Pultr, Aleš

Trnková, Věra

English

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeAleš Pultr; Věra TrnkováOn realization and boundability of concrete categories in which the morphismsare choiced by local conditionsCommentationes Mathematicae Universitatis Carolinae, Vol. 8 (1967), No. 4, 651--662Persistent URL: http://dml.cz/dmlcz/105139Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1967Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCommentationea Mathenaticae Uniteraitatia Carolinae 8,4 (1967) ON REALIZATION AND BOUNDABILITY OF CCNCHETE CATEGORIES IN WHICH THE MQRPHISMS ARE CHOICED BY LOGAL CONDITIONS A. FULIR, V. TRNKOVÁ, Praha The preaent páper ia m contribution to the queation of fuXl embeddability of ooncrete categoriea into categories of algebra* (boundabilitý) aaked by J.R. Iabell in 11]. In [2 3 and [3] a method of embedding of categoriea deacribed by functora is elaboráted. U8ing that method, we prove that al-ao categoriea the object of which are topological apaces and morphÍ8ma aelected among continuou8 mappinga by means of lo-cal conditions given by aome functor, are again boundable. E*g.t let ua aak whether the category of topological apace8 with locally one-to-one continuou8 mappinga ia boundable* We obtain a poaitive anawer using local aelection by means of binary relationa "to be diatinct". Slmilarly we obtain the boundabilitý of, e*g«, the following categoriea: topological apaces with local homeomorphisms, topological apaces with quasi-local homeomorphisms (here ia < different "localiaation" than in previous examplea), topological spaces with open local homeomorphiama, metric 8pace8 with locally Lipschitzian mappinga (here we must, of course, ahow first how the globál Lipschitz condi-tion can be realized by meana of a functor), etc. - § 51 -We use the notation from [33 and L43. Without partiou-lar mentioning we use the assuiqption of non-existence of mea-aurable cardinala (oř, the weaker assumption (M) from C3J). The reader acquainted with the quoted papers will see easi-ly ln which statements we need this assumption and in which not. For brevity, we use the expres si on F-structure on X for sny subset of F(X) where F is a set functor (i.e.f a functor from the category of aets into itself); if rt 8 resp. are F-a truc tur es on XtX resp., we say that a mapp-ing f: X —• X is r s-conpactible if F(f)(r) c s in the covariant čase, F(f)(s) c r in the contravariant one. The páper is divided into three parta. The first one contains definitions and some theorems. The main results are proved in the second part. Finally, the third part deals with applications of these results to concrete caaes. I. Definition 1: Let F be a covariant set functor, r f s F-structures on topological spaces XfX resp. We say that a mapping f: X —* X i s locally r 8-coapacfciblef if: for every x £ X there is an open U 9 x with F(f) ( [ F ( í u ) ( ř ( U ) ) l n r) c s Cwhere ju i s the em~ bedding of U into X . If F i s contravariant, we say that f i s locally r s-compactible, i f for every x c X there is an open U ^ X with F Q U ) ÍFÍfXe)) c F(ju)Cr) . Remark: The definition is motivated as followa: in $52 -covariant (contravariant) ease we reciuest that for every x e X there be au open U containing x such that tlU i (U, CFC£U.)