Simplicial Presheaves by J.F. Jardine

8/10/2019 Simplicial Presheaves by J.F. Jardine

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journa l of Pu re and Ap pl ied Algebra 47 (1987) 35-87

N0rth-Holland

35

S I M P L I C I A L P R E S H E A V E S

J . F . J A R D I N E *

Ma thema t ics Depa rtment , Univers ity of Western Ontario, Lon don , O ntario N6A 5B7, Canada

Com mun i ca ted by E .M. Fr i ed lande r

Received 5 October 1985

Revised January 1986

i s t r o d u c t i o n

T he c en t r a l o r g a n i z a t io n a l t h e o r e m o f s i m p li c ia l h o m o t o p y t h e o r y a s s e r ts t h a t t h e

category S o f s im p l i c i a l se ts ha s a c losed mo de l s t r uc tu re . Th i s m ean s th a t S com es

e qu ip pe d w i t h t h r e e c l a ss e s o f m o r p h i s m s , n a m e l y c o f i b r a t i o n s ( i n c l u s i o n s ) , f i b r a -

tio ns ( K a n f i b r a t i o n s ) a n d w e a k e q u i v a le n c e s ( m a p s w h i c h i n d u c e l ~ o m o t o p y

e qu iv ale nc es o f r e a l i z a ti o n s ) , w h i c h t o g e t h e r s a t i s f y Q u i l l e n ' s c l o s e d m o d e l a x i o m s

CM1 to C M S. Th i s the o re m i s we l l kno wn and wid e ly used ( see [23 ] an d [3 ]) .

O ne c o u l d r e a s o n a b l y a s k f o r s u c h a t h e o r e m f o r t h e c a t e g o r y S S h v ( C ) o f s i m p l i-

d al s h e av e s o n a G r o t h e n d i e c k s i t e C , b a s e d o n t h e i n t u i t i o n t h a t a n y t h e o r e m w h i c h

is t ru e f o r s e t s s h o u l d b e t r u e f o r t o p o i . I m m e d i a t e l y , h o w e v e r , a d i l e m m a p r e s e n t s

its elf. O n t h e o n e h a n d , c o h o m o l o g i c a l c o n s i d e r a t i o n s , l ik e th e V e r d i e r h y p e r -

c ov erin g t h e o r e m , s u g g e s t a l o c a l t h e o r y o f K a n f i b r a t i o n s . F o r e x a m p l e , i f o p l r i s

the s it e o f o p e n s u b s e t s o f a t o p o l o g i c a l s p a c e T , t h e n a m a p f : X - , Y o f s i m p l i c i a l

sh eav es s h o u l d b e a l o c a l f i b r a t i o n i f a n d o n l y i f e a c h m a p o f s t a l k s f x :

Xx--" Yx

is

a K an f i b r a t i o n i n t h e u s u a l s en s e. O n t h e o t h e r h a n d , m o n o m o r p h i s m s s u r e l y

should be co f ib r a t io ns , g iv ing a g lob a l t heo r y .

T he t w o a p p r o a c h e s d o , i n f a c t , y i e ld a x i o m a t i c h o m o t o p y t h e o r i e s f o r a l l

c at egorie s o f s im p l i c i a l sheaves . T he loca l t he o r y fo r the ca t ego ry o f s imp l i c i a l

sh eav es o n a t o p o l o g i c a l s p a c e w a s c o n s t r u c t e d b y B r o w n [ 4] ; t h e c o r r e s p o n d i n g

g lo bal t h e o r y w a s d e v e l o p e d s l i g h t l y l a t e r b y B r o w n a n d G e r s t e n [ 5] . T h e l o c a l

th eo ry f o r a r b i t r a r y G r o t h e n d i e c k t o p o i a p p e a r s i n [ 1 7] . T h e g l o b a l t h e o r y i n t h e

genera l se t t i ng i s a r e su l t o f Joy a l [18 ]. Th e g loba l t he o r y o f co f ib r a t ion s i s pa r t o f

a c lo se d m o d e l s t r u c t u r e o n S S h v ( C ) . T h e t w o t h e o r i e s a r e

distinct,

s ince i t i s no t

true t h a t e v e r y lo c a l f i b r a t i o n i s a g l o b a l f i b r a t i o n . T h e E i l e n b e r g - M a c L a n e o b j e c t s

/((F , n ) c e r t a in ly fa i l t o be g lo ba l ly f ib ra n t i n gen e ra l , e ssen t i a l ly s ince she a f co -

homology is non- t r iv ia l .

