D U A L I T Y OF F U N C T O R S AND D U A L I T Y O F C A T E G O R I E S

V . V . K u z n e t s o v and A . S . S h v a r t s UDC 513.882

Duality of functors can ke defined in a broad class of categories {D-categories) (see [1] and a detailed exposition in [2]7; d~ality of functors in the category of topo'.ogical spaces is d~fined in [3]. We shall es- tablish here the cenrJection between duality of functors and (he duality of categories. J"

We shall assume that the D-category K under consideration possesses a eointegral object J {under this condition, for every functor F there exists a dual funetor DF [2]). We shall use the following notation throughout the art icles: !XI denotes the fu~ctor from category K to the category of sets; this functor exists by virtue of the concreteness of category H {in [1, 2], JXJ is used instead of ~'); X denote~ the functor H(X, J); r X m~d ~X denote the natural morphisms X~X ~ J and X ~.X respectively, where X is an arbi t rary object from K. Unless stated otherwise, the admissible functor F acting in K and the object A E K will be consid- ered fixed.

We examine the morphiam XA:DF(A) ~ ' ( A ) obtained from the composition of the morphism DF{A) ~H(FA), A@A), existing by the definition of a dual functor, with the morphism H(F{A), rA):H(F(~.), A@~,) ~F(---(-~.. We shal.__[l see below that for a sufficiently extensive class of D-categories knowledge of the s truc- ture of object F(~.) and of set [XA l [DF(A) I allows us to reconstruct object DF(A) with isomorphic precision. Thus, if K is a category from th~s class, then in order to have a description of functor DF it is sufficiel,t for each AE K to obtai,~ a description of the set J~A ] JDF(A) J, The main result of this paper is a theorem giving a descrintion of this set. We use the theorem to obtain the description of the daal functor for some concrete examples of D--categories.

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First we indicate the conditions under which knewledge ofob jec t F(.~) and set J~AIDF(A)J allows ,~s to reconstruct object DF(A).

Definition 1. $ A proper subobject of object Y of category K is a pair (X, f) consisting of cbject X and monomorphism (see [417 f :X- -Y (denoted (X, f)c y), if for every object Z and morphism ~ Z ~ Y such that JcpJ JZjc JfJ [X], there is a morphism r - -X for which q,=f~,

It is not hard to see that if (X, f ) c y, the object Y and set [fJ [XJc [Y[ fully character ize object X. Let (X', f ' ) c y, with [f' [ iX'J= JfJ ]X[. Then there exist morphisms a:X' ~ X and g ' :X- -X ' satisfying the relations f ' = f a , f = f ' e ' . Hence we find that f ' = f ' e ' ~ , f=feg ' , ~.d inasmuch as l and f' are monomorphisms, it fol: lows that e ' e = l X, and Ee' =1 x, where 1 x denotes the identity morphism X--X, i .e. , objects X and X' are isomorphic.

We now indicate two sufficient conditions under which [:-~r- any admissible functor F and AE K in cate- gory K the relation

(OF(A), k.t) c F(r * (1)

holds.

A proof (Theorem 14 appearing in [2] asser ts that if J is a unive, sal object (i. e,, if for any X(~ K the morphism t x is normal [4]), then a monomorphism of the focm ~A is normal. But when the definitions of a normal monomorphism and a proper subobject are compared, it is easy to c o n c l t ~ d e that if f :X- -Y is a nor- real monomorphism, then the pair (X, f) is a proper subohject of Y. Thus, if object J is univers--o.l, relation (I) holds.

-t A short exposition of the results of this article appears in [11]. 1: This definition is applicable i.n any concrete category.

~'t The morphl:-m ~A is a monomorphism (see [1, 2]);

Translated from Sibirskii Matematicheskii Zhurnal, Vol. 9, No, 4, pp, 840-856, July-August, 1968, Original article submitted Jaunuary 10, 1967,

627

Another condition requi res that the following asser t ion holds.

PROPOSITION 1. Let D-category K satisfy conditions:

1) ~X, I x ) C X for any X c K ;

2) it follows from {.'~L f) c y that for any (H(Z, X), H(Z, f))c H(Z, YL ZE K.

Then for any admissible functor F and any Af: K. relation (1) holds.

We shall not prove ~.bAs proposition as this may be done through a simple modification in the proof of above-mentioned T h e e : - ~ 1' from [21.

In the sequel we sh~ l assume that category K ~atisfies either the universality condition o=f cointegral object J or tke conditions of pro2ositiol~ 1. We s t r e s s that in either case the relation (X, L x ) : X holds for any XEK.

Let X, Y, and Z be a~bitrary objects of K and let r X | ~ Z be a morphism. The morphisms X ~H(Y, Z) and Y ~H(X, Z) corresponding to morphism q generate two mappings of set [X[ • [Y[ into set !Z[. We denote the mappings by ~(~) and ~(~0) respectively. In all presently known D-categories these mappings coincide. It has not howe,-or been proved that this fact follows from the D-category axioms. In connection with this, we require that g(r =n{cp) for the case in w'hieh q0 is the identity morphis . . I~:I--I ~ l , where I is the unit object of catcg~::-y K (the object for which H(l, X) =X for any XE K). In other words, we assume that the natural Luvolut~oa ~ object I is the identity morphlsm.

We shall show that the equality ~(~0) --77(~) for any morphism r --~ Z, where X, Y, and Z are arbi t ra l7 ob- jects of K follows from the assumption just made. To do this it is clearly sufficient to verify that ~(1X~y)_- ,](1X@y). The mappings ~(1X@y): ~X~ >-:~'l-- [X| and: ~ l x | y): IX[ • [Y[ - - IX| Y[ go from bifunctor [X] • [Y[ to bi- fu~nctors ]X~ Y[ (both fun~or s operate from category K to the category of sets), coinciding on (I, 1), and for any pair of morphisms a : l ~ X and fl:I ~Y, the diagram

I : l x l ~ I -;~"• i X l x I t ' t ~ ,

I .,," I ~,+.++m.. I x | l constructed for each of these mappings is commutative. Hence: if for point (x, y,~ E [Xi x IYl there is a pair of morphisms a : I ~ X and ~:I ~ Y such that r 0 ~ d fl-i(y) ~ ?, then the mappings $(1X| y) and rI(1X| y}

coincide on (x, y). The proposition will thus be proved :'f we ean show that for any object XE K and arn i t rary point xE [X[ there is a morphism a : I - - X such that a - l ( x ) ;~ 0. But the morphism ~X:I - -X, corresponding to the point x under the isomorphism [X[ =Horn (I, X), possesses th isproper ty . 0hrehave [ax[e=x ' where e is the point of lit correspondL'~g to the identity ~o_-phism 11 undo," the isomorphism [It =Horn (I, 1).)