!-*(*>))—* (y^} C-P IU : ( U , F (JUX/L))-*(Y, A>) is compaetible. Def lnltion 2. if r is an F-structure on X we de-fine a P~ F structure r* on X as followa: a e r * if and only if a n r * J0 . lanými V Let F be a contravariant set funět©r. If r, s are F-structures on X,Y respectively, then f: X~* —> X i s r a-compactible i f and only i f i t i s r * s * -compactible. Fgoof: Let F(f)(s) c r f a e r * . Then evidently P T C f H a ) - (p (f ))-* (Q, ) e A>* . Conversely le t r F(f )(*,+ ) c A*, f e ^ > . Let FC-DCf)**-* then C F C - f ) ( £ ) ) e /& * and, eonsequently, P " R * X F ( t )Cf »«CFCÍ)J^CFC«f ) )6 ^*; however ý e e ČFC-ř))"' C FCfKf ) ) ^ >̂ ? which is a contradiction. Lemmat 2: Let F be a contravariant set functor. Let f: X —y X be a mapping, r f s F-structures on XfX res­pectively, U c X • Then Ff*u ) ( F C-ř )(*)) c FCjy X*,) (*.) i f and only i f p-F(f)ílp-F(#u)(P-F(UmnM,*>c*>* . ^ Beafiíi Let ( * * ) do not hold. Then there exiets - #53 -&e P-FC4)(CP-F($U)(P~F(U))3 n * * ) auch that o, „A>*0 . ^ t P"FC*>(Jtr)mCFC-f)riC+) •, Jlre LP~F(?U K P " R U ) ) . ] n / ř * i . e . , /r n H. = 0 and Jtr- (FC2u))~i Cc) where c c F(U) . Then e n n FCŽU ) (K) = 0 . Jí f js a, n A> then FCÍ)Cf)«-f r j Fiiu)CFC*)i$)) e*c . Then F C ^ K F C f K f ) ) # FCfryC/t) , i . e . ( * ) does not hold. Let ( * ) do not hold. Then there eziats 9& £ A> such that f " F(?u)(F(f)(*ť» $ FC-3-u )(*•) . Conaequently (FCJuH^Cf)*"- ' 0 ," *••• (FítuM" (§)*** • One-point set C f ) i s an element of P" F ( (J) má therefore cu e í P" F(£u ) ( P~ F CU )) J n / t * . Using ( * * ) there i s P~FC«f)C^> e 4> * f i . e . CF.CÍ ))"<f ( a ) n * - # . B u t * 6 CFčf »""?<&) n /b . Definitlon 3« Let F be a covariant (contravariant) set řunctor, r,s F-structures on topologlcal spaces X,Y respectively. We say that a mapping f: X —> Y is quasilo* cally r s-compactible if: for every non-void open U c X there is a non-void open V c U such that Fíf)[ t F(#v) (F(S/))2 n n, ) c A> - $54 The arem 1, Let P be a contravariant aet functor. Let f: X —> X be a mapping, P ,8 F-atructurea on topologi-cal apaoea XtX reapectively. f ia locally (quaei-local-ly) P 8-compactible i f and only i f i t ia locally (quaailo-cally) P * a* -compactible. Proof: Thia ia an eaay conaequenoe of Lemma 2« Definition 4» Let XtX be topological apacea. A map-ping f: X —> Y i s termed aemi-open if f (A) has a non-void interior whenever A haa a non-void interior. Remark: Every mapping f such that for every open non-void U ttere ia a non-void V c U with f(V) o-pen, ia , of course, semi-open. Definition 5. The estegory T ( E, , ••• ? 5t ' <**»•"» Qn,) ia defined aa followa: The objecta are ayatema C X f t > fy?