A p o i n t t h a t a l l a u t h o r s ( i n c lu d i n g m y s e l f ) s e e m e d t o m i s s u p t o n o w is t h a t , i n

* S u pp o rt e d b y N S E R C .


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36 J.F. Jardine

t h e p r o o f s o f t h e r e s u l ts a b o v e , i t is n o t s o m u c h t h e a m b i e n t t o p o s t h a t i s c re atin g

t h e h o m o t o p y t h e o r y a s i t is t h e t o p o l o g y o f t h e u n d e r l y i n g s i te . T h e s e p r o o f s may

b e g e n e r a l i z e d t o p r o d u c e l o c a l a n d g l o b a l h o m o t o p y t h e o r i e s f o r s i m p l i c i a l pre.

s h e a v e s o n a G r o t h e n d i e c k s i te w h i c h d e p e n d o n n o t h i n g b u t t h e a x i o m s f o r t h e site.

Th i s pape r p re sen t s the se re su l t s . The loca l t heo ry i s g iven in the f i r s t sec t ion ; the

m a i n r e s u l t t h e r e i s T h e o r e m 1 . 1 3 . T h e g l o b a l t h e o r y a p p e a r s i n t h e s e c o n d s ec tio n,

c u l m i n a t i n g i n t h e p r o o f o f T h e o r e m 2 . 3 . T h e c o r r e s p o n d i n g r e s u l t s f o r s i m p l i c i a l

s h e a v e s a p p e a r a s c o r o l la r i e s o f t h e s e t h e o r e m s . T h e f i r s t tw o s e c t io n s m a y seena

l e n g t h y , b u t t h e i d e a w a s t h a t t h e s e r e s u lt s s h o u l d b e p r e s e n t e d i n a 'u s e r- fr ie n d ly ,

f a s h i o n .

I n r e t r o s p e c t , i t h a s b e e n i n t u i t i v e l y o b v i o u s f o r s o m e t i m e t h a t t h e r e s h o u l d be

s o m e s o r t o f h o m o t o p y t h e o r y f o r s i m p l i c i a l p r e s h e a v e s , s u c h t h a t t h e c a n o n i c a l

map f rom a s imp l i c i a i p re shea f to i t s a ssoc ia t ed s imp l i c i a l shea f i s a weak equ iva -

l ence . Ev idence fo r th i s appea rs in [17 ] , i n f ac t , and i t i s a ch ieved by bo th theo r i e s .

I t f o l l o w s , i n p a r t i c u l a r , t h a t t h e a s s o c i a t e d h o m o t o p y c a t e g o r i e s a r e e q u i v a l e n t .

T h u s , t h e l o c a l a n d g l o b a l t h e o r i e s , w h i l e p r o v e a b l y d i s t i n c t , d e s c r i b e t h e s a m e

t h i n g , r a t h e r l i k e th e w a y t h a t h y p e r c o v e r s a n d i n j e c t iv e r e s o l u t i o n s d e s c r i b e s he af

c o h o m o l o g y .

O n e m a y r e c al l t h a t a p r o p e r d e s c r ip t i o n o f s h e a f c o h o m o l o g y r e q u ir e s b o th o f

t h e s e p o i n t s o f v i e w . T h e s a m e i s t r u e o f t h e h o m o t o p y t h e o r y o f s im p l i c ia l p re -

sheaves o r sheav es , a s is ev ide nced in the th i rd sec t ion o f th i s pap e r . Th e ba s i c idea

the re i s t o app ly the re su l t s o f the f i r s t pa r t o f t he pape r to ge t a new desc r ip t ion

o f & a l e K - t h e o r y a n d t h e c o m p a r i s o n m a p o f t h e L i c h t e n b a u m - Q u i l l e n c on je ctu re.

