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S Ü Fen Ed Fak Fen Derg Sayı 24 (2004) 57- 61, KONYA The Sheaf of The H-Cogroups Ahmet Ali ÖÇAL1 Abstract: In this study we show that the sheaf which is constructed in [1] is a sheaf of H -cogroups and it is an H -cospace if we equip the set of sections of this sheaf with the compact-open topology. Finally we give some characterizations. Key Words: Cospace, Sheaf, Cogroup, Homotopy H-Kogrupların Demeti Özet: Bu çalışmada [1] deki metod kullanılarak H-kogrupların demeti inşa edilmiş ve bu demetin her bir kesitlerinin kümesi kompakt açık topoloji ile donatılmış ise bir H-kouzayı olduğu gösterilmiştir. Sonuç olarak bazı karakterizasyonlar verilmiştir. Anahtar Kelimeler: Kouzay, Demet, Kogrup, Homotopi Introduction Let C be the category of topological spaces X satisfying the property that all pointed topological spaces ( X , x ) , x Î X have the same homotopy type. This category includes for example all topological vector spaces. Let us take X Î C as a base set if Q is any abelian H cogroup, then there exists a sheaf (H , p ) over the topological space X which is formed by a Q H -cogroup. For each x Î X , p -1 (x ) = [Q; ( X , x )] = H x is the stalk of the sheaf which has a discrete topology. (where [Q; ( X , x )] = H x is a set of homotopy classes of homotopic maps preserving the base points from (Q, q 0 ) to ( X , x ) ) [1,7]. 2. The Sheaf of the H -Cogroups Definition 2.1. Let ( X , x ) be a pointed topological space. The diagonal map DX :X ® X ´X is defined by D X (x ) = (x, x ) and the dual of the diagonal map D X , denoted by Ñ X , with ÑX :X Ú X ® X defined by Definition 2.2. Let (H , p ) Ñ X (x, x 0 ) = Ñ X (x 0 , x ) = x. [2]. be a sheaf over X and s Î G( X , H ) [1]. The folding map Ñ Hx : H x Ú H x ® H x is defined by Ñ H x ( [ f ]x , (s o p ) [ f ]x 1 ) = [ f ]x Gazi University, Faculty of Arts and Sciences, Department of Mathematics, 06500 Ankara, Turkey The Sheaf of The H-Cogroups and Ñ H x ( (s o p ) ([ f ]x ) , [ f ]x ) = [ f ]x . Now let us define a comultiplication on H x as follows: if [ f ]x , [c ]x Î H x then vx : H x ® H x Ú H x v x ( [ f ]x ) = [ Ñ X o ( f , c ) o v ]x , where v is an operation of a Q H -cogroup and c :Q ® ( X , x ) is a constant map. It follows from this definition that the comultiplication v x is well-defined, closed, continuous and with this comultiplication H x is an H -cogroup. In fact, the constant map cx . H x ® H x which satisfies c x ( H x ) = [ c ]x is a homotopy identity of a pointed topological space ( H x , [ c ]x ) . That is, each of the composites ( 1H x Ú cx ) vx ® Hx Hx ÚHx is a homotopic to 1 H x , that is, ( ® ( cx Ú 1H x ) Ñx Hx ÚHx ) ®H ( x ) Ñ x 1H x Ú c x v x ~ - 1H x ~ - Ñ H x c x Ú 1H x v x . The continuous map fx :H x ® H x is a homotopy inverse for H x . That is, each of the composites ( 1H x Ú f x ) vx Hx ® Hx ÚHx ® ( f x Ú 1H x ) Ñx Hx ÚHx ®H x is homotopic to c x , that is Ñ x 1H x Ú f x v x ~ - cx ~ - Ñ H x f x Ú 1H x v x , ( ) ( ) where f (x ) is defined by fx ( [ f ]x ) = [ f o f ]x and f is homotopy inverse of a Q H -cogroup. It can be shown also that v x is a homotopy associative. Now we have the following results: Result: 1. H x is an H -cogroup. Result: 2. The sheaf which is constructed in [1] is a sheaf of H -cogroups. Let G ( X , H ) denote the set of global section of H with the compact-open topology. Then the mapping v ¢: G ( X , H )® G ( X , H )Ú G ( X , H ) which is given by v ¢ (s )(x ) = v x ( s (x ) ) = v x ( [ f ]x ) = [ Ñ x o ( f , c ) o v ]x ( s Î G ( X , H ), x Î X ) defines a comultiplication on G ( X , H ) such that v ¢ is well-defined, closed and continuous [3]. In fact, if U is an open neighborhood of v ¢ (s ) in G ( X , H ) Ú G ( X , H ) 58 Ahmet Ali ÖÇAL for every s Î G ( X , H {M ( C i , Oi ) }iÎJ ( J ), then there exists is finite collection of open sets in the subbasis, is finite) such that v ¢ (s ) Îæç I M ( Ci , Oi ) Ú I M ( Ci , Oi )ö÷ Ì U . iÎJ è iÎJ ø Since the comultiplications v x are continuous in H x , we can choose neighborhoods U i of s (x ) such that if si ( x ¢ )ÎU i , then v ¢ ( s i )(x ¢) = v x¢ ( s i ( x ¢ ) )Î ( Oi Ú Oi ) Thus, s ÎI M ( C i , U i ) and if s ¢Î M ( C i , U i ) then V ¢ (s )Î( I M ( Ci , Oi ) Ú I M ( Ci , Oi ) ) since v ¢ ( s ¢ )(x ) = v x ( s ¢ (x ) ) £ I ( Oi , Oi ) for all x ÎI C i . Let I : X ® H be the section (p o I =1x ) satisfying I (x )ÎC ( c x ) Ì H x . Such an I exists