Seminarberichte Nr. 22

deposit_hagen

Publikationsserver der Universitätsbibliothek

Mathematik und

Informatik

Seminarberichte aus dem Fachbereich Mathematik der FernUniversität

22 – 1984

der Mathematik (Hrsg.)

Seminarbericht Nr. 22

Feuilletages Co-Isotropes Principaux et Couple de Champs Self-Orthogonaux quasi Concourants sur une Variete

Pseudo-Sasakienne

von F. Arnato

Connectivity Spaces and Component Categories

1

von Reinhard Börger 15

The Geometry of Buekenhout Ovals. A Survey

von Giorgio Faina

Topological Universes and Total Spectral Dualities for Differential Analysis

von L.D. Nel

Algebra of Constructions I.

The Word Problem for Partial Algebras

von Adam Obtulowicz

Separation Properties in Algebraic Categories of Topological Spaces

von Günther Richter

Motivations and Preparations to Develop a Cohomology Theory foP Subalgebras and Submodules

41

99 -

-

129 -

- 183 -

-

195 - von Klaus Werner Wiegrnann

Introduction.

DE CHM!PS SELF-ORTHOGONAUX QUASI CONCOURANTS .,, .,,

SUR UNE VARIETE PSEUDO-SASAKIENNE de F. AMATO (*)

Les varietes pseudo-Sasakiennes M(U,j,v[,g) ou. 9.L, j , ~ et g sont r~

spectivement l'operateur para-complexe, le vecteur canonique (ou le ve~

teur de Reeb), la 1-:forme de contact et le tenseur metrique de signatu re m+ 1, m ont ete def'inis recemment par R. Rosca [ 1] .

Taute CR-sous-variete [2],[)J M qui est eo-isotrope a ete denommee dans

[4] une CICR-sous-variete.

Soient S et S~ les deux distributions self-orthogonales (abr. s.o.) canoniques sur M. ~

On demontre dans ce travail que les distributions

~* ,,.., l

L = s* ~ {j

J

definissent deu.x (m+1)-feuilletages eo-isotropes denommes principau.x.

Les feuilles MI: et Mr..;(< respectivement de ~ et i:. ~ sont des CICR-SQ us-varietes de defaut m, qui sont totalement geodesiques et de presque

courbure moyenne constante dans M. ~

En outre chacune des sous-varietes Mi: , ME.* est feuilletee dans le sens [ 4]. En:fin la 2-forme de Ricci § et 2-forme f ondamentale _n_ de

M ^{(.fi. =} d~/2) appartiennent a la m@me classe d'homologie H 2

(M,R).

Au paragraphe 3° on suppose que M

N

porte un couple de champs vectoriels

XE S , i~E s~ s.o. quasi concourants de contact, dans sens de R. Rosca [5].

L'existence d'un pareil couple e~t assure par la fermeture d'u.n systeme

(~) Lavoro eseguito nell'ambito del • G.N.S.A.G.A. del C.N.R ••

differentiel exterieure.

On montre que:

(a) X et x* sont des directions geodesiques et tx,x~,j} est un 3-feuiJ letage;

(b) X ^(resp. X~) est une section parallele dans s* ^(resp. S);

(c) la condition necessaire et suffisante pour que X (resp. x~) ^definis

, V

sent une trasfonnation infinitesimale conforme sur M est que -1-for

~ ~

me de Ricci y- sur

M

soit une relation integrale d'invariance pour

X ^(resp. X*).

Au paragraphe 4°, on montre que le relevement complet ..n.

⁰

^{de ..?i. sur la}

,..,

,..,

variete fibre tangente TM ayant M pour base est une forme homogene de de gre -1.

D'autres considerations mettant en jeu certains operateurs differentiels sur TM sont aussi faites.

~

-Preliminaires.

Soit M une variete pseudo-riemannienne a (2m.+~)-dimensions d'index m+1 (c'est-a-dire de signature m+1, m).

En chaque point p EM on a la decomposi tion standard (

~

.1)

ou T~, 7'-f..~ et 1,~ sont respectivement l'espace tangent, l'espace neu

p p p

tre a

2m

dimensions et une droi te genre temps en p f M.

Soient S~, s'!c1f,~ les deu.x distributions self-orthogonales (abr. s.oQ)

p p p

(cbacune am dimensions), qui definissent un automorphisme involutif 'U, de carre +1 (t~ est l'operateur para-complexe definit dans [6].

Soi t JE jp , q ^{E .11.}

¹

^(M) le couple qui definit une structure de contact

( i , q) ^{sur M et soi t} V l 'operateur de differentiation covariante defini par le tenseur metrique g.

Soit f"1TM =:xM l'espace des sections des cbamps vectoriels du fibre

..., ~ ~

Si pour tous champs vectoriels Z,Z' E ~M les tenseur de structure ('LL, 'i, q, g)

r ^{~~: Zl = z_ - _Vl< Zl j} g('U.z, '11.z • )

~ " " " " . . . , w , v ~

-g (

Z, z ' )

⁺

^{Vl (} Z) Yl ( Z' )

. ₌

( 1. 2) g(Z,j) = vi_(Z) . _7z ' _{1 = 'lLz·}

d~(Z,Z') = -2g('U.z, ' z') . _{q (j) = ~}

'

la variete M(tL,'j ,q,g) a ete denommee dans [--f] une variete pseudo-Sas!!:

kienne.

Rappelons aussi que puisque le (1,1)-tenseur de structure iL ^satisfait a ^'l.L-U=O

3

on peut dire que toute variete M(tL,1,q,g) est une para

f-variete (7] .

,.,

Dans le but d'etudier des immersions impropres dans M nous considerons sur M un champ de reperes de Witt adapte [4], soit W ⁼ {hA; A,B,C = O, 1, ..• ,2m} ou {ha; a =1, .•. ,m}

= \ , ,

{ha~; a~= a+mj = s; ^{sont des vec}

teurs isotropes (ou nulles) et le vecteur de structure S= h 0 est aniso tro-pe.

On a comme on sait:

(1.3)

et

( 1 . 4)

j g(ha thb"'") = -~b

lg(J,ha~) ⁼

^{0 ,}

, g(j',ha) = 0

g(j ,j) = 1

'l~h = h

a a ' ^'l,(,

^h

^{a .w-=}

^-h

^a

^~

'

Si W*= f ^c.0A J ^{e st} la eo base de W, on :po ^,...

se W

^~

⁰

= yt

^{, V}

et la f orm.e de soudu re d~ (dp est une ~-fonne differentielle canonique a valeurs dans R2m+ 1

sur le fibre W(M) ^{[8]), s'ecrit}

( ~. 5) dp =

I l resulte de (1.3) que le tenseur metrique g est exprime par

( .Ä •

_-, 6) _g

^w _{= 2} ~ _{L -}

_w ~ a

^IQ'-, _{\CY ~} ."":""

a * ₊ ~ _{vi_}

_@

~ _yt

0

^{"'A ~A ~C ,..,A c a:,}

~

^;;_A

D'autre part si B

= fBc

C-0

(f-Bc ,;.;

^{C (M)) et ~B}

sont respectiv~

ment les formes de connexion et les 2-formes de courbures sur le fibre

W(M), on a les equations de structure (E. Cartan), sous fonne symbolique:

(1.7) Vh = e ^{® h}

(,1.8) dw= -e-/\w ~

,..,

,.., ~ ~

(-1.9) d 8' =

^_fJ/\0'+

^e

Les deu.x distributions Set ~ S~ etant involutives on sait que l'on a:

t ^{e: + ~a* 0b~ =}

⁰

^{, -a* eb =}

⁰

^{, -a eb~=}

⁰

(1 .~O)

_{,,., 0}

,.,a~ ~a

"'0

Ba ⁺ eo ⁼

⁰

^{, eo + e}

a: ^,14,=

^o

On deduit encore de (1.2), (1.7), (1 • 8) et (1.10)

d~= 2n

ou ^{l'on a pose}

~ ~

et nous conviendrons d'appeler _n_ la 2-forme fondamentale de M.

