S E M I A D J O I N T F U N C T O R S A N D K A N E X T E N S I O N S

M. Y a . M e d v e d e v UDC 519.48

A func to r U �9 A --* B wil l be c a l l e d a r igh t s e m i a d j o i n t func to r if for some func to r G : B --* A the re ex - i s t n a t u r a l t r a n s f o r m a t i o n s go : 1 B --* UG and r GU ~ 1 A such that U~b- g0U = 1 U. In this case G will be ca l l ed a base func to r for U, and go and ~ base n a t u r a l t r a n s f o r m a t i o n s � 9

A left s e m i a d j o i n t func to r is def ined in a dual m a n n e r ; na me l y , a func to r F : A --* B is sa id to be left s e m i a d j o i n t if for some func to r V: B ~ A the re ex i s t n a t u r a l t r a n s f o r m a t i o n s ~ : 1 A ~ VF and/3 : F V ~ 1 B such that flF �9 Fc~ = 1 F. The func to r V and the n a t u r a l t r a n s f o r m a t i o n s a and/3 a r e a l so ca l l ed base�9

If U is a r igh t s e m i a d j o i n t func to r with base func to r G, then any func to r T for which the re ex i s t s a n a t u r a l r e t r a c t i o n T ~ G is a l so base for U.

But if for a r igh t s e m i a d j o i n t func to r U t he r e ex i s t a base func to r G and base n a t u r a l t r a n s f o r m a t i o n s go and r such that G is a lef t s e m i a d j o i n t func to r with base func to r U and base n a t u r a l t r a n s f o r m a t i o n s go and r then U is r igh t ad jo in t to G. Note a l so that an n - a d j o i n t func to r (see [1]) is s emiad jo in t .

It is e a s y to ve r i fy that the c o m p o s i t i o n of r igh t (left) s e m i a d j o i n t f unc t o r s , if it is defined, is a r igh t (left) s e m i a d j o i n t func tor .

R e m a r k 1�9 Not eve ry r igh t (left) s e m i a d j o i n t func to r is a r igh t (left) adjoint functor .

Indeed, c o n s i d e r the ca t egory C with a s ing le object C and a s ing le non iden t i ty m o r p h i s m ~ : C ~ C such that ~ ~ 1 c and c~ �9 ~ = ~. Let C be the subca t ego ry of C c o n s i s t i n g of C and the ident i ty m o r p h i s m I c. Then the re a r i s e s a un ique p a i r of f u n c t o r s F : C ~ C and E : C - - C. It is obvious that F is r igh t and left s e m i a d j o i n t with base func to r E, but F is not ad jo in t to E.

P r o p o s i t i o n 1. F o r a func to r U : A ~ ]3 to be r igh t s e m i a d j o i n t it is n e c e s s a r y and suf f ic ien t that for some func to r G : B --~ A the r e e x i s t s a n a t u r a l b i f unc t o r t r a n s f o r m a t i o n ~ : H o m A (G(-), -) ~ HOmB(-, U(-)) that is a r e t r a c t i o n .

P roof . Let U : A --~ B be a r igh t s e m i a d j o i n t func to r with base func to r G and base n a t u r a l t r a n s f o r m a -

t ions go and r Define t r a n s f o r m a t i o n s 7: HOmA (G(-), -) -~ HomB(- , U(-)) and ~: HOmB(-, U(-)) --~ Horn A �9 (G(-), -) . F o r a r b i t r a r y ob jec t s A 6A and B 6 B and a r b i t r a r y m o r p h i s m s f : G(B) ~ A and g : B ~ U(A) se t

7/B,A(f) = U(f)- go B and ~B,A(g) = ~OA -G(g). It i s e a sy to ve r i fy tha t ~7 and ~ a r e n a t u r a l . Mor e ove r ,

~ ~" "~. ~ (g) ='1, ~ ( ' ~ C (g)) =Ur ~r (g) . ,p~=U%, ,~V~ g=g,

i. e . , ~ is a r e t r a c t i o n with r ight i n v e r s e ~.

C o n v e r s e l y , fo r some func to r G : B ~ A suppose the re ex i s t s a n a t u r a l r e t r a c t i o n ~: HomA(G(-), -)

HOmB(-, U(-)) with r igh t i n v e r s e ~. Define a t r a n s f o r m a t i o n go: 1 B --* UG by se t t ing ~0 B = ~B,G(B)(1G(B)).