The condition just ix:posed may be used to establish the form of the mappings [~.X]:[X[ ~[~[ , Since the morphism ~X corresponds to the identity morph~sm X ~ X . it is clear that for each point xE [X], r is

a morphism . X ~ J such that l~[a= [a[x for any r J).

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We come to the solute_on of our main task -- a description of the set [),A[ [DF(A)]. We shall examine category L. dual to category K. To each object XE K there corresponds an object X*E L and to each object Y E 15 there corresponds a.~ ooject Y.E K, with iX*)+ =X and (Y.)*= Y. Since the cointegral objec~ J exists in K. we can in a natural ~-ay provide the objects of category L and the sets of morph ' sms of one object of category L into another tt~,e lat ter means that L is t ransformed into a K-based category, using Linton, s terminolc%~ [5]) with the ~ r u c t u r e of objects of category K. We can define the covariant functor ~ by mak- ing object ~=H(Y*.J-') correspond ~o object Y E L ; we can further define functor H'(Y', Y") contra- variant with respect to Y" and covariant with respect to Y". assuming that H'(Y'. Y") -=H(y.., y . , ) . It is c lear here that IH'(Y'. Y~'~=Hom(Y' , Ym). By m~:ing the set i~I correspond to Y~: L, we convert L into a concrete category. We nc~e that functor Y. coincides with functor H'(Y,I*) In fact H'(Y I*) --IFI Y - =H(I. Y.) =Y. . It is also relevant to note here that the ~.mbedding [t~cl of set ]X[ into'set ~X[ can b~e'ah?.~,~ with the sequence of corre-~pondences . . . . [X[ = llom (I. X)--Horn (X*.. I*) -~" Hom(XV .. I*) -Horn (X, J) = I X [ . " - - ~-~ ~ . . . . .

A functor G, opera-:trig from category L to category K. will be called admissible (strong, in the te r - minolog-y of [b!} if for eac.k ~air CY, Y') 6 L there is a morphism g:H+(Y, Y')~H(G(Y), G(Y')) such that the mapping [g[ coincides wit?, ;~e natural mapping of (Y, Y') into Horn (G(Y), G(y~)).

We suppose a family {G~}~e.~ of admiss ible functors G)` with values in K to be defined in L, with a fanfily {.~,),e,z~ of mappings ~rp of the functor G)`(Y~ into the functor Y to be def i ,ed for each ),EA. For Y E L and fixed )`E A, each point tE [G~Y}[ defincs a family {[rplt} with index set M), consist ing of points of [YI. We shall call such famil ies )`-specific. For any L - m o r p h i s m 3~:Y1 ~Y2, _each ),-specific family {[Tr/z[t} of set [Yl[ is c lea r ly t r ans fo rmed by mapping ]yI into a )`-specific family {[zrkt[s } of set IY21.

Definition 2. We shall call a sys tem {Gx}~eA of functors G)` and mappings 7r/a full if for each pair of ob- jec ts Y1, Y2 E L, a L - m o r p h i s m 5:y! ~Y2 such that ~,=~ cor responds to every K-morph i sm "~.Yl ~Y2 if and only if for all )`EA the mapping [3'[ t r a n s f o r m s any )`-~pecific family in Yl into a )`-specific family in [Y2[-

We requi re a full sys tem {G h} of functors G X with famil ies {a,},e.~f~ to be defined in Y. Category L may thus be represen ted as consisting of objects of ca tegory K (of the form .VO, these objects supplied w;_th additional st~ ucture determined by )`-specific fmnil ies ,

We now examine an auxil iary ca tegory ~ . Here, K-object X will be conaidered an ~,-objeet if a set S(X, )`) of famil ies {p,},~.~ (which we shall also call )`-specific) of points p/~E IX[ is chosen in the set IX] for each :~EA; we shall call the K-morph i sm ~':X 1 ~ X 2 an L - m o r p h i s m ff t h e ~ - s t r u ~ u r e is introduced into ob-; jec ts X~ and X 2 and if m o r p b i ~ T agrees with this s t ruc ture , i .e . , if for any )` EA, every family of s y s t e m S(X1, 29 is mapped by [TI into a family_of sys tem S(Xz, )`). Catet ;ory '~ is c lear ly an extension of L, and the re fore eve ry K-object of the f o r m X = X is provided natural ly with an ~,-object s t ruc ture . In partic ' .flar, object J is provided with ~ - s t r u c t u r e , s ince -2 =I.

We introduce an q~-object s t ruc ture into object F(~,) by constructing for each ),E A and each point q E IF(G~.(A*})[ a ~-specif ic family according to the following rule: p/a = [F(Tr/zA ) lq, where ~rpA:G)`(A *) ~ A is the morphisr,~ genera ted by the mapping 7r#. We shall call the s t ruct 'Jre defined here the basic ~ - s t r u c t u r e in object F(A).

THEOREM. The morphismy:F{~,) ~ J belongs to the image of [DF(A)[ under the mapping ])`A]:DF(AI ~ H o m (F(A), J) if and only if 1~ is m~ "~-morphism re la t ive to the basic "/._,-structure in F(.~).

Before commencing the proof of the theorem, w~ shall prove t~'o auxiliaxT proposi t ions .

We introduce the notion of a X-X-family in the set [F~,) i as a f~,~'nily of points constructed for fixed )`EA _and XE K according to ~he formula _~..~[F(q~p) Ix, where xE iF(X)[, and the family of morph i sms ep~::X -o~, is , , -specif ic i n t h e object H(X,~.) =X@A. Thus, an L - s t ruc tu r e , which we shall call a L -X-s t ru c tu r e . is constructed for each XE K in object F(.~).