Enfin il vient aussi

Ce qui exprime que J est un champ de Killing ( comme dans le cas Sasakian;

Soit maintenant M une feuille d'un feuilletage eo-isotrope F sur M. On

tv

C

a montre dans [4] que toute pareille

^SOUS

variete M est une CR-variete de

M (dans le sens [2] , [3]).

Les

CR-sous-varietes eo-isotropes ont ete denommee dans [4] les CICR-sous varietes.

Si M est une CICR-sous variete, alore la distribution verticale D~ sur M est un feuilleta~e isotrope.

2 - Distributions eo-isotropes principales sur M. ~

Considerons sur M les deux (m+1)-distributions ~ = S@{.JJ,i.*= S,\<${4J

qui ont ~n cornmun la droite :J.

,.., ,.,, ,.., :,J,,

,,.,*

Puisque orth. "z:

=

s, (resp. ortb. "2: = S ) il resul te en vertu d 'une defi

nition connue que

~

~ _{et L} ~ .... sont des distributions eo-isotropes que nous conviendrons d'appeler principales.

,-V

Conformement a ^(1.1) la distribution com-plementaire de L est s""'. ^Soit

*

^{-1;1,- m~ , V}

1l'

= u., /\ ••• /\

w la forme simple uni taire qui correspond a ^S * et notons par f la ~-forme de Ricci sur M, soit

(2.-1)

A l'aide de (1.8), (1.~0) on trouve (2.2)

,<J

et cette relation montre que T(*est exterieure de recurrente [9], avec

f- mq comme ~-forme de reeurrence. Alors en vertu d'une proposition con nue, il resulte que L ~ definit un feuilletage eo-isotrope.

,-.J

q ,,, e

Si H (~,R) est le q espace de cohomologie du feuilletage

^~

, alors on sai t [1 o] que F, - ^{m vi_} definit un element de _E

¹

_(y: ~ _,R).

Notons par M'i'.. les feuilles a. ^m+-1 dimensions du feuilletage L . ~ La for me de soudure indui te dp (nous notons le s elements indui ts sur ME. en su_E

primant ,,...,) est:

( 2. 3)

Il resulte aussitSt de (2.3) que le tenseur metrique g = g/ME. de la sous variete eo-isotrope M~ est:

( 2. 4)

On a donc rang g = ^-1 et dim ML = m+1 • Il resul te done en vertu d 'une de fini tion connue que ME est une sous variete eo-isotrope de defaut m.

,.._,, ,...,, .-..., I'\,/ , V

Puisque SC5-::: definit le fibre normal au feuilletage L , et l'on a'lLS=S il resulte eonformement a [4] que Mi: est une CICR-sous variete donc la di stribution vertieale Di est le feuilletage isotrope;~,

D'autre part il est elaire en vertu de (2.2) que la forme simple unitaire

~

^~

7( qui correspond a S est aussi exterieure recurrente.

L'on eonclut de la qlle la distribution horizontale D de ME (qui au fait

se redui t a 2)}) e st at1ssi invol1.1ti ve.

Conformement a. ['11], ^{[2] ,} [4} , on peut donc dire que la sous variete ML est feuilletee et les feuilles de ML sont respectivement les sous va rietes isotropes Set les droites genre temps {J}.

Soi t mantenant

TI € T'"ME: @ T*Mr

®

r-1-M~

la eeconde forme quadratique ve ctorielle de M~ (IT est un ~-morphisme de T~(M~), dans Met ne depend pas de la connexion normale v.J. ^{[8]). Eu egard} a [8], ( 2.3) et (1.9) on trouve rapidement rr ^{= O,} et l'on conclut que ML est totalement geodesi que.

Notons comme dans [12], par Aq(M,TM) =rL\.q T~M le module des q-formes vectorielles sur Met par dv : Aq(M,TM)---+ Aq+ ¹ (M,TM), l'operateur de differentiation exterieure covariante par rapport a la connexion ~ '7

v

²

^V

^V

(d = d o d n'est pas toujous zero).

L

a sous var1e e

·

' t '

M

r

^~

.t.t ^an t ^or1en · t.t. ^{~e sO1en} · t

^~

""' ·.

^{~ i}

,...q

T*M _ _ _ ! , . ^,~~

"'m+-1 -q

^I ~M

^et

'"[M

respectivement la "star" operateur et la forme volume sur ML •

r-

Eu egard a ^{(2.3) et (~.3) il vi~nt}

-::- ( )a-,f "1 ,,... a tm.

h ( 2. 5) * ^dp

^{=G.. -} <:.A.J / \ ••• /\ CA) / \ • • •

I\

c:.A.:>

® a~ +

• 'Wl. ~

+t:..AJ/\ •.. /\cu ®.3

(les termes chapeautes sont omis) et l'expression de ~dp montre que c'est un element de Am(ML~,TMr-»)•

On a : ^{dv (*dp)} =J'® ^{t'M ou.} ;p est un champ vectoriel invariant. Confo!

mement a ^[4] nous dison que

L:

:P/m+-1 est le vecteur de presque courbure moyenne

de M

L •

A l'aide de (1.7) et (1.8) on deduit de (2.5)

(2.6)

"7')

~

^~

(

²

te

On voit donc que

J-

est une section dans L et l'on a

~

J",'J')) = m = C . Nous disons dans ce cas que Mz: est de presgue courb11re moye?}lle con5tante.

Nous finiron ce paragraphe avec les considerations auivantes Notons par

( 2. 7)

~

On sai t [-13] que fr ^{definit (.a.} une constante pres) la '1ere classe de Chern

,.,

sur M.

A l'aide de (1.9) et (~.~O) on trouve apres calcul

~ ,..., ~

(2.8) dj<-= 8-.n.

L' equa tion ci de ssus montre que (9 ~ et ..ii appartiennent a la meme classe d'homologie H2 (M,R).

Theoreme: Seit M('lk,1,~,g) une variete pseudo-Sasakienne a 2m+1 dimensions et soient Sets~ les 2 distributions self-orthogonales canoniques sur M.

,..,

Toute pareille variete M est feuillete~ par deux CIOR-sous varietes pri~

cipales Mi: et M!:~ telles que ML/l Mi:~ ={1}. ^Ces sous varietes joussent des proprietes suivantes:

(a)

ML

(resp. ML.;t-) est totalement geodesique;

(b)

ML (

resp.

ML*)

est eo-isotrope de defaut m, et est de presque cour bure moyenne constante egale am; •

( c) ML. (resp. ME:*) est feuilletee par {jJ ^et S ^(resp. s~).

3 - Couple de champs vectoriels quasi concourant de contact.

