condition (i) holds via maxfi; i0g, while condition (ii) follows by 0 the fact that given any u; v : 휆i0 ! 휆i of Λ, i ≤ i , the compositions uui0+1 = vui0+1 : 휆i0+1 ! 휆i coincide with the inclusion 휆i0+1 ,! 휆i. (Moreover, it is readily 0 seen that for every pair i < i , the set Λ(휆i0 ; 휆i) consists of two elements|the inclusion and the function that is the identity at each element but bi0 , which goes to bi0+1.) In certain considerations, condition (ii) seems to be too restric- tive. Therefore, we introduce a weaker one obtaining a more general type of “cofiltered” category in the following way. INVERSE HYPERSYSTEMS 257 Definition 2.3. A category Λ is said to be weakly cofiltered (or pairwise cofiltered) provided condition (i) and the following condi- tion are fulfilled. 0 0 0 0 00 0 0 0 (ii) (8u1;u2 2 Λ(휆 ; 휆)(9u1;u2 2 Λ(휆 ; 휆 )) u1u1 = u2u2 ; i.e., the following diagram commutes 0 휆 0 u1 . - u1 휆 휆00 : u - . u0 2 휆0 2 The dual notion is a weakly filtered (or pairwise filtered) category. Example 2.4. Let A be a set such that card(A) ≥ 2, and let 휎 : 2 A ! A be a permutation such that 휎 = 1A and 휎 6= 1A. For every i 2 N, put 휆i = A. Further, for every i 2 N, put u1 = 1A : 휆i+1 ! 휆i and u2 = 휎 : 휆i+1 ! 휆i. Let us define a category Λ by putting Ob(Λ) = f휆 j i 2 Ng, Λ(휆 ; 휆 ) = f1 g = f1 g, Λ(휆 0 ; 휆 ) = i i i 휆i A i i 0 ? whenever i < i, and let Λ(휆i0 ; 휆i) be the set of all possible 0 compositions of members of fu1; u2g = f1A; 휎g whenever i > i.
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