LEMMA 1. The morph ism y:F(.~,) ~ 7 is an element of the set [)`A ~ [DF(A) I if and only if it is for each X and L - m c r p h i s m relat ive to the ~ - X - s t r u c t u r e in F(~,),

We naed enly one (the first) a sse r t ion of the l e m m a to prove the theorem, so we shall prove it in one direct ion only. That is, we sb.~l use the assmnption that the morph i sm y agrees with L - X - s t r u c t u r e for every X to show that yE ]~,A [ ] DF(A)I.

For a r b i t r a r y X(~ K, we examine the morph i sm U x : F ( X ~ H ( X ~ ' F(~,)) corresponding to the natural morph i sm H(X, .~) ~H(F(X), F(~,)). To each point x E !F(X) I there cor responds ~ morph t sm [u~)x = Vx:X@ A ~F(A}. Using the equality iF(~) [x = lVxl ~p, where q~ is an a rb i t r a ry point of Horn (X, ~.)= [X~--AI, it is e a sy to see that v x is an ~ , -morphism re la t ive to the L - s t r u c t u r e in==~X~ and the ~ - X - s t r u c t u r e in F(~.). We ~ - so examine the morph i sm v~ and H(X| J)-H(X---~A, F(A))~X@A. W.e denote by >r the image of point xE [F(X) [ under " " the mapping [Vx]:]F(X)[ ~ H o m (X~A, J). It is c lear that ~r and as the composition of two L - m o r p h i s m s it is a n L - m o r p h i s m and, consequently, it i s an L - m o r p h i s m of object (X| object I*. There fo re , noting the ea r l i e r given descr ip t ion of ~nappings of the fo rm iX, we conclude that the m o r - phism ~'~x:X | i s the image of some point z E ~ | under the mapping [iX| ~ | ~ ~X-"@.g, !. Thus the morphismvx:F(X')~X='=~'~ is such that [VX[ ]F(X)[~[tX@A[[X @A [. Hence using (X6~A, t X ~ A ) c ~ , it follows that there e.xists a morph ism f x : F ( X ) ~ X | such that VX=tX@Af X.

r �9 �9 ~ " ' to ~o ,.,~,,~,-,,,-~,~a T~- is easy to Thus, for each XEK we can consider the morphisrr, ,::.F(X) -A~.V)~ . . . . . . . . . . . . . . . . . . see that these mci-phisms define a mapping f of functor F into functor z2 A.

We need yet to show that lEA[f=% It follows from the definition of morph ism ~A that [XA[g, where g is an a r b i t r a r y mapping of functor F into functor ~A, is a morphism represent ing the composition of g~:F(A) - - .~ �9 and VA:A�9 - - j . Consequently, we need to ver i fy that vAf~='~. For the proof we note f i rs t of all that for a r b i t r a r y object XE K and a rb i t r a ry morph ism It:X ~ J , the d iagram

629

h

is commutative. By u~L,~g this assertion it is sufficient to show that ~ ~ = % where • or equiv- alently, that ]2](]vZla)=}y]a for each aEIF(~,)j. But ]v'~la=~EHom(A-WX, J). and Ixltl~alrn)=lzalr A. Therefore, since !'~'.'a[7_k = {'fla, we see that equality Xv~ = ' / i s satisfied. This concludes the proof of the assert ion.

LEMMA 2. For a.~y B, C E K and ,~,E A there exists a morphism w:GA((B @ C)*) ~ flB(GA(C*)) such that for any t E G~jB r C)*) a::d ~E MA, the morphism ~n t = i~,,[t of object B into object C can be represented in the form of a composition r t=ztzCwt, where co t $]wltE~m (B, GT.(C*)).

We denote by o tee morphism GA((B~C)*)~H(H'((B~C)*,C*), GA(C*}) corresponding to H'((B~C}*, C*) -~ H(GA((B$ C)*}, G)..~C*)); this l ~ t morphism exists by virtue of the admissibili ty of functor G~; fl de- notes the morphism B ~H'({B | C)*, C*) corresponding to the identity morphism under the isomorphism B $ C ~ B @ C for isom~-phism H(B, H'(B@C)*, C*)) =It(B, H(C, B~:C)) =H(B~C, B~.C}. The morphism H(H'((B@~C) *, C*), Gx(C~)--* H(B, GA(C*)), whose con,position with a defines the desired morphism w :Gx(B~C)*) ~ flB(Gk(C~)), corresponds to ~.

We now show that this morphism satisfies tl~e requirements ol the lemma. Let b be some peint of [B[, let [fl[b =fib, and to the L-morphism'flb let there correspond the K-morphisms ~,:H(B, C ) ~ C and GA(.Sb):GA((Bs --GA(C*), which together \vith morphisms =B@C and ~r/aC for fixed/~E M A form the commut~ive diagram

G~((B | C)')~ th(C}

i 1 ~(~, ~1 ~ -+~ : .

It is character is t ic of fl% that for any dEHorn (P C), the ~qualitv [flbl.5= 16lb. For a rb i t ra ry "~IGA({B ~ C}*.) [ we have [wt[b = ~at[Pb= [Gx(~)It, where ott = [a[ t i s a m o r p h i s m of i, '(B• C}*, C*) into CA(C*),

Hence, since the diagram is commutative, we find that [~/C[ [wtlb= lflbir t- But [flblCp,t = [r [b a~d this .means that [r/.C~ct[ = [r is a morphism of H'{(B~C) ~, C*} into GA(C*). '

We now go immediately to the proof of the theo~-em. We examine the A-X-family in the set [F(AI; this fantfiy is-~r with respect to ai"bitrarily selected XE K, XE IF{X} [, andthe ) specific family {~} ~ e MX from the set Horn (X, A) (we-shall in the sequel censider the A EA fixed). Let t E [GA((X~ A)*) [ be ,t point defining the A-specific family {~0p.}. By appiying Lemma 2 for B=X and C =A, we s:.all have the morphisms ~Z,:GA((X~ " .~*) ~.qx(GA(A*)) a.pd c0t,:X ~ GA(A* ) connected such that w t ' = lw,]t. For each At E M x we find r = 7'~A~ *- ttenee iF(~Pp) ix = [F(,r# A) I [F(wt') fx, i .e . , t he A-X-family under consideration is a L-specific. family con.~t ~ d c t e d ~ with rcspeot~ to point q = IF{ w], .{x. Thus.. . every morphism *~F(A)- ~ J consls-" tent with the basic L-s t ruc tvre also is consistent with the L-X-s t ruc ture for each XE K, and consequently is a point of s e t ]~A ] • ~DF(A)[.