continuous [4], where C ( c x ) is the path component of c x in H x . Thus, I is the identity of G ( X , H ) . Hence, we have the following result: Result: 3. Since H x is an H -cogroup, G ( X , H ) is an H -cospaces with comultiplication v¢ . Let S be family of supports on X and V Ì X . Then GS V ( V , H ) is the collection of and sections s ÎG ( V , H GS V ( V , H ) satisfying s = { x Î X : s (x ) ÏC (c x ) } where S V = { A Ì V : AÎ S } . The collection ) is closed under the comultiplication of G ( X , H s ÎGS V ( V , H ) then which restricted to GS V ( V , H ) for if s Î S V . Since v ¢S V (s ) it follows that v ¢S V (s ) Ì s c { = xÎ X : v S V ( V , H )ÎC ( c x ) and so v ¢S V (s ) Î S V { xÎ X : I (x )ÏC ( c x ) } = Æ ) and hence g U V ( S )= S c , . Also I ÎGS V ( V , H ) (because I = ) is an H -cospace. X , G ( X , H ) is an H -cospace (Result 3). If V open subset of X such that V Ì U we can define a map restricted map, that is }É s GS V ( V , H Thus for any open subset U is of If U ,V ) g U V : G ( U , H ) ® G (V , H ) is another which is called V. and W are any three open subsets of X such that W Ì V Ì U then one can U V U U ì ü , X ý is called direct system [5]. and so í G ( U , H ), W W V V î þ 3. The Characterizations Let Q be any H -cogroup and X 1 , X 2 be two topological spaces in Category C . Let observe that g =g og g H 1 , H 2 be the corresponding sheaves which is given Result 2, respectively. Let us denote these as the pairs ( X 1 , H 1 ) and ( X 2 , H 2 ) . Definition 3. Let the pairs ( X 1 , H 1 ) and ( X 2 , H 2 ) be given. We say that there is a homomorphism between these pairs and write F b * , b : ( X 1 , H 1 ) ® ( X 2 , H 2 ), ( ( * if there exist a pair F b , b ) such that ) 1. b : X 1 ® X 2 is a surjective and continuous map. 59 The Sheaf of The H-Cogroups 2. b * : H 1 ® H 2 is a continuous map. 3. b * preserves the stalks with respect to b . That is, the following diagram is commutative b* ® H1 p1 ¯ H2 ¯p2 b ® X1 X2 4. For every x1 Î X 1 the restricted map b * H x1 : H 1x1 ® H 2 b ( x1 ) is a homomorphism. Theorem 3. ( X 1 , H1 ) Let the pairs and ( X 2 , H2 ) be given. If the map b : X 1 ® X 2 is surjective and continuous, then there exists a homomorphism between the pairs ( X 1 , H 1 ) and ( X 2 , H2 ). Proof. Let x1 Î X be arbitrarily fixed point. Then b ( xi )Î X 2 and [ Q : ( X , x1 ) ] = H 1x 1 Ì H 1 , [ Q : ( X , b (x1 )) ] = H 2 b ( x1 ) Ì H 2 are corresponding H -cogroups or stalks. Since ( X 1 , x1 ) , ( X 2 , b ( x1 ) ) are pointed topological spaces and f1 , g1 are base-points preserving continuous from ( Q, q 0 ) maps ( X 1 , x1 ) then there exists f 2 , g 2 base-points ( X 2 , b (x1 ) ) can be defined as f 2 = b o f1 , g 2 = b o g1 , to preserving continuous maps from ( Q, q 0 ) to respectively. Moreover, if f1 ~ g1 rel.q 0 , then it can be easily shown that f 2 ~ g 2 rel.q 0 . Thus the correspondence [ f ]x 1 ®[ b o f ]b (x ) 1 is well-defined [6] and it maps homotopy classes of base- points preserving continuous maps from ( Q, q 0 ) to points preserving continuous maps from ( Q, q 0 ) to there corresponds a unique element ( X 1 , x1 ) , to the homotopy classes of base( X 2 , b (x1 ) ) . That is, to each element [ f ]x 1 [ b o f ]b (x ) . 1 Since the point x1 Î X is arbitrarily fixed, the above correspondence gives us a map b * : H 1 ® H 2 such that b * ( [ f 1) b * -1 b * ] ) = [ b o f ]Î H 2 , for every [ f ]Î H1 . is continuous. Because if U 2 Ì H 2 is any open set, then it can be shown that (U 2 ) = U 1 Ì H 1 is if U 2 Ì H 2 is an open set, then U 2 = Ú s i2 (Wi ) and an open set. In fact, iÎI p 2 (U 2 ) = Ú Wi , where the Wi ’ s are open neighborhoods and the si2 iÎI are sections over Wi . Thus U Wi Ì X 2 is an open set and b -1 æç Ú Wi ö÷ Ì X 1 is an open set since b is a surjective and è iÎI ø iÎI continuous map. Furthermore, since b -1 (Wi ) , i Î I are open in X 1 , there exist sections s1i : b -1 (Wi ) ® H 1 such that ( ) Ú s 1i b -1 (Wi ) Ì H 1 iÎI is an open set. It can be shown that ( ). U 1 = Ú s 1i b -1 (Wi ) iÎI 2. b * preserves the stalks with respect to b . In fact, for any (p 60 2 [ f ]x Î H 1x 1 1 ( (b op 1 ) [ f ob * )( [ f ] x1 Ì H1 , )= b (p ( [ f ] ) )= b (x )= x )= p (b [ f ] )= p ( [b o f ] )= x . ]x 1 1 1 x1 2 * 2 x1 2 x2 2 Ahmet Ali ÖÇAL 3.It can be easily shown that b * H 1x1 is a homomorphism. ( ) As a result F = b * , b is a homomorphism. 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