On peut decomposer la forme de soudure dp exprimee par (1.5) en (3. ,1) _d p ⁼ d p S

$

d p

3 ..-

(±)

q ^® 1

ou

~ dP -

,va

® h

s -

u.) a

( 3. 2)

dp~=

N

Et"

w © h*

_a

Confomement a la definition de R. Rosca [5] (voire aussi R. Rosca - K. B~

cbner [14]) nous disons que les champs vectoriels X ^f S ; i*E s-1t-, fonnent un couple s.o. Quasi concourant de contact si l'on a:

(3.3)

ou

~ r . , , - ,.,, ,..., <'V

f,f*E C

⁰⁰

(M). On sait que si Z 6

^~1

est un champ vectoriel quelconque,

alors VZ est une section de End TM. Poson maintenant

(J.4)

et notons suivant [12], par p: TM-? T~ l'isomorphisme musical par ra~

port a. g.

Posons alors:

a ~

^a-.lt

f

~ = p (x) = ^{~ a} x

^u..:,

~

~a-i.

~

^a

~* = p (X*)= ~ a

X c..(..)

(3.4)

Compte tenu de (1.7) et (1.~0) on deduit de (3.3) (3.5)

A l 'aide de ( 3. 5) la differentiation exterieure de J ^{et ;l.} ~ donne

(3.6) J ^d~ = f Ji ⁺ q /\ ~

( d~~=

f~.Q + vi, /\ d..~

et compte tenu de (1.11) la differentiation exterieure de (3.6) donne

\ df ⁼ f q - 2d.

( 3 • ?) l

^df~=

f)+,-q-

²

~ ~

On verifie facilement que le systeme differentiel exterieure (3.6)+(3.7)

, V "':11<

est fermee et ceci assure l'existence du couple (X,X ).

Posons:

(3.8) ~ = ~/f , ~~-= 'd-*;f~

On trouve facilement a l'aide de (3.6) et (3.7)

,.., "'* _,

dr ⁼ dp ⁼

^.0.

On donc d (p- ~) ^{= 0 ce} qui montre que la difference r - ^f4f- e st un cocycle.

D'autre part on deduit de (3.3)

,..., ' V / V , V

[

y'X

X=

f

X

,.., ~~ ~,..,~

V rx ⁼

^{- I ' X}

et puisque g(X,X) = 0, g(X*,X*) ⁼ 0 les relations ci dessus prouvent

que les chams vectoriels X, X~ sont des directions geodesiques.

[ J [ ^{~ ~*1}

Si indique le crochet de Lie on trouve aussi X,X J= 0 ce qui montre que x ^et x* commutent.

Enfin in vient aussi

[ x, ^{!] =} x - ^{j' ,}

^[~.t-

^x

^,..3

^{-eJ =}

^{-X -~114-}

f

et ces relations et [x,x~]= O, permettent de dire que les trois champs vectoriela X, xlll-, ^$ definissent un 3-feuilletage.

Soi t maintenant ü*e s* ^(resp. u ^ES) une section quelquonque dans s9.t-(r~

sp. S). ^{De (3.3)} il vient auesitSt:

~ ~ _{Vu~X = 0 ,}

^{,V ,..,/lt}

(resp. ~u X= 0)

et par consequent X ^(resp. x*) est une section parallele dans s~(resp. S).

ta variete ~

M

etant orientee, seit

(3.9) ~ ⁼ t.Z./

A ••• /\ ~ 1

/Wl. / \

~

l'\J

l'element de volume sur M.

Par rapport a une orientation locale de Mon a pour tout tout champ vecto ~

riel Z ^G 1EM r~ ^2]

( 3 • ~

0)

~ div Z ^{= (}

^d

ⁱ

^V

Z) ~

= d

~ z f = i' z f

(;z:': derivee de Lie).

En faisant usage de (J.10) et de, (3.5) on trouve

, V ""'"""V

{

div X = mf - J-<-(X) (3. -1 1) div _xill"= -mf*+ f,(X*)

Ainsi la condition necessaire et suffisante pour que X ^(resp. X*) ^defini_f!

se une transfonnation infinitesimale conforme sur M, est que la 1-fonne de Ricci f- ^{seit une} relation inte~rale d'invariance pour X ^(resp. X~).

Considerons maintenant la CICR-sous variete ME discutee au paragraphe 2°.

En faisont usage des equations (~.9) et (3.7), la differentiation exteri eure de (3.5) donne a~res calculs

(J.12) ^b a a

X (&)b - 2f @

0

=

0

. _{, a,b =-1, ••• ,m}

La relation ci dessus montre que la sous variete M~ est a courbures recur rentes.

Il est claire que la

^SOUS

variete Mt:=*jouit de la meme -propriete.

Theoreme: Sur toute variete pseudo-Sasakienne l'existance d'un couple

(X,X~) de champs vectoriels s.o. quasi concourant de eontact est ass~

re par l'existence d'un systeme exterieure fennee. Les vecteurs X et x~

jouissent des proprietes suivantes: .

(a) ce sont des directions geodesiques;

(b) il definissent avec le vecteur de structure J un 3-feuilletage;

,..,, ,...,,, ,..,...~ /V

(c) X (resp.X~) est une section parallele dans S (resp. S);

(d) la condition necessaire et suffisante pour que X ^(resp. X~) ^definis

sent une trasfonnation infinitesimale confonne sur M, est que la

N -

1-ferme de Ricci f soit une relation integrale d'invariance pour

X

( resp. ~~)

^{X •}

4 - R~levement complet sur TM.

Soit TM la variete fibre tangente a une variete para Sasakienne et so!

ent f ^{(vA) et B·= { ;;:;A , dvA}} respectivement le champ vectoriel de L!

ouville et une cobase adaptee sur TM.

Notons suivant [15] par ( )

⁰

le relevement complet sur TM.

Si~ est une q-forme sur M, alors u.:Jc est un isomorphisme lineaire de A q M --- • A TM.

Soi t alors ..n. la 2-forme fondamentale definie par ( ~., 12).

Conformement a ^{[15] on a:}

( 4.'1)

et a l'aide de

(4.2)

et puisque il vient aussi

(4.3)

De (4.2) et (4.3) on trouve

(4.4) = ^SL

^rvc

ce qui prouve comme on sai t [ '16] .I{ e st une 2-forme homogene de degre

^,f.

Considerons maintenant comme dans [17], sur TM le champ scalaire 1 ^definit

par

(4.5) ~ ...

^{a ~a~ ·}

1 = L

a v v

E c

⁰⁰

'li

Soient mantenant d et i respectivement les operateurs de differentia

V V

tions et de derivation verticale sur

..(i.TM. 'V

Ces operateurs definissent res pectivement une anti derivation de degre

~

sur ...tl TM [16] .

et une derivation de de~re

⁰

On obtient de (4.5) la forme semibasique

~

*

(4.6) ~ (

,.,a .., a ..,a

d l=L°

V

w

^{+ V}

V

a

On trouve de (4.6) la 2-forme de rang 4m

u.Ja ) .,,

~

*

~

( 4. 7) 'f = L

_a

(

^{dva /\}

^{U:, a} + dv

⁸

^I\

^{(;Ja )}

⁺

Par la suit nous coviendrons d'appeler le champ scalaire 1

^€

^C

⁰⁰

TM ^la _---. nor me para-henni tienne de 'f . ~

La 2-forme \.f' ~ etant fermee, on trouve rapidement

~

' ⁱ

_V

4' ⁼

⁰

En vertu d'une definition connue et puisque 'r' ~ n'est pas de rang maximale

~ ~

( dim Ker '+' = 2) nous di sons que 't' e st une forme pre-finslerie:i;me.