Conversely, let TE [AA[ [F(~,)[. We ne~-d to verify that the morphism T is consistent with the basic L-s t ructure of object F(A). We apply Lemma 2 for B =A and C =I . We obtain the morphlsm )'~'~A(A*) ~ -QA(GA(IO). We denote by h# t_he morphisms ~A(r,/~*):.O.A(GA(I=)) ~A. It is not difficult to see

,'rp ~ h/aw- In fact, [h/,I [~" it = [h~[~0~, = 7,ul wt'=[~]~, where t is an arbitrary., point of ]Gk(A*)]and r = [u~'~. We select an arb i t ra ry A-specific family (r~}~e,~ x in IF(7*)I and iet q EtF(GA(A"))I be the point specifying this family. It is clear from what has been said that the family {pp} coincides with the family {F~}~x of points r/z defined by

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~ = t~(a~)It , (2)

where r = IF(w.)]q~ 1F(~A(G~(I*)))[, We shall show that the mapping IT[:IF(A{--lJ{ t r ans fo rms may family {r.}.~,~ of points ,'p ,z-;efined by (2) for a rb i t r a ry r ~]F(F~A{G~(I*)))I into a X-speclfic family; this will com- plete the proof. The re exis ts by supposition a mapping f:F - -ZA for which 7 = rA]X, and therefore there exis ts a mapping g:FP., ~ for which ~,=gj (see [1, 21 under the i somorphism {F--ZA}--= {F.qA--ZI}). Let I g~x("~lr= s 6 lax(r) I ~ C l e d r l y , [gj Ir/a = l~,ls, and therefore the family { r ~ is mapped into the ~-specif ie family {I,pls}. This p roves the theorem.

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We shall now exa-7~ine a s e r i e s of concrete examples of D-ca tegor ies . By applying the theorem just proved to these D-ca t egor i e s , we shall obtr:n some asser t ions regarding the descript ion of dt.al functors .

1. Let K denote :;Se category of all vector spaces (the morph i sms a re l inear tran.sfcrmations). J denotes the one-dimen-~i, onal space. Then for each X~ K the object X is the set of all l inear functions on X. We construct re la t ive t~ each directed set M the functor G M operating the ca tegory L to category K by a s - sociating the space GM:'Y) (consisting of all general ized sequences {} ~}, weakly converging to zero, with index set M, of points Y:om the space Y=Y) with object y~ L. For each ~s M we define the functor mapping 7r~.'GI~L(Y) - -Y pairing type sequence {~ v} ~GM(Y) with i ts/~-th component. Then the general ized sequences f rom Y with di rected ir~Sex set M, weakly convergent to zero, are M-specif ic familez in the objects Y,

LEMMA 3. The _~ystem of ftmctors G M in the eatego~-y of vector spaces is full.

Proof . it is knou~ (for example, see [61) that a l inear t ransformat ion r is the image of a point x ~ X under the natural ~ app i ng X - -X if and only if it is continuous in the weak topology .X. Hence, using the commutat :ve diagra~a

Xt L ' X , "~- " ~S for~rly g,

we can conclude that th.~ l inear t ransformat ion f:X2 ~X1 is ~ontinuous relat ive to weak topologies if and only if it has the form g for s a ~ e l inear t ransformat ion g:X 1 --X~, Thus, we may represen t ca tegory L by spaces of the form ~.; these sr..~ces a re provided with a topology of pointwise convergence and of continuous l inear t r ans fo rmat ions . Bu," fc:~r l inear topological spaces , ~.eneraiized sequences will converge to zero under a l inear t r ans fo r rmt ion ~ a~d only if the t rans format ion is contiruous, whence the l e m m a follows,

The bas" : ~ - s t r u c ~ u r e in object F(,~) is defined by the sys tem of genera l ized sequences P~t got f rom

p~ = F(a~)q (3)

for points q~ F(GM(A*))~. We introduce into F(~.) ~he s t rongest of the l inear topologies in which each of these sequences converges t~ zero; we call tl'as topology specific. It is c lear then that the ~-morphi :sms-F(A) ~ J a re l inear function~_ on F(A) and are continuous in the specific topology. Thus, applying the theorem to this case gives uc

PROr~OSITION 2. ~"The space DF(A) is isolnorphie to the space iinea r functions on F(A) continuous in the specific topology.

2. We now denote ~y K the category of all discrete abelian groups and their homomorphisms, and by J the group of re3! numbe'Ts rood 1. For each x E K, the object ,~ is the characteristic group ofgroupX. Anal- ogously to the precedin~ Lns~ance, we construct for each directed set M the functor G M which pairs an ob- ject YE L and the group ~f generalized sequences from Y= y. with index set M, weakly cnnverging to zero; to each p5 M we pair tS~ mapping ~.-GM(Y ) --y, which carries each sequence {}~} into the point ~/~. Here, the ge ,neralized sequenc:es, weakly converging to zero, with index set M are M-specific families in the ob- ] e~s Y.

LEMMA 3' . The -~ystem of functors G M in the category of d i sc re te abelian groups in full.

It follows f rom tS~e Pontryagin duality theorem that for any X1, X2EK , a homomorph i sm f:X2-'*X1 has the form g for some h o ~ o r p h i s m g: X 1 ~ X 2 if and only if it is continuous re la t ive to weak topologies in and X1- It i s proved in ~:~e same way that the morph lsm ~1 --~r2 cor responds to some morph i sm Yl "-~Y2 if and only if it i s conttnuc~us xelative tc weak topologies in Yt and Y2- It is now c l ea r that the sy s t em of func- to r s G~M is full.

631

The bas ic L - s t r u c t a r e in group F~) is again defined by specifying a sys tem ef genera l i zed sequence P u accord ing to (3) for poin:s qE F(GM(A*)) . By examining in F(A) the s t ronges t of the topologies which a re consis tent with the group s~ructure and in which all the sequences p# converge to z e r o (the specif ic topoi- ogy), and by applying our t heo rem, we get

P R O P O R T I O N 3. The group DF(A) is i somorph i c to the c h a r a c t e r i s t i c group, continuous in the specif ic t opo logy of g roup F(A).