I n p a r t i c u l a r , t h e r e i s a n i s o m o r p h i s m

K ~ t _ l (S ; Z / l ) -~ [ . , l - 2 i K / l l ] s , i>_0 ,

f o r d e c e n t s c h e m e s S a n d p r i m e s I n o t d i v id i n g th e r e s i d u e c h a r a c t e r i s t ic s o f S . The

s q u a r e b r a c k e t s d e n o t e m o r p h i s m s i n th e h o m o t o p y c a t e g o r y a s s o c ia t ed to the

c a t e g o r y o f s i m p l i c i a l p r e s h e a v e s o n t h e & a l e s i t e d t l s for S , and i s the te rmina l

o b j e c t o n S P r e ( d t l s ) .

K / I 1

i s n o t a t i o n f o r o n e o f t h e m o d 1 K - t h e o r y p r e s h ea v e s on

d t l s . T h i s r e s u l t i s T h e o r e m 3 . 9 o f t h i s p a p e r ; i ts p r o o f u s e s t h e m a i n r e s u l t o f [281.

W e t h e r e f o r e o b t a i n y e t a n o t h e r d e s c r i p t i o n o f & a l e K - t h e o r y , a t l ea s t i n degrees

a b o v e - 2 . T h e p o i n t i s t h a t , w i t h t h e r e s u lt s o f t h e f i r s t t w o s e c t io n s i n h a n d , 6 tale

K - t h e o r y m a y b e r e g a r d e d a s a g e n e r a l i ze d c o h o m o l o g y t h e o r y f o r s i m p l ic i al p re-

s h e a v e s o n 6 t l s . I n s o f a r a s t h e m o d l K - t h e o r y p r e s h e a f i s d e f i n e d o n a n y o f the

sch em e- the o re t i c s i te s wh ich a re av a i l ab le fo r S , w e a re en t i t l ed , v i a th is de sc rip tion ,

t o o b j e c t s l i k e ' f l a t ' K - t h e o r y o f ' Z a r i s k i ' K - t h e o r y . T h e s e i n v a r i a n t s m a y a ll be

r e l a t e d t o t o p o s - t h e o r e t i c m e t h o d s . Z a r i s k i K - t h e o r y i s t h e o b j e c t o f s t u d y o f [ 5 ] .

A n o t h e r c o r o l l a r y o f T h e o r e m 3 . 9 is t h a t t h e c o m p a r i s o n m a p r e l a ti n g m o d l

K - t h e o r y a n d & a l e K - t h e o r y n o w h a s a v e r y s i m p l e d e s c r i p t i o n . I n p a r t ic u l a r , the

L i c h t e n b a u m - Q u i l l e n c o n j e c t u r e r e d u c e s t o a ' f l a b b i n e s s ' a s s e r t i o n f o r t h e s i m p l i -

c i a l p r e s h e a f K / I ~. T h e r e a s o n f o r t h e c o n c e n t r a t i o n o n s i m p l i c i ai p re s h e av e s w ill


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Simplicial presheaves 37

b ec om e o b v i o u s t o t h e r e a d e r a t t h i s p o i n t . T h e c o n j e c t u r e i t s e l f a s s e rt s t h a t t h e

h o m o to P y g r o u p s o f t h e s i m p l i c i al s e t o f g l o b a l s e c ti o n s o f K / I 1 a r e is o m o r p h i c t o

the groups [*, ~ 2 i K / l l ] , a t l e a s t i n h i g h d e g r e e s . I t i s t h e r e f o r e i m p o r t a n t t o u s e a

h om O to py t h e o r y w h i c h s e es s i m p l i c i a l p r e s b e a v e s r a t h e r t h a n t h e i r a s s o c i a t e d s i m -

p lic ia l sheaves ; t he can on ica l m ap f ro m a s imp l i c ia l p r e s he a f to i t s a ssoc ia t ed s im p l i -

d a l s h e a f m i g h t n o t i n d u c e a w e a k e q u i v a l e n c e i n g l o b a l s e c t i o n s .