Nous nous proposons de revenire avec une autre occasion sur les proprietes

; v ,.J

des relevements verticaux et complets dei champs vectoriels X et X* discu tes au paragraphe 3° ainsi que des differents automorphismes par rapport

et 'f'. ~

-c ~

Theoreme: Le relevement complet

...0:.

sur la variete fi bree tangente TM

N

~

d'une variete pseudo-Sasakienne M, de la 2-forme fondamentale ..n.. est homo

,.., ~

gene degre 1 . Si 1 est la norme para-hermi tienne du cbamp de Liouville 'f

s1.1r TM alors la derivation exterieure de la differentiation .verticale de

~ 1 definit une 2-forme pre-finslerienne sur TM. ~

BIBLIOGRAPHIE

I R. Rosca:

On

Pseudo-Sasakian Manifolds. Rendiconti di Matematicao ( 1984) ( t.o appear) .

2 A. Bejancu: CR Submanifolds of a Kaehler Manifold II. Trans. Amer.

Math. Soc. 250 (I979), p. 333-345.

3 K. Yano -M. Kon: CR Submanifolds of Kaehlerian and Sasakian Mani folds. Birkbauser, Boston (1983).

4 V.V. Goldberg-R. Rosca: Contact co-isotropie CR Submanifolds of a pseudo-Sasakian Manifold. International Io urnal of Math. and Math. Sciences (1984) -

(to appear)

5 R. Rosca: Variete Lorentziennes a structure Sasakienne admettant un champ vectoriel isotrope ~-quasi concirculaire. C.R.

Acad. Paris, t. 291 (1980), Serie A.

6

P. Libermann: Sur le Probleme d'Equivalence de Certaines Structures Infinitesimales. Ann. Mat. Pu.ra

e

Appl. 36 (I95I).

7 B.B. Sinha: A differentiable manifold with para f-struct11re of rank r. Ann. Fac. Sc. de Kinshasa, Zaire Math-Phy. 6 (1980) N° I-2, p. 79-94.

8 J. Dieudonne: Elements d'Analyse, 4. GuthieE-Villars, Paris(I97I).

9 D.K. Datta: Exterior recurrent forms on a manifold. Tensor N.S. 36 (1982) p. II5-I20.

IO A. Lichnerowicz: Varietes de Poisson et feuilletages. Annales Fac.

des Sc. Toulose, vol. IV (1982), p. 195-262.

II M. Kobayashi: CR submanifolds of Sasakian manifold. Tensor N.S. 35 (I98I) p. 297-307.

12 W.A. Poor: Differential Geometrie Structures. Mc Grow-Hill Comp.

(I98I)

IJ · s._Goldberg: Curvature and homology. Academie Press, (I962).

I4 K. Buchner-R. Rosca: Sasakian manifolds having the contact quasi- concurrent property. Rend. del Circolo Mat.

di Palermo, Serie II, T.XXXII, (I983),p. 388.

I5 K. Yano-S. Ishihara: Tangent and Cotangent Bundles. Dekker N.Y.

(!973).

I6 Cl. Codbillon: Geometrie Differentielle et Mecanique Analytique.

Hermann, Paris (I969).

I7 F. Amato: Sur une classe de variete para complexes possedant la

propriete concirculaire. Riv. Mat. Univ. Parma (4) 8

(I982), p.223.

C O N N E C T I V I T Y SPACES A N

D

C O M P O N E N T C A T E G O R I E S

REINHARD BÖRGER

AMS (MOS) code

Pr~~ary: 18A40, 54AOO secondary:16A99, 54A05

Abstract:

connectivity spaces are obtained by axiomatizing

the notion of the "set of all connected subsets" (e.g. of

a topological or uniform space). This leads to generalization

of cornponent categories, which ha~e been studied by several

people, especially by Graciela Salicrup. The results are

taken from the author' s thesis [ 2 ].

e

§O. Introduction

In this paper we are concerned with generalizations of connectivity of topological (or uniform etc~) spaces into two different directions. At first we consider an abstract notion of the "set of all connected subsets" of a given space. This leads to the notion of a connectivity space.

The connectivity spaces forma topological category Zus over Set, but Zus is not cartesian closed.

The second generalization is an abstract notion of the

"category of all connected objects" of a given category.

This leads to the notion of a "component category" which has been studied at first for the category topological or uniform spaces by Preuß [7] and Herrlich [4] and later for arbitrary topological categories by Strecker [10] and

Salicrup and Vazquez [91,0ur notion works for arbitrary category together with a set-valued functor. The generaliza- tion was motivated by Purnplün's unpublished idea of defining component categories by "quotients of the forgetful functor".

For an arbitrary functor, component categories need not have many interesting properties, but for mono-fibrations

the theory becomes quite smooth.

§1. connectivity Spaces

1.1 Definition: A connectivity space is an ordered pair X=(V(X) ,J(X)), where V(X) is a set and J,(X) is a collection of non-empty subsets of X with the following properties:

(i) ~x}E}(X) for all xEV(X).

(:ii.)

u('E,3(X) for all c:J(X) with f{~,n'f{~

in this case 3<x) is called a connectivity structure on V(X). A connectivity space X is called connected,iff V(X)f}(X) .If X and Y are connectivity spaces, a connectivity morphism is given by a map V(f) :V(X)

•

V(Y) such that the set-theoretical image V(f) [Z]

of any ZE)(X) under V(f) belongs to }(Y).

Zus denotes the category of Connectivity spaces and connectivity morphisms with the obvious composition (i.e. such that

V: Zus

•

Ens becomes a functor) ..

(Zus is an abbreviation for the German word "Zusarnmenhangsräume". We use this

abbreviation in order to prevent confusions with convex sets, convergence spaces,

contigual spaces etc.)

i c:2

Theorem:

(i)

V:Zus•

Set is

d

topological functorc

( *)

A

source (X,

(fi:X^•

Yi) iEI)

,J (

^{X) =; Z E ~( (V}

^(X))\·'.

^{\7)} l}

'lt,

i EI

in Zus is

initial

iff

V( f. \/ [

z l

E)(

y ) }

^C

l - p l

( iii) Zus is cartesian clos.ed "

Proof: ( i) Let X be a set, let (Y i) iEI: be

a.

f:amily of connectivity spaces, and let

(X, (f.

:X

^• V(Y.) ·er) be a source in Set. 'I'hen

a.

l l ls;.

V-initial lifti.ng (X, (

f.

~ X^• .Y.) . E.-I) can b:e g:i.ven by

1. .1. l

V(X) :=X,V(f.)

:::;:;.f.,

whereas 2(X) j_s. defined_ b_y (*).

l l f ·

'rhen the proof of ( i) is straightforward, and: ( iL) also follows immediately. In particular, ( i) implies, tha.t Zus is cornplete and that limi ts i.n Zus can be obtained_

as initial liftings of limits in Set. (see[6])

(ili)

Asswne that Zus is cartesian closed and consider the connectivity space X with

V ( X) ~

=

^{~ 0, 1 r 2 },}

} ( X) ~

= [{

0 } , { 1 } , { 2 }, { 0, '\ }, { 1 , 2 } , { 0, 1 , 2 } }.