3. Let K be the c a t e g o r y of Banach spaces and continuous l inear ope ra to r s with norm ot exceeding 1, and let J be the one-dime~.sional Banach space. If XE K, then .X is the space of continuous l inear function- als on X. We defiue a s y s t e m of functors G M (M again runs through the col lect ion of d i rec ted sets) in which each fanctor GhI pa i r s o b j e ~ YE L and the space of bounded genera l i zed sequences {~,},<M f rom ~ with no rm I! {~.~.} II ~-- supl]~_~II, weakly converging to ze ro . To each point p E M the ftmctor mapping 7r/~:GM(Y} ~ ~

c o r r e s p o n d s in a natural m a nne r . The bounded g e n e r ~ i z e d sequences v, eakly converging to zero , with in- dex set M he re a s s u m e the ro le of the M-spec i f ic fami l i es in the oLjects Y.

LEMMA 3". The s y s t e m of functors in the ca t ego ry of Banach spaces is fulI.

Prooff. The natural o p e r a t o r X ~ .~ is known to be i some t r i c . By Baaach~s duali ty theorem, a con- tint,'~us l inear functional r o~ the space ,~-~ is the i maga of a point x EX under the mapping x -*:~ if and only if it i s weakly cont inuous. Hence we conclude that for any X1, X~ EK the opera to r f :X2~X 1 (a K - m o r p l u s m ) c o r r e s p o n d s to some o p e r a t o r g:X1--*X ~ such tha t f = ~ if and only i f it i s cont inuous re la t ive to weak topologies in X~ and X 2. The l e m m a follows ~nen L is r ep re sen t ed as the c a t e g o : y fo rmed by spaces of the form X, which are supplied with a weak t o p o i c ~ and with bounde~ weakly continuous l inear opera to r with norm not exceeding 1.

The s y s t e m of genera~2_zed sequence p/~, defined by (3) with _-'esf~ct to points qE F(GM(A*}) , d e t e r - m ines the bas i c L - s t r u a u r e in object F(A). By defining the spec i f ic topology in the space F(~,) as the s t ronges t loca l ly convex topology for which all sequences p# converge to zero , we obtain

PROPOSITION 4. (This was obtained e a r l i e r b y o t h e r means in [7],) The space DF(A) i s i s o m e t r i c to the subspace of Y()[) consistLng of function~ls cont inuoas in the speci f ic topology.

4. We now take as K the ca t egory of se t s with base point [~]. T h e morph~sms a re mappings c a r r y i n g base ~ i n t into base point. -_,he base point in each set will be denoted by 0. The two-p~int object {0, 1} will a s sume the ro le of object J . Fo r each XEK, we can c l ea r ly identify object ,X wi.~h the se t of all subse t s of X not containing a bane point', with '.he e r r , ~ set as the base point in .X.

Fo r each power m, we cons t ruc t a functor G,n which a s soc i a t e s object YE L and the set of all collec-~ t ions {H ,} ,~ of elemen*.s of ~ having index set ~ w i t h power m ( we cons ider the col lect ion {M u} in which MI,=0 for each u to be the base pointvof set Gm(Y) ). For each functor G m we define a family {n,,z.u,,nn} of mappings of fanc tor G m into functor Y, a s suming ~r#{M~M#, with .nu/.M,},~----UM~ and .nn{M~},e~ ---- flM, . The 2ol lect ions {Mp} of e l emen t s of ~ with index set of power m, supplemented by the join an.] mee t of all the i r e lements , ftmction he re a s the m-spec i f i~ fami l ies in the se t ~.

LEMMA 4. The s y s t e m of functors G m in the ca t ego ry of se t s with base point is full.

P r o o f . We f i r s t show ~hat under .'he mapping t x : X - - ~ the image of X cons is t of just those mappings ~ : X - - J which p r e s e r v e set ur.ions and in te r sec t ions r view of the i somorph i sm J =~, we cons ider the c o r - responding opera t ions on e l emen t s of J to be defined).

For the given mapping ~ : :X--J let there be a point x E K such that qx)= ~. It is c l ea r that for hi E X , x E hi c a r r i e s ~ into 1, an,:'_ ":rAo 0 for x~M. It follows eas i ly that for any col lect ion {Ma} of e lements Ma EX the re la t ions ~(13Mr ~ D~(M~) and ~(~M~) = ,q ~(.~L,)~ " a r e sat isf ied. Converse ly , let ~ be consis tent

with the join and meet oper-ations. If ~--tX(e) is not t r iv ia l , then the re is a one-point set M(o)={x0} in ~- such that ~(M 0) = 1, f o r otherw~_'se, since unions a r e p r e s e r v e d , ~ would c a r r y any M E X into 0. But then, if x0E hi, then ~ ~ ) = 1, since M =M 0 k/hl; if, however , x 0 ~ hl, then {(M)-- 0, s ince M0tJM = 0. This means that ~ = ~X-(.<0).

It fol lows_easily f rom ~his proof thai if X1, X2EK and ".f mapping f..X~--*.~2 p r e s e r v e s the join and mee~ opera t ions , then tLere exis ts a g:X~ ~X2 for which f = g . It is a lso not difficult to see that for any g:X1 ~ X ! the mapping f =~:_X~ ~ X~ preser~,es the join and meet opera t ions . Indeed, suppose for exaorp!e

that f does not p rese rve the join operation. Tb~s implies that there is a set { ~ } of e lements of X2 for which U/(M~t -/= ](UM=). There is a point x in one of the sets Uf(M=) ~, ](UM=) not be[:m~ng to the other,

tx ~ ~t 4t

and then the mapping f~-~l - -}2 ca r r i e s element tXl(X) ~ ~, into some element o~ set :~2 ~ t lying in ",.xe(Xe), which is impossible.

T h u s , category L is realized as a subcategory of the categories of f,~ll la t t ices t ~,-~d of mapping which are consistent w~.th the s t ructures of full lattices, after which the Iemma is clear .

A

When category L is constructed, we provide objects XEK with a s t ructure such t.Emt, for some col- lections of elements of X, the operations of "join" and "meet" are defined, ~boi t perhaocs not uniquely.