To p rove that ~0 is n a t u r a l , f i r s t note that s ince ~ is n a t u r a l , for an a r b i t r a r y m o r p h i s m f in Horn A

�9 (G(]3), A) we have ~B,A(f) = U(f)" go B, and for an a r b i t r a r y m o r p h i s m w: C ~ B in B we have 7?C,G(B)(G(w))

= r Then

~ - ( 0 =n~, ~(~) (G (,o)) = U G (o)) . ,~ ,

i. e . , go : 113 --~ UG is a n a t u r a l t r a n s f o r m a t i o n .

Def ine a t r a n s f o r m a t i o n r GU --~ 1A. F o r each A6A le t SA = OU(A),A(1U(A) )" Exac t ly as for go, it can be p r o v e d that ~b is n a t u r a l . M o r e o v e r , fo r an a r b i t r a r y ob jec t A ~A we obta in

T r a n s l a t e d f r o m S ib i r sk i i M a t e m a t i c h e s k i i Zhu rna l , Vol. 15, No. 4, pp. 952-956, Ju ly -Augus t , 1974.

Or ig ina l a r t i c l e s u b m i t t e d J a n u a r y 12, 1973.

�9 19 75 Plenum Publishing Corporation, 22 7 West 17th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.00.

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Hence U is a r i gh t s e m i a d j o i n t func to r .

Let the f u n c t o r s T: A ~ B and S : A - - K be given. Reca l l (see [2]) that a r igh t Kan ex t ens ion of S by

T i s def ined to be a func to r f r o m B to K, denoted by RanT(S), and a n a t u r a l t r a n s f o r m a t i o n c~ : RanT(S ) " T S that sa t i s fy the fol lowing u n i v e r s a l i t y p r o p e r t y : if we a re g iven ano the r fu~nctor P : B ~ K and n a t u r a l

t r a n s f o r m a t i o n fl : P T -~ S, then the re e x i s t s a un ique n a t u r a l t r a n s f o r m a t i o n 7: P -* RanT(S) such that c~

. T T = f i .

If in the def in i t ion of a r igh t Kan e x t e n s i o n we drop the r e q u i r e m e n t that the n a t u r a l t rmas fo rma t ion

, / b e un ique , we ob ta in the def in i t ion of a weak Kan ex tens ion , which wilI be denoted by W-RanT(S) .

Suppose we a r e g iven the f u n c t o r s

B ~ - A & K ~ T

and a r igh t Kan ex t ens ion RanT(S) with n a t u r a l t r a n s f o r m a t i o n c~ : RanT(S)" T ~ S. We shal l say that a f u n c - to r F p r e s e r v e s th i s r i gh t Kan ex t ens ion if the fune to r F �9 RanT(S) and the n a t u r a l t r a n s f o r m a t i o n F~ : F �9 RanT(S) �9 T -,- FS a r e a r igh t Kan ex t ens ion of FS by T.

THEOREM 1. A r igh t s e m i a d j o i n t func to r p r e s e r v e s any r igh t Kan ex t ens ion .

P roof . Let U : A ~ B be a r igh t s e m i a d j o i n t func to r with base func to r G and base nat~aral t r a n s f o r m a - t ions r and $. In addi t ion , for s o m e f u n c t o r s T: K ~ T and S : K - - A suppose the re ex i s t s RanT(S) wi~h n a t u r a l t r a n s f o r m a t i o n a : RanT(S ) �9 T ~ S. Let us p rove that the func to r U" RanT(S) and the n a t u r a l t r a n s - f o r m a t i o n Ua a r e a r igh t Kan ex t ens ion of US by T. If we a r e g iven a h m c t o r P : T -* B and a n a t u r a l t r a n s - f o r m a t i o n fi : P T ~ US, we have a func to r GP : T -* A and a n a t u r a l t r a n s f o r m a t i o n 6 = ~bS" Gfi : G P T -* S. Consequent ly , t he r e e x i s t s a un ique n a t u r a l t r a n s f o r m a t i o n # : GP - - RanT(S) such that the d i a g r a m

6PT ~ R~,r(~)r

2

c o m m u t e s . Let co denote the n a t u r a l t r a n s f o r m a t i o n U~ �9 ~oP : P --~ U" RanT(S). Then

Ucz. mT=Ucz. UF, T. epPT=U6 . ~PT=Ur . UGh. q)PT =U$S. (pUS. ~=~.