'Then there is a connecti vi

ty

.strucb.lre. on the hom-set Zus (X,X), such that the evaluation rnap can be lifted to

a

Zus morphism (cf

[3]),

i.e.there is a connec- tivi ty space Y and a Zus-morphism e ~ XTrY• X 'IV'i th V (Y) =Zus (X, X)

V(e) (x,u)=V(t) (x) for all x€V(X) ,tC::V(Y), such that for any connectivi ty space R and any Zus-morphisrn f: x·nR^• X

there is a Zus-morhphism h:R^• Y with

W ~\!i.)) (GJl)) ::=1l7((1l!l)) ((1\ )) :=w((Wiw :Zi:)) -:;,=W«Vl)) «,O)) :=O.,.

V ( 1 il )) -:.==117 ((w)) ((~)) :=V/ ((W!)) ~:O)) :=V/ ((W!] (("11» ::=11 .,.

VJ-((lhl.o)l (( o)) =\i]

w

((®o (( 11

l)

=W/ ((®11

ll

((O)) -=Mt"

'71 ((®11 )) (( 11 )) =Wi

[1U1,..w]=W((lbJ

0)

Ho„

11} ]E~Y» _ -[w„W!]-=w((lbi.

11)) [[Ow111']EJ{(Y».

As ~t!!Jlww]n-[w„w]-,,

11.

11. ((:ii..:ii..)) :Fi.elds-[u •.

w.-WT1Ej00 „

^by

11 „ '72.. ~- (i..ii..)) W!e geit

~==[«<OJ„1U1»,,[11,,w»,,«~,,w»]E}<(x~w».

'R'lbiemi. Wie geit

me

col!llitrad:ii..c-tt.:ii..0J111.

i!Ow2}=W[e)) [~]E,((X].

IBlJf it:.ll:u.e dima.].:ii..-tty

tln.eoremru

for

itOJP<Ol.log:ii..cal

f1ilmc-ttors [ 16i]

descr:ii..pit.:ii..ol!ll of tlnose f:ii..na]. s-ttrucitUl[es. At f:ii..rst ~e

1.3. Definition:

A non-empty collection [ of non -empty sets is called chain-linked,iff for any S,S

¹

E0there are a natural number n and sets S=S

0

,s ₁ , .. ,Sn=S'E0 such that Si Si+ 1 +~ for all iE{o, .. ,n-1}.

1.4. Theorem:

(i) If Xis a connectivity space and ;fcJ<.X) is chain- 1 inked, then UbEJ( X) .

( ii) Let X be a set and let U. be a collection of non- empty subsets of X. Then there is a smallest

connectivity structure U-* on X with 'U.cU*. A subset Z<=X belongs to U* if4-z=1 (#Z denotes the cardi- nality of Z) or

u,n~(Z) is a chain-linked system with U(U,,~(Z))=Z.

(ili)

A sink (Y,(f.:X.

^•

Y) 'EI) in Zus is V-fi.nal, iff

l. l. i - -

J ( Y) is the smallest connectivity structure on V(Y) containing

U: = { V ( f . ) [ Z ]

¹

i EI , Z E ( X . ) }.

l. l.

(iv) A sink (Y, (f. :X,

•

Y) 'EI) in Zus is a coproduct

l. l. l.

sink,iff (V(Y), (V)fi)) iEI) is a coproduct sink in Set and

holds. All coproduct-injections in Zus are

V-initial.

(v):In Zus, quotients are heridi tary,

i.

e. if f:X

^•

Y is a V-final Zus-morphism, with V(f) surjective and if Bdl(Y) is a subset, then the Zus -morphism g:A

•

B is final where f:=V(f),

·_ - --1 -

V(B) :=B,

V(A)

:=A:=f (B)

J(A)

:=)CX)('i'2(Ä)

]CB) :=.5(Y)t1

(P(B),

V(g) (x) :=f(x) for all xEA.

Proof: (i): As uf:j:~, we choose a fixed aEu.t'.

Then there is an s

0 Er with aES

0 . If xEU;f is -arbi trary, there are sets s

1 , .. , s ED wi th xE s

n n

and s.ns.+

1 ~~ for Q<i<n-1. For any i with O<i<n

J. J. T - -

we get by induction U{S .jO<j<i}E1 (X), since for

J - - ;J

i<-n u{s.jo<j<i}E

¹

(X), s.

1Efc@-(X)

and

- J - -

,! i+

~+sinsi+ 1cu{SjjO~J~i}nSi+1 by 1.1 (ii) implies U{S. j0<j<i+1 }=u{S.

\O<j<i}nS

·+1C)(X).

J - - J - - i

.r

In particular, for Z:=u{s. \O<j<n}, we get

. J - -

a,xEZcuf.This proves u,f'=uf' for

1 . 1 ( ii) implies uU=u'l.L' E}(X) .

(ii) :By (i) we getu..*c7Jtfor any connectivity structure 7..J/"on X with

7.l,c1:,':

If ZEU, s

0 ,s

2 EU"@{Z) we get for s

1 :=z,s ₀ ,s ₁ ,s ₂ E , s ₀ ~s ₁ =s 0 t~

s 1 ns 2 =s

2 t~- ^{Thus ll~~{Z)} is chain-linked with

ZEU~~(Z)f~· Z=u({Z})cu{U~~(Z)}cuf(Z)=Z and thus

ZEU*. This proves lcU*, and it suffices to show

thät-U* is a connectivity structure on X itself.

By def ini tion, 'U * satisf ies 1 . 1 ( i) . Now let J'd(_* be a system with :f"=1=~

1

n.f9=~ e.g. aEn{'. Then

for every xEUJ""there is a ZEöwith * xEZ=t=U (1.in@(Z) )cu (Un<f(u ) . This proves

u ( U

⁰ (y.(. u ) ) =U

·-=~ and hence in particular U

^'l

1( ^uÖ) +~.

Obviously we have ~$:(=il

¹¹

~(U u), i.e. 1l n;;i(ut) is a non-empty collection of non-empty set. In order to complete the proof thatU.

¹¹

fi.(u.) is chain- linked, let S,S'EUn&(uf) be two sets. Then there are sEScU {"' and s' ES' cu{. Thus there are Z, Z' Er c ~(""

wi th sEZ, s' EZ' • Then ·lL n~(Z) , and 7J.

¹¹

f-( Z') are chain-linked. As ZE1c"'we have aEZ=U ('tL_n~(Z)), and

• thus there are s

1 , •. ,snE'U'"'@(Z)cU

⁰

~(ut"> with sEs 1 , aESn and sinsi+ 1 f~ for all i with 1<i<n.

In the same way there are rnElN,

sn+ 1 , .. ,Sn+mEU~~(Z')cUt1~(Uf)with aESn+ 1, s'ESn+m and sinsi+

1 +0 for all i with n+1:i<n+m.Defining

s ₀ :=S, sn+rn+ 1 :=S' we also get sfs 0 °s 1 t~,

aESn°Sn+ 1 f0 ,s'ESn+mnsn+rn+ 1 • Thus we have S.EU~~(uf) for all i with O<i<n+m+1 and S.'"'S.

1 ~0

1. .. 1. 1.-t- i

for all i with O~i~n+rn. This proves that 'U_n~(U6) is chain-linked, i.e.uJ'EU.*. This proves that

tl* also satisf ies 1. 1 ( ii) .