In part icular , the join and meet are defined in Y(A) for families {p/~ which can be: ~efined with r e s - pect to some point qEF(Gm(A*)) according to the formula

p~ ~- F(~)q. (4)

where m is the power of the set of indices /2. We have here that

bP. : F(~u)q, HP. ---~ F(~n)q. (5)

We can show that the operation~ introduced into F(A) in the scope of each family S of points e; F(~J for which these operations are defined possess the ~ollowing proper t ies : a) both operat~:,~ns are defined for any subfamily of family S; b) the results of success ive application of the operations in a..~y order are de- fined; c) the commctat ive, associative,, and distributive laws hold. We wiAl not require~ these propert ies , however.

Applying the theorem to this instance yields

PROt~SITION 5. The set DF(A) i s i somorph ie k) the set of mappings F(A)- -J whic~ a re consistent with the above introduced partial multivalued john and meet operations.

5. Now let K be the category of partially ordered sets with base point. We consfcher all partially ordered sets X with base point 0 satisfying 0-<x for any xE X to be the objects; we cons~fier isotonic map- pings car ry ing base p~'int into base point to be themorphisms . We can show that all tb~ D-ca tegory axioms ~re satisfied in this category. The order r e l ~ i c n in the sets Horn (X, It') i~ introduced Lm the following way: ~-<fl (~, fl EHom (~. Y)) if and only if c~(x) -<fl(• for any xEX. We single out trivial m~pp'ngs in the s e ~ �9 ,~m iX, Y). The *.ensor produc~ of o~,j~c~=~ �9 X and Y is defined as the set c,~alained from the direct product X • Y by "sewirg" the sets consisting of points of the form (x. 0) and (0, y), with (x, y) = (~.~. y~) if x -~x 1 and Y-<Yl, to one point (which is considered to be base point). J denotes the set consisting of the two points 0 and I and is c lear ly both a cointegral w}d unit object in K. For each X E K we can identL-ry the set X with the collection of ideals of X not containing 0. The empty set serves as the base point in .~. :and the order ing coincides with the one already specified. It is clear that if each of the ideals Mc~cX of ~ome collection {M a} is an element of .X, then the ideals U Ma and ~ M~ a r e also elements ot .X, i . e . 2 full lattice s t ruc -

ture is introduced into X.

In a manner analogous to that of the preceding instance, we construct for each pcw,er m a functor Gin, pairing object YE L and set Gm(Y) of collections {My} of elements of Y with index set offpower m (natural ordering). T h e f a m i l y {n., ~u, nn} of mappings ,~f each functor G m into functor Y is a l~o def inedby: ~r/z{M~} =hip, nu{M.}= 0 M., nn{M, } = 13 M, . All collections of elements of ~ with index se.~ of power m, sup-

plemented bv the join and meet of all their elements, are once again the m-sr~ecific fam~fiies in the objects

LEMMA 4' . The system of functors G m in the category of part ial ly ordered sets ~'ith base po}nt is full.

Proof. We shall show that under the imbedding tx :X ~X, the images of the poiat~ x~.X are just those mappings ~:X ~ J w h i c h are consistent with the full lattice s t ructure .

~ A-~'uli lattice is a set in which for each collection of elements the operations of meet ~,nd join are defined, subject to the law of the corresponding operations on sets.

633

F o r .~:.X ~ J , let t h e r e be a point xEX such that tX(X) =~, Then ~ m a y be d e s c r i b e d thus : ~(M) =1, i f x E 5I, and ~(M) = 0, if x ~ M. It fol lows e a s i l y f rom th i s that ~ p r e s e r v e s the jnln and mee t o p e r a t i o n s .

C o n v e r s e l y , let ~ :~ ~ J be -ons i s t en t with full l a t t i c e s t r u c t u r e . If ~ is t r i v i a l , then we have tX(9}=~ Suppose ~ i s not t r i v i a l . We examine the f a t a l ly of a l l e l e m e n t s hi of "X for which ~t, hl) = 1. Then by the mee t p r e s e r v a t i o n condi t ion ~ ' e a l so have ~(M0)= 1 for ideal M 0 r e p r e s e n t i n g the i n t e r s ec i i o i : of a l l such s e t s . We note that M 0 is the p r inc ip l e ideal g e n e r a t e d by s o m e point x 0. F o r if th is wei 'e not ~o, then, choesi~,g for each xE M 0 the p r inc ip l e idea l g e n e r a t e d by x, we would obta in the se t of e l e m e n t s of X such that ~ a s - umes the va lue 0 on each one of them and a s s u m e s 1 on t h e i r join. If now hi E,K con ta ins the point x~, the~ 3I 0 c M and consequen t ly ~(M) = 1. If x 0 ~ M. then M0(~ M i s the subse t of Me not e o ' nc id ing with t h e m . : B u t then ~(M 0 riM)=0, and this means that ~(M) = 0 . Thus , tX('X 0) =~.

As now follows e a s i l y f rom the proof, the m o r p h i s m f: 'X2--'X! can be r e p r e s e n t e d in the fo rm f =g fo r some g:X 1 - -X 2 if and only if the mapping f is cons i s t en t with the full l a t t i c e s t r u c t u r e s in Xt and Xz. Th i s m e a n s that L can be r e p r e s e n t e d as a c a t e g o r y c ons i s t i ng of full l a t t i c e s and the mapp ings that p r e s e r v e t h e i r s t r u c t u r e . That p r o v e s the l e m m a .

Thus , the s u p p l e m e n t a r y s t r u c t u r e in ob)ect F(A) is such that the jo in and mee t a r e def ined for fmc~- /:Lies of points~of F(A) in th is ins tance a l so . T h e s e f a m i l i e s a r e def ined by p a i r s (m, q), w h e r e m is s o m e power and qE F(Gm(A*)) is a point, acco rd ing to (4), and the r e s u l t s of the o p e r a t i o n s a r e g iven by (5). We can p rove that the o p e r a t i o n s on the points of each ~uch s e p a r a t e l y c o n s i d e r e d f ami ly S p o s s e s s the fo l - lowing c h a r a c t e r i s t i c s : a) ".he r e s u l t s of both o p e r a t i o n s a r e def ined for any sub fami ly R ' o f S; b) the r e - su l t s of s u c c e s s i v e app l i ca t ion of the ope ra t i ons in any o r d e r a r e defined; c ) the c o m m u t a t i v e , a s s o c i a t i v e , and d i s t r i b u t i v e laws hokl. But we do not need ~his fac t .