*"; <*„,,, fy , *"x ťm.) where If i s a topology on X t r^ F̂  -atpucturea and r^ 0^ -atructurea, the morph-lams fporn f X , t , fy,.. • , /%^ , /t^, , , , y /t^ ) into C Y , & ? ^f ? »•»; - ^ ? -̂r j * *' ? ̂ ^n, ) are the continuoua map-pinga fpom C X , f ) into C y, & ) which ar© r^ a^ -compactible fop i » lf**«» n and locally r^ â  -compactib-le for 1 = 1 , . . . , m • The category T^ £Pr» •• •; PVv • Gf # • • • 7 §nJHf>'"> H*^ is defined as followa: The objecta are ayatema C X 9 f , V " > **,i* fV"> >£,;*"*•••>'*£ > w h e r e "* ia a topolo-gy on X f p^ F̂  -atructurea, r j Ĝ  -stpuctupea and r'* H^ -atpuctupea on X . The morphiama fpom CX , f , fy7.*» • ' •^>^V-> /C> <<»•••> *£ ) i n t 0 c y ; *> *<iy* '-> i*m,i *ii'*'><*'m>) *>ii>'> > ^fl ' are the continuoua - « 5 -eemi-open mappinga frea í X , r ) into C Y, &) whioh are r̂  â  -coapactible for 1 «* 1, . . .» n , locally r j e^ -compactible for 1 * l 9 . . . t m and quaeilocally rV aV -compactible for i * 1, . •«, p. II . Deflnltion 6. Let F be a eovariant aet functor, r an F-atructure on a topological apace X . Befine a P" F-s truc ture r and r on X aa followa: a e F i f and only if for every x e X there exiats an open U B x with a, n si n L F(j>u )(F(U))J m 0 a e v i f and only if for every non-void open U the-re exiata a non-void open V c U with ou ft H, n n í ? (ju )(FCU))3 * 0 -LfimTO 3 ! A continuoua mapping f: X —• X IQ local-ly r a-coapactible if and only if i t ia F ?-compactible. Proof: We muat prove that for a continuoua f the f o l -lowing statements are equivalent: I . V x e X B U s x open with F C * ) C C F(<yu)CF(U))2s> *> > c A> II . P - F ' ( f ) ( S ) c ^ Let I . hold. Let a e i" . We muat prove P~Fí4) Cou ) -• C FC-ny^Ca) € /t . Let x e X ; then there exiata U, 3 K open auoh that F(* ) i í FCfy)CFCU, )) Jn /tO c -O. Sinoe a e T there exiata \/ 3 t (x) open with [ F ( ^ ) ( F ( y ) ) J A ^ n a * ^ ( Pat U » U, o 4~<CV). Then U i X ia open and l t ia aufficient to prove 056 <1> <Ft*)r'(a,)n LFCiy)<T<U))3n * - $ -Sinoa iu m i^ * iUfl/i thara la F(?U)(F(U)) C c F C ^ ) C F f ř ( / f « f ) Í F ť U ) ) c F C f y K F C U , ) ) , conaaquant-ly FC + >£CFCá,>C'F«/>)3n*, )cFff>UFfy K F O í , » 3 n / * ) c * • If (1) does not hold than there axiata ^ € Ff U) with ? - F (*u ) C *i) e K n CFíi)T*Ca), i.e. F (+ ) C f ) € a , FCf) C f ) € * . Thara ia f P ^ - *y « 9-for a convenient g , conaequently F f f ) « F C ^ ) * f r ^ ý ^ R ^ ) # I l l • f t F C f ) f ř ) • F ř * ) f F % K í ) ) • F f ^ ) f F ^ > ^ > ) t f F ^ ? f F ^ ; , • Conaaquantly, Fí-f)ff )e C F(#v )(F(V))3 n /o ň a, whioh la m contradiction. Lat I I . hold. Chooaa x e X . Slnoa FC y ) N, A> e 3> thara ia C Fí-ř ))""f C F ( / ) N * ) e £* . Conaaquant-1 * thara axiat» U 3 X opan with ÍCF CDT^CFCY)^*^ n * ntFC^ )CF(U))1 ** 0 , i r f € í F(^u )CF(U))]n n> n, than f 4 C F C-f » - " C F c y > v ,& ) ? i . e # F C f ) í f ) * F(Y)\ * i . a . FC-ř)Cf ) 6 * • Lcmnjn 4. A continuoua aaml-opan mapping f: X —> X la quaailocally * a-coaipactibla i f and only i f i t ia ř 1 -coai-pactible. ££&££: We mat prove that the