T h e t h i r d s e c t i o n a l s o c o n t a i n s t e c h n i c a l r e su l ts w h i c h l e a d t o t h e G o d e m e n t a n d

B r o w n -G e r st en d e s c e n t s p e c tr a l se q u e n ce s f o r m o r p h i s m s i n t h e h o m o t o p y c a t e g o r y

on S P r e ( 6 t ] s ) . B o t h a r e e s s e n t i a l l y s t r a i g h t f o r w a r d a p p l i c a t i o n s o f w e l l - k n o w n

to w er o f f i b r a t i o n s t e c h n i q u e s , m o d u l o t h e t e c h n ic a l p r o b l e m t h a t t h e i n v e r s e li m i t

f u n c t o r o n s u c h t o w e r s m i g h t n o t p r e s e r v e w e a k e q u i v a l e n c e s i n t h e s i m p l i c i a l

p re sh ea f s e t t in g . I n b o t h c a s es , o n e g e t s a r o u n d t h i s p r o b l e m b y a s s u m i n g a g l o b a l

b ou nd o n 6 t a l e c o h o m o l o g i c a l d i m e n s i o n . T h is a s s u m p t i o n i s b ec o m i n g s t a n d a r d

practice [28].

T h e p a p e r c l o s e s w i t h a r e s u l t t h a t a s s e r t s t h a t g e n e r a l i z e d c o h o m o l o g y g r o u p s o f

a s im p lic ia l s c h e m e X o v e r a b a s e s c h e m e S m a y b e c o m p u t e d e i t h e r in t h e h o m o t o p y

c ate go ry f o r t h e b i g 6 t a le s i te o n S , o r i n t h e h o m o t o p y c a t e g o r y a s s o c i a t e d t o t h e

~tale s i te wh ich i s f ib re d ov e r X. Th i s r e s u l t gene ra li ze s the c oho m olog ica l r e s u l t o f

[17] w h i c h l e d , i n p a r t , t o a s t r e a m l i n e d p r o o f o f S u s l i n ' s t h e o r e m o n t h e K - t h e o r y

of a l g e b r ai c a ll y c l o s e d f i e ld s [ 2 6 ,2 7 ] . I t a l s o i m p l ie s t h a t T h o m a s o n ' s t o p o l o g i c a l

~ l ui va r ia n t K - t h e o r y [ 3 0 ] m a y b e i n t e r p r e t e d a s g e n e r a l i z e d s i m p l i c i al p r e s h e a f

cohomology o f a su i t ab le ba lanced p roduc t .

1 . L o c a l t h e o r y

T h r o u g h o u t t h i s p a p e r , C w i l l b e a f i x e d s m a l l G r o t h e n d i e c k s i t e . S P r e ( C ) i s t h e

c ate go ry o f s i m p l i c i a l p r e s h e a v e s o n C ; i ts o b j e c ts a r e t h e c o n t r a v a r i a n t f u n c t o r s

fro m C t o t h e c a t e g o r y S o f s i m p l i c ia l se t s, a n d i t s m o r p h i s m s a r e n a t u r a l t r a n s f o r -

ma tions . Reca l l t ha t t h e top o lo gy o n C i s spec if i ed by fa m i l i e s J ( U ) o f s u b f u n c t o r s

RC C ( - , U ) o f r e p r e s e n t a b l e f u n c t o r s , o n e f o r e a c h o b j e c t U o f C , a n d t h a t a n e l e -

ment R o f J ( U ) i s c a l l e d a c o v e r i n g s i e v e . E a c h s u c h R m a y b e i d e n t i f i e d w i t h a

s ub ca te go ry o f t h e c o m m a c a t e g o r y C ~ U , a n d s o e a c h s i m p l i c i a l p r e s h e a f X r e s t r ic t s

to a func to r o n each cove r in g s i eve . I de f ine

X ( U ) R = l i m X ( V )

4 - - -

~: V-~ U~R

and ca l l th is th e s im pl ic ia l se t o f R - c o m p a t i b l e fami l i e s in X ( U ) . T h e r e i s a c a n o n i c a l