PROPOSITION 6 . Objec t DF(A) i s i s o m o r p h i c l e t h e subse t of the p a r t i a l l y o r d e r e d se t F(A) c o n s i s t i n g o f the mappif igs F(A) which a r e r with the p a r t i a l jo in and the part ial , mee t s t r u c t u r e e x a m i n e d above .

6. Las t of a l l we c o n s i d e r the e a s e ~n xvhich K is the c a t e g o r y of ful ly r e g u l a r t opo log ica l s p a c e s wi th ba se point . The m o r p h i s m s in th is c a t e g o r y . a r e mapp ings which a r e cont inuous on b i c o m p a c t a a~.~l which c a r r y 0 into 0 (0 again denotes the base point of each X E K}. Func t e r H{X, Y) can be def ined by introducip_g the c o m p a c t open topology into t h e set of m o r p h i s m s of s p a c e X into space Y. We can v e r i f y without e s - pecia l d i f f i cu l ty that c a t e g o r y K s a t i s f i e s D-ca tegory . a x i o m s C1 through C5 [21 ( i m m e d i a t e v e r i f i c a t i o n of the o r i g i n a l s y s t e m of D - c a t e g o r y ax ioms E1 through E4 for c a t e g o r y K is not as s i m p l e a s for the c a t e g o r y of func~ion~ Hausdor f f k - s p a c e s examined by D.B. Fuks [31; the p roof of the c u r r e s p o n d i n g a s s e r t i o n s in [31 con ta ins an e r r o r , which led to the a p p e a r a n c e of [:01 where in a D - c a t e g o r y c ons i s t i ng of t opo log ica l s p a c e s is cons t ruc t ed ) . The t e n s o r product in ;'~ can be d e s c r i b e d thus: the se t IX'~'Y! is ob ta ined f rom the d i r e c t p roduc t X • Y by ident i f ica t ion at the point 0 of the " coo rd ina t e c r o s s ~ X • U 0 x y; the topolowy in X ~ Y can be spec i f i ed as the weakes t topology for whinh a l l n u m e r i c a l funct ions on IX~ Y[ a r e cont inuous , with t h e s e funct ions g e n e r a t i n g functions on X x y that a r e cont inuous on bico_mpacta.

We take the s t r a i g h t l ine as the co in t eg ra l ob jec t J . LCt X E K be an ob jec t . Tl ' en .X i s the se t of n u m e r i c a l funct ions on X continuous on b i compac t a , and a s s u m i n g the value 0 a t point 0, wi th the topo logy cf uniform conve rgence on b i c o m p a c t a . Toge the r with X, we sha l l f u r the r exaanine the s p a c e N(X) of a l l funct ions o n X , cont inuous on b ' co r~pac ta (and not n e c e s s a r i l y c a r r y i n g 0 into 0):and tl?e space -N~X) of a l l cont inuous funct ions on X a l so provided with the c ompa c t open topology. We note that each of the s e t s ,X. N(X), and N'(X) can be p rov ided with a r ing s t r u c t u r e in a na tu r a l m a n n e r . F i r s t we p r o v e the fo l lowing.

L E M M A 5. X is ison.,,JrPhi_c (as an object of c a t e g o r y K) to the s p a c e M(X) of cont inuous , add i t ive , aa,d mu l t i p ! i ca t i ve func t iona ls on X with a po in twise c o n v e r g e n c e topology. (The anMog of th i s a s s e r t i o n for the s p a c e N'(X) in p l ace of .X and the i s o m o r p h i s m c o n s i d e r e d as a h o m o m o r p h i s m is p roved by Michae l in

The c o r r e s p o n d e n c e X--h!(.X) is e s t a b l i s h e d r a t u r a l l y : to the point xE X t h e r e c o r r e s p o n d s the ft,.nc- t ional ~ o n X, a s s u m i n g on e v e r y ~ ~ X the va lue ~(~) = (p(x). It is c l e a r that the funct ional ~ is add i t ive and m u l t i p H c a t i v e . It i s a l so c l e a r that ~ is weakly cont inuous and m o r e o v e r cont inuous in the c o m p a c t open topology o f ) [ . T o see that the e s t a b l i s h e d c o r r e s p o n d e n c e is biunique it i s suff ic ient to v e r i f y that e v e r y cont inuous addi t ive , and mu l t i p l i ca t i ve f t ,nctiohal ~ on .X can be def ined acco rd ing to ~(r = r for s o m e point x ~ X.

634

We note f i r s t of "all that eve , 'y funct ional ~:X ~ J which is an e l emen t of M(X) ear. be ex tended to a con- t inuous , addi t ive , and m u l t i p l i c a t i v e funct ional def ined on the space N(X). In fact , the funct ional 7 : N ( X ) ~ J , for which 7(~) = ~ ( q - t2(0) 4-q(0), p o s s e s s e s a l l the e n u m e r a t e d p r o p e r t i e s (it is not d i f f icul t to ve r i fy t h e s e p r o p e r t i e s ) . T h u s , to show that the nonze ro funct ional ~ i s def ined by a point xE X, it is suff ic ient to s e e that the funct ional ~ is def ined in a c c o r d a n c e with the s a m e ru le .

To do th i s we show that t h e r e e x i s t s in X a point x 0 such that for any e l e me n t q6N(X) s a t i s f y i n g r =0, the r e l a t i o n 7(~) =0 ho lds . We a s s u m e that for any x E X t h e r e e x i s t s a function q x : X ~ J cont inuous on bicor,~pacta and such that ~x(X) ~ 0, but 7 (q~x) = 0. Let Q be ~ b i compac tum in X. F o r each x ~- Q we s e l e c t an open ne ighborhood U x of poil~t x in Q, on which th~ function qx does not vani3h. The~'c e x i s t s a f ini te num- b e r of poin ts xl, x 2 . . . . . x n of Q such that the s e t s Uxi , Uxz . . . . . UXn cove r Q. But then function r = ~xl 2