following atatamanta are equivalent: I. For every non-void Opan U c X thara exiate a non-void open V e U with FCfK LFCjy )( FC V))Jn* ) c *>. - 657 -XI. PmFC4) (X) c £ Lat I . hold* Lat a e H . Wa muat prove P~ F C-f) (&)*> • CFCf )Y* Ca ) e £ Lat U be open non-void in X . There exiata non-void opan Uj c. U with F C f X C F C ^ ) ( F ( ^ ) ) ] n / t ) c > b , f ia aemi-open, theref oře there exiata non-void open W c f ( U^ ) • Sinca a e 0 t there exiata non-void open W1 c 1 with £ F C$w ) CF(VJ,, ))1 n a, n A> *> 0 . t ia con-tinuous, V * U, A f - *( W^ > ia non-void open, V c U . Now i t ia euffieient to prove: (1) C CFC-P )r1(a)3 n í FC?V )(F(V))2n* - 0 Sinoe ^ « ^ o ^ ^ there ia F(#y)CF(V)) m • F"í^ KP^nn)(F(V))>c FCfy )CF(U1 )) , conaequently, (2) F Í - ř ) C C F y y ) ( F f y ; ) 3 n A ) c * . Lat (1) do not hold. There exiata [ c [ F(fy)CF(V))3 n H> •uch that FČ-řKf ? € a . Then, uaing (2) f there ia K C+ ) C f ) c a n /s> . Sinca +(V) a W, thera ia ^iv • 9** '̂  # ?V * o r convenient g • Put f = « F # K ) f*fc > • Tban Ff í ) Cf ) - F f # H F f ^ ) ty )) ** s F(^, HFCg,)) 6 Ff ^ ) ( F f ^ )) whioh ia a oontradic-tion. Lat I I . hold. Chooaa U non-void open in X ; F CY) ^/fr € é % , ©onaequently íF(4)T C F CY ) ^ A>) e % , Then there exiata non-void opan V c U with -#58— iCF(f))-''(F(y)s/0)2n*.niF(4y)(F(V»2 - 0 . if f e CF(?y ) (FCV)) 2 n h, then §4 í C F m r V F f y ) v * ) eonaaquently F< + )(f) + FCy)s N/S> , i . a . F C - ř K f ) e A> . Aa an eaajr conaequenoe of Lemmaa 1,3 and 4 wa obtain the following two theorema: where Gj « (^ for covariant (5- ? (?/ « P~ G% otherwiae. Tfrepren j» T^ ČEJ,..,, F^ I ©,, . . . , Q^ l H„..., H^ ) z ^ =* scp-, P+, i? f...,5,p-^,.-vP-^ > P - « „ . . . , P"** ) , w h e r e G^ and H^ are defined as in the previous theorema Reqiarfo: Here and in the following, the top o logy ie al-ways represented by the set of all open subsets* Corollars; If F19.,., F^, Gi,"*> §*,' Hi9...9 H^ are constructive functors (set 141)9 then there exists an iateger k and a set A such that TCFi , . . . , f * i <?„•••> <v>=* SCCP-)*-. VA ) T%Cfí,..., F^ I &„... , Q J H„..., H^)=í SCCP-)*C VA ) Conseojaently the categaries TCÍ^?.^; F^ I <J^.,., G^ ) and T^CE,,...) (^ i ^1?—; Qw ' H7>"v H^ ) ars in suoh a čase boundable* - P$9 -in i n l n dř2 &% fou xxx* The ořem 4. The following categoriea Si ?' •' ? ̂ r are boundable: objecta are alwajpe topological epacee, tha morphiame ara in <&i : locally one-to-one continuoua mappinga, open local homeomorphismaf loeal homeomorphiama, quaaiopen continuoua mappinga (i. a. continuoua which ara open with the exeeption of a nowhere danae set), in <£s : quaailocal homeomorphiama (i.e. continuoua map-pinga which are local homeomorphiama with tha exeeption of a nowhere denae aet). Remark: The reader aeea certainly other varlationa. P£oof• It aufflcea to repreaent quoted categoriea aa full aubcategoriea of the categoriea from Theořema 2 and 3. Á^ zdt T ř - I & ) (endow the object by the Q-atructu-7 T <&z z% T C P"*" í Os ) (the