(ili)

follows imrnediately by comparing U* (as under (ii) and J{ Y) •

(iv) By a general theorern for topological functors,

(Y, (fi) iEI) is a coproduct sink in zus, iff i t is

V-final and (V(Y), (V(fi))iEI) is a coproduct sink in Set. So the "only if" part irnmediatey follows

from (

ili) ,

and for the "if

¹¹

-part i t suff ices to show that 7,l:= {V(f.) (Z] liEI, ZE2(X.)} is a connec-

.1. • ,/ .1.

tivity structure on V(Y) provided (V(Y), (V(f

1 ) )iEI) is a coproduct sink in Set. But 1.1 (i) irnmediately follows from V(Y)=U{V(f.) [V(X.) JliEI}. As

.1. .1.

V(f.) [V(X.) Jnv(f.) [V(X.)]=!2) for ifj, for any (ck

.1. .1. J J

with f+!l),n['f~ there must be a single iEI with

;fc:U{V(f.) [Z) jZE2(X.)} and then 1.1

(ii)

follows

.1. ,f .1.

easily. The injectivity of V(f.) together with

.1.

1. 2

(il.)

easily yields that the f. are V-initial.

].

(v) :Consider 'U:={V(f) [Z) j·zE}(X)} and

?.t:={V(h) [Z]

j

ZE3(A) }={V(f) [Z] j ZEJ(X)"'~(A) }=

={V(f) [Z) I ZEJ(X) ,V(f) [Z] c:fil:::U.n~(B). Now consider an arbi trary SE

}(B)

= }(Y)

n3?(B) •

Then by (

ii )

and

(ili)

we see that /tS=1 orur'"'~(Z) is a chain-linked with u('l..lfr18-(Z) )=Z. But Zc:B gives

'Ufi "9,( Z) =·u„ ^{1'-(B) n} @(

^z)

=Un t(

^{z) •}

By (

ü)

and (

ili)

this proves that g is final. The hypothesis that V(f) is surjective is even superfluous.

Finally, we are going to decompose a connectivity

space into connected ones.

1 .5 Theorem and Definition:

Let X be a connectivity space. Then the following assertiones hold;

(i) :The reiation ~ on V(X) is an equivalence relation, where x~yif:f there is a ZE}(X} with x,yEZ. The equivalence classes are called components of Z in the sequel. The set of all components of Z is always denoted by P(X), and ~(X) :V(X)

^•

P(X) always denotes the canonical projection.

( ii) : P (X)

c),(X) .

(ili)

:For kE (x~(K, J(X)

n

Al(K)) is a connected connectivity space ,- and ukis a Zus-rnorphism frorn this space to X, where G(uk} :K

•

V(X) is the inclusion map. More- over (X,(uk)kEP(X)) is a coproduct sink.

(iv) :If Y is another connectivity space and f:X

•

Y is a Zus-morphism, there is a unique rnap P(f) :P(X)

^•

P(Y) with P(f}

e(Z:(X)

)=(s(Y))

eV(f).

In this way,

P:Zus

^•

Set becornes a functor and Z::V

^•

P becornes a natural transformation. In the sequel P i s always considered as a functor in this way.

(v) :If rn:XTIX

•

X is a Zus-rnorphism such that V(X) is a group under the operation xy:=V(rn) (x,y), then there is anormal subgroup NcV(X} with

3,(X)={ZE~(X)\{~} \3xEV(X),zcxN}.

(vi) :Aproduct of arbitrarily rnany connected connectivity spaces is connected.

Proof: (i}: We only have to prove the transitivity.

If x~y, y~z, there are Z,Z'EJ(x) with x,yEZ, y,zEZ', thus yEZ~Z'f~ and hence x,zEZ0Z'Q(X), therefore x~z.

(ü)

:Then we have K=U[for {:={ZE'J,(X)iaEZ}. Since {a}EJf~ and aEnJ'"i~, 1.1 (ii) yields K=UfEJ(X)

(.ili)

:By 1.2(i), (K,~(X)n](K)) is a connectivity space

such that uk is a V-initial Zus-morphism. By (ü), K is connected, and by 1 .4(iv) ,(X, (uk)kEP(X)) is a coproduct sink.

(iv) :As s(X) is surjectice, the uniqueness of G(f} is trivial. In order to prove the existence, we just have to show that s(X) (x)=s(X) (y) implies

s (Y) e(V(f)) (x) = s(Y)

⁰

(V(f)) (y) for all x,yEV(X).

But for all such x,y, there is a Ze3(X) with x,yEZ, and then G(f) (x) ,G(f) (y)EG(f) [Z]EJ(Y) proves the desired equation. If g:Y

^•

W is another Zus-morphism (for some W), we get

P (gf)

^{6 (}

s(X)) = r:(W)

e

(V(gf)) = s(W)

~

(V(g))

e(V(f))

=

=P(g)o(s(Y))·(V(f))=P(g)o{P(f));jl,(X), and as c;;(X) is surjective, this implies P(gf)=P(g) · (P(f)).

Obviously P preserves identical morphisms.

(v) :Let N be the component of X that contains the unit element e. Then (ii) gives NEJ(X), and our hypothesis

yields xN:={xy\yEN}E)'X)

for all

xEV(X).

Now let x,yEN be arbitrary. Then we get exENEj(X),

x,xyExNe:

2

^(x)

and hence

s(X) (e)=C:(X)

(x)=~(X)

(xy), i.e. xyEN.

-1 -1 -1

For any xEN we have e=x x and thus x , eE:x N ,hence

-1 . -1

z;(X)(x )=c,;(X)(e), i.e. x EN. This proves that N is,a subgroup of V(X). For any xEV(X) we have

-1 -1 1 ^-1

e=x xEx NxE_,.-(X) and hence x Nxc::N. Thus

N

is a normal subgroup. Now let zE3(x) be arbitrary. Then there is an xEZ, andwe get e=x-

¹

xEx-

¹

zEJ.(X) and thus x

-1

Zc::N, i.e. Zc:xN. On the other hand, assume xEV(X), ZcxN, Zf0. W.l.o.g. we can assume xEZ, because there is an x'EZcxN=x'N. Now we consider M:={ (zy ,y-

¹)

!zEZ,yEN}cV(X)xV(X)=V(XnX). Then the image of M under the first projection from

V(X) xV(X) to V(X) is NEJ(X). For the image

L={zyjzEZ,yEN} under the second projection we get xNcLc{ZYI ZE N,yEN}={xwy!w,yEN}cxi.~, hence L=xNE (X).

By 1.2(li) this.implies M~J(XnX), and now our hypo- thesis yields Z={zyy-

¹

)zEZ,yEN}=V(rn) [M]~j(X), q.e.d.

(vi) :Let (Xi)iEI be a family of connected connectivity spaces. For every iEI we have V(Xi)Ej(Xi) ,~(?-(Xi) and thus V(X,)f~- The image of V( n X.) under the

1

iEI

¹

i-th projection then is V(X.)EJ(X.), and now 1.2(ii)

1 .

.J'

1

gives V( TI X.Je}( IT X,) ,q.e.d.

iEI

¹

iEI

¹

~2.Component Categories

In this chapter we give a generalization of the

usual component categories. It turns out that a

~uite abstract setting is convenient for the de- finition.