+qbc2 ~+" " ' + ~ f f n is s t r i c t l y g r e a t e r than 0 on a l l Q and 7(~) =0. We s h ~ l f ind a a > 0 such that g,(x) > e for any x 6 Q . We e x a m i n e the funct ion g,' def ined a c c o r d i n g to the fol lowing r u l e : g~'(x) = a , if ~(x) > ct and r =~(x) ~f ~(x)_-z~. The funct ion r can be c e p r e s e n t e d in the form of a compos i t i on ~] ' gh, w h e r e h is d e - f ined as fo l lows: b(x) =1 i f g ( x ) - <rv and h(x) =a/q(x) i f 4(x) > a . T h e r e f o r e ~(ff') =0. When if' i s mu l t i p l i ed by a cons tan t , we obta in for the given b i c o m p a c t u m Q a function f Q : X ~ J such that fQ(x) =1 for e v e r y x ~ Q , with ~(fQ) =0. The se t of aH b i c o m p a c t a of s p a c e X is d i r e c t e d (by a s sumpt ion ) . The g e n e r a l i z e d sequence fQ c l e a r l y c o n v e r g e s in the e c m p a c t op,~n topo lcgy to a function i d e n t i c ~ l y equal to 1. I t fol lows f rom the cont inu i [y of 7 that 7(1) =0, which c o n t r a d i c t s the n o n t r i v i a l i t y of 7. Th is c o n t r a d i c t i o n shows that point x 0 e x i s t s .

Noting now that it r is cons t an t and e v e r } ~ ' h e r e t a k e s the value c, then 7 ( r =c for a r b i t r a r y tp6 N(X), we obta in ,-/(~) = ~(~0 - tP(x0) + tp(x 0) =O(r q (x 0) + ~)(,r = r Th i s c o m p l e t e s the p r o o f of the biunir4ue- h e s s o f X ~M(X') .

We r.ow p ro , ' e that the topo log ,e s of s p a c e s X and M(.X) a r e i s o m o r p h i c . It is c l e a r that the mapping X --M(.X) is con t inaous on b i c o m p a c t a s ince if {:./~ is a converg ing g e n e r a l i z e d sequence lying in s o m e b[ - c o m p a c t u s of X, then e v e r y function tp ~ .X carr ie .~ th is sequence into a converg ing g e n e r a l i z e d sequence {~;(x/.t)}. We n o w t a k e s o m e c o n v e r g i n g g e n e r a l i z e d sequence {~#} of the s p a c e M(.X). Then the ~equenee of poin ts x#~ X c o r r e s p o n d i n g to the poin ts ~$ a ! so c o n v e r g e s . In fac t , c o n v e r g e n c e in M(~'~ m e a n s that e v e r y lunch'.on r c a r r i e s "he sequence { ~ into a conve rg ing sequence . But if the point x6 X c e r r e s p o n d i n g to the l imi t ~ of the sequence {~k~ is not the l imi t of the sequence {x~}, then we m a y s e l e c t f rom {xlz } a c o - f inal s u b s e q u e n c e nc point of which l i e s in any r e ighbo rho~d of point x. And s inc~ X is ful ly r e g u l a r , we can find a cont inuous function q on X (a l so an e l e m e n t of YO Ior which the i m a g e of th i s subsequence does not c o n v e r g e to the i m a g e of x. Th i s p r o v e s the l e m m a .

Since subspace ~x(X~. 6f space X coincides as a set with space M{X) and since ,x(X)~M(.~) is here- with continuous, it follows at once from the fact thatX_and M(X) are isomorphic that X and tx(X ) are also isomorphic, i.e., the condition (X, tx}cX holds. (We note that it has not been possible to prove that this~ relation holds in the category of functional H2usdorff spaces.}

We examine ~he functor P, operating from L~o K, for which P(Y) = Y x ~, and we also examine the family {,7 i} (i =1, 2, 3, 4) of transformations of functor P into function Y, carrying z =(x, y) ~ P(Y) into the points r,(z) =x, ~r2(z) =y, ra(z ) =x +y, and ~4(z) =xy respectively. For every directed ~et M we construct al- so a functor G M such that GM(Y) is the space of generalized sequences of Y, uniformly converging to zero on bicompacta, and with index set M; this space is provided with a termwise convergence ~.opnlogy. We de- fine for each functor G M a family n/~ mapping it into functor Y; these mappings pair each sequence {xv} ~GM(Y) with its/z-th component.

LEMMA 6. The system of functors formed by the functor P and the family of functors G M in the cat- egory of fully regular top~:ogical spaces with base point is full.

Using the fact that the images of the point x~ X under ~X: ~-~ are continuous multiplicative functions on .~, it is rot difficult to see that f:.X~ ~-~I can be represented in the form g for some g:X I --X~ if and only if f is continuous, additive, and multiplicative. Sufficiency follows from the fact that with the conditions imposed on f, the subspace LxI(XI) of "~I is carried by~ inte subspace Lx~(X2) of ~ . Necessity is just aa

clear: for every morphism g:X I ~X2 the mapping f i." clearly additive and multiplicative: continuity of f Is proved by the fact that if an open set U I in "~I is specified by a bicompactum Y.icX and by an open set VcJ , then the preimage of U I is an open set U~ of ~ defined by the biccmpactum K~ =g(Kl) and the open set V.

635

T',us, an L-morph i sm Yt ~ Y i corresponds to the K-morphism f:~a ~ z if and only if f is s imul ta" neously continuous and a ring homomorphxsm. The l emma now follows easi ly.

We now examine the auxilia_,-y topology in the space F(A); this topology is descr ibed ,as the s t rongest topology in which every general ized sequence whose t e r m s pp can be obtained using pp =F(r~)q~dth respec t to s,)me point qEF(GM(A*)) converges. Fur ther , we consider as ,2::!.~'.ing the sum pt+pz and the product FlP2 of pairs of points Pi, P2 E F(A) for which there exis ts a point qE F(P(A*)) such that pl =F(;rl)q and P2 = F(Tr2)q; we assume here that p: + P2 -- F(r'a)q and PiP2 = F(~4)q.

The result concerning the description of dual functors which we obtained assumes the following form in the instznce under consideration:

PROPGSI-rION 7. The spaceDF(A) is i somorphic to a subspace of the space F(,~); this subspace is formed ~y mappings of F(~,) into J which are consistent with s t ructure in F(A) iutroduced above, i .e . , they are continuous ia~ the atLxiliary topology and p : e s e r v e the sum and px-oduct of those pa i rs of po in ts for which these operat ions are defined.

L I T E R A T U R E C I T E D

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