P^-atrueturea ara repaated to­pol ogy, Q-atructurea aa above) <£3=* T ( - I fl , Q • P+ ) (the Q-atructure aa above; i f (Xt ir ) ia a topological apace, define a Q. • P*-atructu-re r on X aa followa: ( A , Ca,)) e * <=» a. 4 A ) <&, "=± T ř - I - I P* ) ^ t h c p * -etructurea are repaated to-pologlea) JCfzX T% ( - I - I Q , a * P-t ) analogically aa with <#3 . - 060 -1HHHHI S - **•* (X)jD ) ba a metric epace. Define a p o d -atructura ( R ia the aet of reál numbera) p aa followa: Lat C X , ^ ) / ^ ^ p ' ) be matric apacee. A mapping f: : X —* X# ia Lipechitziaii i f and only i f i t ia p p ' -com-pactible. Proof: I . Lat -P i C X, <p ) —+ C X% <p' ) be Lip-echitzian. Then p'(* C X >> -PC/^J) éi K-p Cx, /y. ) for every x t y € X . Lat Qf e p ' j then for every ut v € 6 ť l i 91 C^, 1/* ) ét -le^ • & (AL 7 <v ) for eon-venient k^ . Coneequently C CC f=> * Q >C«P ))Cy >3 C x , ^ * -i.e. PR • Q C * Kcy>€ p II. Let * : (Kyp^—t (K% <p') be p p ' -compactible. Since ď" e ď there ia ď o C - p x ^ e p . Theořem 5« The categoriea of matric apacea with Lip-achitzian and locally Lipschitzian mappinga are boundable. Proof: Firat of them ia by lemma 5 realizable in S C Pa- • S ) • The aecond one, conaidering of courae the inr dueed topologiea, i* realizable in T C— I fĵ  •• (J ) • It ia only neeeaaary to verify that the localization from Defini-tion 1 (for F contravariant) really correaponda to the u-eual localization of the Llpachitz condition. If (Xfp ) ia & metric apace, ( U ̂  fy ) it a subapace 9 then p£ » fjj o • Gity, ) (p >. For, if cf € p then ?K * Q (^ ) (y ) » - 661 -•9^^^'^Pi;^ V e PÍ ****** 9 <*>**>>-m ip č * , ^ ) for x , y e U , Cf f * , ^ > * tf ©therwiaa; then 9? € JO and l̂  * Q C ^ ) fcp ) s f , A mapping f ; CX, p ) ~ * CX'? <p' ) i a , by Def inl t ion 1 , l o c a l l y 5 p ' -contract ib le i f and only i f for evtry x € X tha-ra ex iata U 3 X opan with ( C ^ fl^))^^ fltř » ^ c c PR • a <*<, > Cp > * $0, , i.a. 9 • ff ^ x f 4 >*p, whenevěr <y € jo ' . But i t holda i f and only i f f ^ ia Lip8chitzian# R a f a r a n c a a : t i ] J.R» ISBELL: Subobjecta,adtquacy,completeneaa and ca-tagoriea of algebraa, Rozprawy aatamaticzna XXXVtWarasawa 1964. I a) Z. HEDKLÍN, A# PULTR: On f u l l embedding of categoriea of algebraa, I l l i n o i s J . o f Math.,10(1966), 392-406. [3 ] Z. HEDRLÍN, A.PULTR: On categorlal ambaddinga of topolo-g i c a l atructurea i n t o algabradc, CMUC 7(1966), 377-400. 14] A* PULTR: On aalacting of morphiama among a l l mappinga between underlying s e t s of objeeta in con-orata catagoriaa and rea l laa t lon of theaa. CMtlC 8,1(1967) ,53-83. (Raealvad Mařen 21,1967) - #62 -

On realization and boundability of concrete categories in which the morphisms are choiced by local conditions

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On realization and boundability of concrete categories in which the morphisms are choiced by local conditions

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Institute of Mathematics AS CR, v. v. i.