It r e m a i n s to p rove that the n a t u r a l t r a n s f o r m a t i o n co such that Uc~ �9 co T = fi is un ique . 7Let ~ : P -~ U - RanT(S ) be a n a t u r a l t r a n s f o r m a t i o n such that Uc~ �9 ~ T =/3. R e p l a c i ng p by the n a t u r a l t r a n s f o r m a t i o n SRanT(S)" G~ : GP - - RanT(S) in (1), we again ob ta in a c o m m u t a t i v e d i a g r a m . In fact ,

a- ~Ranr (S) T. G~) T=r GUn. G~.~ T=ll~S. G~ =5.

But s ince p i s un ique , we have SRanT(S ) �9 Gw = # . Then

=U~Ranr (S) �9 tpURanr (S). if) =Ut~Ranr (S) �9 UGff) �9 qJP=UFx'cpP=oJ.

This p r o v e s the t h e o r e m .

COROLLARY. A r igh t s e m i a d j o i n t func to r p r e s e r v e s i n v e r s e l i m i t s .

P roof . The r e s u l t fol lows i m m e d i a t e l y f r o m T h e o r e m I and the fo l lowing s t a t e m e n t (see [2], The - o r e m X.7.1): for a func to r S : K -~ A the r e ex i s t s an i n v e r s e l i m i t if and only if t he r e e x i s t s a r igh t Kan e x - t en s ion RanE(S), where E : K ~ 1 is a f anc to r into the ca t egory with a s ing le ob jec t and a s ing le iden t i ty m o r p h i s m .

If a m o r p h i s m of some c a t e g o r y is a m o n o m o r p h i s m , i ts k e r n e l p a i r e x i s t s and c o n s i s t s of two i den - t i ty m o r p h i s m s . By v i r t u e of th i s , f r o m the p r e c e d i n g c o r o l l a r y we obta in tha t a r i gh t s e m i a d j o i n t ~anctor t akes m o n o m o r p h i s m s into m o n o m o r p h i s m s .

The fo l lowing c r i t e r i o n for the e x i s t e n c e of ad jo in t func to r s is known (see [2], T h e o r e m X.7.2): a func to r U : A --~ B has a lef t ad jo in t if and only if t he r e ex i s t s a r i gh t Kan e x t e ns i on RanU(1A) p r e s e r v e d by U. In th i s c a s e Ranu(1A) is a lef t ad jo in t to U, and the r igh t Kan e x t e ns i on Ranu(1A) is p r e s e r v e d by an a r b i t r a r y func to r T : A --- K.

The fol lowing t h e o r e m is va l i d for r i gh t s e m i a d j o i n t f unc t o r s :

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THEOREM 2. F o r a f unc to r U : A --- B to be r i g h t s e m i a d j o i n t i t i s n e c e s s a r y and su f f i c i en t tha t t h e r e e x i s t a weak r i g h t Kan e x t e n s i o n W - R a n u ( 1 A ) p r e s e r v e d by U.

P r o o f . Le t U : A ~ B be a r i g h t s e m i a d j o i n t f unc to r with b a s e f tmc to r G and b a s e n a t u r a l t r a n s f o r m a - t i o n s ga and <b.

Le t u s p r o v e tha t G and r GU ~ 1 A a r e a weak r i g h t Kan e x t e n s i o n W-RanU(1A) . Indeed, i f a func - t o r P : B --- A and a n a t u r a l t r a n s f o r m a t i o n [9 : PU --- 1A a r e g iven , we def ine a n a t u r a l t r a n s f o r m a t i o n . Then

r ~t U=r pGU. P~U=~ . PU r P,pU=P.

Note tha t if oJ : P - - G i s a n o t h e r n a t u r a l t r a n s f o r m a t i o n such tha t ~ �9 ~U =/~, then

=~G . P~=~G . o~UG . P~=r . G~ - o~.

The last equality characterizes the "degree of weakness" of the given right Kan extension. The fact

that U preserves W-RanU(IA) follows from Theorem I.