2.1 Definition: In the sequel

~

always denotes a category, G:~

^•

Ens denotes a functor, and x

0 c~ is the full sub- category of all X with G(X)f~- Fora full subcate- gory AcX and an XEOb(_X), the connectivity space

- -o

TA(X) is defined by V(T~(X)) :=G(X); and i(T~(X)) is the smallest connectivity sructure on G(X) containiD:g <P(~,X) :={ irnG(h) IAEOb(~) ,hE~(~

1

X) }.

imG(h) :=G(h) [G(A)] denotes the set-theoretical image).

A G-quotent is a pointwise surjective natural trans- formation a:G~ (i.e. a(X) is surjective for all XEOl:(~)) where Q:~

^•

Set is an arbitrary functor. If a,1 are G-quotients, then 0~1 means that there is a natural transformation s

^wi

^th s ·

^a=1

(pointwise

product). Fora G-quotient

0

:G~, an XEOb(~) is called 0-connected, iff#Q(X)=1. Conn (a)c~ denotes the full subcategory of all 0-connected objects.

2.2 Theorem: Let

~-0 be a

full subcategory. Then the following

assertions

hold:

(i) :TA:~

^•

ZUS is a functor with V•TA=G

(ii)

:For every AE Ob(~), TA (A) is connected.

(ili) :Let F:X

•

Zus be a functor such that F(A) is

connected for every AEOb(~). Let o:G

^•

VoF be a G-quotient. Then there is a unique natural trans- formation r:TA

^•

F with Vey=o.

Proof: (i) is straightforward, and (ü) follows irnrnediately from the definitions.

(ili):

As V is faithful, Y(X) is uniquely determined for every XEOb(~), and i t suffices to show that

Y(X) :TA(X)

^•

F(X) with V(y(X)) :=o(X) is a Zus-morphism.

In order to do this, i t suffices to show that the initial connectivity structure

{Zc:G(X) io(X) [ZJE}(F(X))} contains the smallest connectivity structure JoT~(X) containing ~(~,X).

We only have to prove o(X) [im G(h) ]E F(X) for all

AEOb(~) ,hE~(A,X).

But o(X) [im G(h) ]=im(o(X)

)o

(G(h)))

=im ( (V&E' (h) )

o ( o

(A))) = (VcF (h) ) [ imo (A) ] =(VoF (h) ) [V c,F (A)]

The last equality is valid, because o is a

G-quotient and thus o(A) is surjective. As F(A) is connected by hypothesis, we have VaF(A)EJ°F(A) thus o(X) [imG(h) ]=VcF(h) (V::F(A) JE}F(X)

and

2.2 enables us to establish a Galois adjunction between full subcategories of

~

and G-quotients.

2.3 Proposition: Let

~c:~~

be full subcategories and let

0,1

be G-quotients with

0<1.

Then the following assertions hold

(i):~c: Conn(o) is equivalent toC:~'.E_A~o.

( ii):

~cConri ( s e,T A) .

(ili):

s~TConn(cr)::

⁰

(iv): i';!>TA:: z:~TB

(v) :Conn(cr)cConn(~).

(vi) :Conn(s~Tc - - onn (cr))=Conn(cr) - - ( v

i i ) :

T Conn s6TA ( ) =TA .

Proof: ( i) "~" Assume ~cConn

(a) , -

cr:

G• Q.

Then we consider the functor F:~

^•

zus with VoF:=Q and

jeF(X)

:={{q}jqEQ(X) }for all XEOb(~) ,(i.e.F assigns to X the "discrete connectivity space

Q

(X)") . For every AEOb (~) , we have #

Q

(A) =

1 ,

and henc e F (A) is connected. Now 2.2.

(ili)

gives that there is a unique natural transformation y:TA

^•

F with Vey=cr.

Then we get the following commutat.ive diagram of natural transformations

G=Vc1TA

cr

=Vr,y

POF

By the definition of F, it follows immmediately that

r;eF

is a natural isomorphism. Thus we have

a= ( r;~F) -1 •

(P,:y)"'(r;"'TA), in particular r;eTA,:cr.

11 <:= ¹¹

Assume

r; eT

A_::0, A EOb(~) . Then there is a natural transformation s:P·TA

^•

Q with s· (l';dl'A)=0.

Let A be an arbitrary ~-object. Then 0(A) and thus s(A) :P~TA

^•

Q(A) is a surjection. But TA(A) is

connected by 2. 2. ( li), and hence we get 4/ P

t>T

A (A) = ¹

and this implies Q(A)=1, i.e.AEOb(Conn(0)).

(li): Apply (i) in the special case 0=r;eTA.

(ili):

Apply (i) for A:=Conn(0)

( i v) By ( li) we get ?:_c~cConn (

r; oTA) ,

and now ( i) ( for B instead of A and seTA instead of 0) yields the assertion.

(v) :By

(ili)

we get r;~TC ( ) <0<-r, and

(i)

(for Conn (0)

onn a - - - -

instead of A and

T

instead of 0) leads to the conclusion (vi):":::i" follows from (ii)(for~:=Conn(0)), and

(ili)

together wi th (vi) ( for sc TC ( ) instead of a and 0 onn a

instead of -r) proves "c".

(vü) :Let XEOb(Zus) be arbitrary. Then we have to prove .}(TConn(i';()TA) (X) )=],(T~ (X))~ The inclusion

¹¹::::i"

follows immediately from ( li).

For the other inclusion i t suffices to show

<ll(Conn(i'.;eT~) ,X)cj(T~(X)) for all XEOb(~) (by 1.4(li).

But for every ZE9(Conn(l';eTA) ,X) there are

an AEOb(Conn(r;oTA) and an ~-morphism h:A

•

X with Z= im(G(h)). Then TA(A)is connected,

G (A) = (VoT

~ (A)}'f~(T ~

(A)

~

, and as T.~ (h)

i.

e.

is a Zus- morphism Z=imG(h) =im(VoT~ (h) )Ej(T~ (X)).

Now we want to show that our generalized component

categories cannot be expected to have "nice"

properties without any hypothe 9 is about G.

2.4 Proposition:

(i) :Every component categories is closed under the formation of retracts.

(ü) :Let X be small and let M be a set with M_2. If Gis the canonical functor with G(X) :=U{~(Y,X) ,YEOb(~_)}x

M

the every full subcategory, which is closed under the formation of retracts, is a component category.

Proof:

(i) :Let~ be a component category and consider AEOb(~), BEOb(~) e:A

^•

B, s:B

^•

A with es=1B. Then we get

(PeTA (e)) (PeTA (s))

=id POT (B}. Thus PQTA(e) is

A -

surjective, and then PeTA(A)=1 ·implies PeTA(B)=1.

(ii):Let

~c~

be full and closed under the formation of retracts, and assume XEOb(Conn(~!TA)). Then we have JFG(X):,#c{1x} )= M:_2, and thus <i?(~,X) is chain-

linked with U<P(~,X)=G(X). For an mEM this implies (1 ,m)EG(X)=U<P (A,X). Thus there are AEOb(~), h:~

^•

X

X -

m'EM, YEOb(A) ,t:Y

^•

A) (1 ,m)=G(h) (t,m')=(hm,m'),

- X

thus Y=X and ht=1 . As Ais closed under the

X

formation of retracts, this implies XEOb(~).

Now we want to look at component categories in the

case that the functor G has nice properties.