Conversely, for U:A ~ B suppose there exists a weak right Kan extension W-RanU(IA) with natural

transformation (~ : W-Ranu(I A) �9 U -- 1 A preserved by U. Then we have a weak right Kan extension W-

R a n u ( U ) wi th n a t u r a l t r a n s f o r m a t i o n [3 : W-Ranu(U) �9 U --* U and a n a t u r a l i s o m o r p h i s m 6 : U" W-RanU(1A) --* W-Ranu(U) such tha t fi- 5U = U a . By the de f in i t ion of a weak r i g h t Kan e x t e n s i o n , t h e r e e x i s t s a n a t u r a l t r a n s f o r m a t i o n ~: 1 B - - W - R a n u ( U ) such that fl �9 7U = 1U. Def ine n a t u r a l t r a n s f o r m a t i o n s

q) : iB --~ U. W-Raau ( I Q and ~ : W-Ranv (IA). U-~ tA,

by s e t t i n g s 0 = 6 - 1 . y a n d r Then

U~.~U=U~.8-'U. ~U=~. TU=I~,

i. e., U is a right semiadjoint functor. This proves the theorem.

Remark 2. By using the proof of the necessity in the preceding theorem, it is easy to prove that if

U : A -- B is a right semiadjoint functor, then the weak right Kan extension W-RanU(IA) is preserved by

any functor T: A --- K.

We sha l l s ay tha t a c a t e g o r y A i s l e f t - c o m p l e t e if each f u n c t o r F : I ~ A f r o m an a r b i t r a r y s m a l l

c a t e g o r y I ha s an i n v e r s e l i m i t .

P r o p o s i t i o n 2. F o r a f u n c t o r U : A --- B to have a l e f t ad jo in t , i t i s n e c e s s a r y , and if A i s l e f t - c o m -

p l e t e su f f i c i en t , tha t U be a r i g h t ad jo in t f unc to r .

P r o o f . The n e c e s s i t y i s obv ious . Now l e t A be a l e f t - c o m p l e t e c a t e g o r y and U : A - - B a r i g h t s e m i - ad jo in t f u n c t o r wi th b a s e f u n c t o r G and b a s e n a t u r a l t r a n s f o r m a t i o n s q~ and r Above we e s t a b l i s h e d that U p r e s e r v e s i n v e r s e l i m i t s . If we p r o v e tha t U a d m i t s a so lu t ion s e t (see [2], p. 117) fo r e ach o b j e c t B~B, then by F r e y d ' s t h e o r e m ([2], T h e o r e m V.6.2) U wi l l have a l e f t ad jo in t .

Le t u s show tha t fo r an a r b i t r a r y o b j e c t B ~ B the so lu t ion se t c o n s i s t s of only a s i ng l e o b j e c t G(B). Indeed , l e t f : B --" U ( A ) b e an a r b i t r a r y m o r p h i s m in B. Then the fo l lowing d i a g r a m i s c o m m u t a t i v e :

8 " u8(8)

In fac t , Ur A ' U G g ) . ~B = UeA" ~UA" f -- f" QED.

The nex t t h e o r e m fo l lows f r o m T h e o r e m 2 and the p r e c e d i n g p r o p o s i t i o n .

T H E O R E M 3. L e t A be a l e l % - e o m p l e t e c a t e g o r y and U : A ~ B a f a n c i e r . If t h e r e e x i s t s a w e a k r i g h t Ken e x t e n s i o n W-RanU(1A) p r e s e r v e d b y U, t h e n t h e r e e x i s t s a r i g h t Kan e x t e n s i o n R a n u (1 A) p r e s e r v e d b y any h m c - t o r T : A ~ K. In p a r t i c u l a r , f o r any f a n c i e r T : A ~ K t h e r e e x i s t s a r i g h t Ken e x t e n s i o n R a n u ( T ) .

1.

2.

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

J. M. M a r a n d a , " C o n s t r u c t i o n s f o n d a m e n t a l e s de d6gr6 s u p e r i e u r , " J. Re ine and Angew. M a t h . ,

243, 1-16 (1970}. S�9 M a e L a n e , C a t e g o r i e s fo r t h e W o r k i n g M a t h e m a t i c i a n , S p r i n g e r , B e r l i n - H e i d e l b e r g - New York (1971) �9

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