2.~ Theorem:Let G be represented by a terminal object. Then every component category

~

is closed under the for- mation of finite products,

Proof: Let EEOb(~) be term.i:nal,w.1.o.g. we can assurne G=~(E,-). Let{:: be a component category_

Obviously

we have,4-q.(E)=fr.(~(E,:E):)=-#{E}:=l, and thus

E:Eüb (A) , i" e. !!_

contains

a l l

empty products. Now

n

let

B=

;1

A.

n>O be a finite product (with i=O

^l

projections

p. :B• A.) and

assume- A.EOb(A) for all

l. l'. - l

i.<n.

Let e.

:A. ^• E

be the untque mo-rph1sm -( for i:::_n)

l l

As G

is representable,

^G

preserves.

f

.ini te

^products,

and thus we have G(B)+~- Now let u,vEG(B)=~(~,B) be arbitrary. Then we have to show (c;~TJ (B)

(ul=

( c;e::TA) (B) (v) , In order to do this, we cunsider

thA unique morphisms

t.

:A.

•

B

l J_

p.ue.

^{' f} '<'

I l~ J l .

l

^{J l}

p t.=

_'-^/ 1

if

j=i.

j ^l _\ ^{A. f}

l ^{pjve 1}

1.

^{, lf}

^{j> i .}

for all j~n. Moreover, we consider the unique maps

p,u,

if

^{. <.}

1 J L

1

^J

p.w = -<

'J ^{l \ l} _l

!

/"-- ^{p.v ,}

if j:::L

J

In particular

' we have

w _o

=v and w

n+·i ' '

=

^{11 Fc,.·L·}

^all

{ p.ue.p.u=p.u=p.w.+ 1 if j<i,

J l l J J l

p.t.p.u = 1A p.u=p.u=p.w.+ 1 if j=i,

J l l . l J J l

J_

p.ve.p.u=p.v=p.w.+

1 if j>i,

J l l J J l

thus w. +

1 =t. p . u=G ( t. ) ( p . u) E im ( G ( t . ) ) . In the sarne

J_ J_ J_ J_ J_ l

way we get w.=t.p.vE irn(G(t.)).

l l l l

As A.EOb(A) and t. :A,

^•

B, i t follows that

J_ - J_ J_

irn(G (ti) )E <ll(~,B) and thus ( s~TA) (B) (wi) =

= ( sCT A) (B) (wn+ 1) . By induction we get ( s()T A) (B) (w

0 ) = ( s~T A) ( B) (wn+

1 ) . Since w

0 =v and wn+ 1 =u, we are done. Cornponent categories behave very well in the case that Gis a rnono-fibration, i.e. for every XEOb(~) and Mc:G(X) there are A

^•

Ob(~), rn:A

^•

X such that rn is G-initial, G(rn) is injective, and irn(G(rn) )=M. If G(rn) is injective, then rn is G-initial if for ever:y BEOb(~), h:B

•

X with

irn(G(h) )cM there is an e:B

^•

A with rnl=h.

At first we need the following

2.6 Lernrna:Let rn:A

^•

X be a G-initial rnorphisrn such that G(m) is injective. L e t ~ ~ be a full subcategory. Then each of the following statements implies the next one (and thus all they are equivalent, if Gis a cornponent category):

(i) :AEOb(~).

(ii): irn(G(rn) )E cj)(~,X).

(ili):

irn(G(rn)) E)(T~(X).

(iv) :AEOb(Conn(soTA)).

Proof: (i)

^•

(ii)

^•

(ili) is trivial. (iii)

^•

(iv): By the

initiality we get J"':=<ll(~,X)l'~(im(G(rn))=

={im(G(h)! BEOb (~) ,h:B

^•

X, (G (h)) Cim (G (m))

}=

={ im (G (ml)) BE Ob

(a) ,

1 :B

^•

N =

={G(m) [im (G(h))!BEOb(~} ,l:B

^•

A) }=

={G(m) [K]

1

KEcp(~,A)}.

By 1. 4 (ü) we know that #im(G (m)) =1 or f ^{is chain-}

linked withu f=im(G(m)). As G(m) is injective, this implies that G(A)=1, or ~(~,A) is chain-linked with ~(~,A)=G(A). By 1.4(ü) this means

V•TA(A)=G(A)E (TA(A)), i.e. TA(A) is connected, and this implies (i).

Now we can give acharactarization of component categories in the case that Gis a mono fibration.

2. 7 Theorem: (cf.

[8])

Let G

be

a rrono-fibration and let/4

be

the class of all G-ini tial morphisms wi th G (m) inj ecti ve. Let

~

c ~ be a full replete suvcategory, which contains all objects A with G{A)=1 Then the following

statements are equivalent.

(i) :~ is a component category.

{ü) :~ is )l-multicoreflective, i.e. for every XEOb(~) there are a class I (whisch is even a set) and a sink (X, (m.

:A.• X)

·Er) of )(-morphisms m. such that

l. l. l. ].

the following properties are satisfied:

(1)

A.EOb(A) for all iEI.

].

-

(2) If BEOb(~), h:B

^•

X, then there is a unique pair (i,l) with iEI, l:B

^•

A. such that h=m.l.

]. ].

(ili.)

:A satisfies the following properties:

(3)Let XEOb(~) and -let (u.:B.^• X) "EJ be a non-void sink of J J J

/1{:-morphismswith B.EOb(A) for all j . Assume

J -

n{im(G(u.))ljEJ}f~ and U{im(G(u.))ijEJ}=G(X). Then

J J

XEOb(~).

(4)If XEOb(A) ,e:X• Y and G(e) 1.s surject.ive, then YEOb(A).

Proof: (i)• (ii) :Define I:=PoTA (X), and for iEI let m.:A.• X be in such that im (G(m.))=(i:;;~T (X))-¹[{i}J.

l. l. l. a

Then (1) is satisfied by 2.6. (iii)• (iv). Now let BEOb(~) ,h:B• X be arbitrary. Then we have

~fG(B)Ej(TA(B), thus ~fim(h)EJ(TA(X)) and hence im(h)c(i:;;8TA~X))-I [{i}]=im(G(mi))-for a unique iEI.

By m.E)'(_ there is a unique

l:B

^• A. with h=m.l. Since

_l. l. l.

G(B)f~, the uniqueness of i is obvious and thus (2) is satisfied.

(ii) • (ij i) : (3) Consider aEn{im(G(u .) ) } jEJ ,and let t :C• X be in

..J{

J

with im(G(t))={a}. Then we have G(C)=I, hence CEOb(~).

Thus there is a unique pair (i,s) with iEI, s:C• A.

0 . 1.0

t=m. s. For every jEJ there is are. iEI, v.:B.• A.

1.0 J J l.

with u.=m.v .. As u.EJ(, there is also an r:C^• B. with

J ^l. J J J

t=r.u .• But this implies t=u.r.=m.v.r. and thus i=iO'

J J J J l . J J

v.r.=s. and hence im(G(u:))=im(G(m. v.))cim(G(m. )) .

J J .J

"'o

J 1.0

This proves G(X)=-U{im(G(u.))ljEJ}cim(G(m. ))=G(X).

J 1.0

Thus we haveimG(m. )=G(X), i . e . m. is G-initial and

10 l.Q

G(mi) is a bijection. But then m

0 has tobe an 1.s0-

0

morphism, (4) :By (ii)

and this implies XEOb(~).

there are iEI, l:Y• A. with e=m.l and hence

l. l.

G(X)=im(G(e))cim(G(m.)). As otove, i t follows that

l.