~ap

rR: X ( U ) -~ X ( U ) R

fore a ch U a n d R . A n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r X t o b e a s im p l i c i a l s h e a f

t h a t t h e m a p r R i s a n i s o m o r p h i s m f o r e a c h U i n C a n d e a c h c o v e r i n g s i e v e

R C C ( - , U ) . S S h v ( C ) i s th e f u l l s u b c a t e g o r y o f S P r e ( C ) w h o s e o b j e c ts a r e t h e s ir e -


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38 J.F. Jardine

p l i c ia l sh e a v e s. R e c a l l t h a t th e i n c l u s i o n j : S S h v ( C ) C S P r e ( C ) h a s a l e f t a djo in t

X ~ L2 X, c a ll e d t h e a s s o c ia t e d s h e a f f u n c t o r , w h e r e t h e f u n c t o r L : S P r e ( C ) .

S P r e ( C ) i s d e f i n e d b y

L ( X ) ( U ) = l i m X(U)R.

R C C( - , U) cove ri ng

T h e c o l i m i t d e f i n i n g

L ( X ) ( U )

i s f i lt e r e d , a n d s o L , a n d h e n c e

L 2 ,

preserves f ini te

l i m i ts ( w h ic h a r e f o r m e d p o i n t w i s e ). T h e c o n v e n t i o n i s t o w r i t e ) (=L2 (X ) . There

i s a c a n o n i c a l m a p

fix : X ~ L X ,

a n d L X i s a s ep a r a t ed p r e she a f i n t he s ense t ha t r /L X i s a po i n t wi se m o n i c .

A m a p p : X - , Y o f s i m p l ic i a l p r e s h e a v e s i s s a id t o b e a

localfibration

if for each

c o m m u t a t iv e d i a g r a m o f s im p l ic ia l s et m a p s

f q

A n

, x ( u )

p(U)

, Y ( U )

t he r e i s a cove r i ng s i eve R C C ( - , U) such t h a t f o r e ach (o : V - , U i n R t he r e i s a com-

m u t a t i v e d i a g r a m

ot ~*

Ank ' X (U ) ' X (V )

~ I p(V)

An B ' Y( U) ~* ' Y (V)

I n o t h e r w o r d s , p ( U ) sa t i s f i e s t he l i f t i ng p r ope r t y o f a Kan f i b r a t i on , up t o r e f i ne -

m e n t a l o n g s o m e c o v e r i n g s i e v e . I r e f e r t o t h i s a s a loca l rig ht lifting property, so

t h a t p : X - , Y is a l o c a l f i b r a t i o n i f a n d o n l y i f p h a s t h e l o c a l r i g h t l i f t in g pro pe rty

w i t h r e s p e c t t o a l l s i m p l i c ia l s e t i n c l u s i o n s o f t h e f o r m A~CZ I n, n > 0 . O f cou rse,

/ 1 " i s t h e s t a n d a r d n - s i m p l e x g e n e r a t e d b y t h e n - s i m p l e x t., and A~ is the sub-

com pl ex o f A n wh i ch i s gene r a t ed by a l l f a ce s o f z . excep t d k l . . A s im p l i ci a l p re -

s h e a f X i s s a i d t o b e locally ibrant i f t h e m a p X ~ i s a f i b r a t i o n , w h e r e * is the

t e r m i n a l o b j e c t o f S P r e ( C ) . E x p l i c i t ly , . ( U ) is a c o p y o f t h e s t a n d a r d 0 -s im p le x A .

O b s e r v e t h a t i s a l s o a s i m p l i c i a l s h e a f .

I f q : Z ~ W i s a s i m p l ic i a l p r e s h e a f m a p w h i c h is apointwise Kanfibration in the

s e n s e t h a t e a c h m a p o f s e c t i o n s q : Z ( U ) ~ W ( U ) , U ~ C , i s a K a r l fi b r a t i o n , t h en q

i s a l o c a l f i b r a t i o n ; i n e f f e c t , n o r e f i n e m e n t s a x e r e q u i r e d . O n t h e o t h e r h a n d , n ot


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S im p l i c ia l p re sh e a v e s 39

eve ry l oca l f i b r a t i on i s a po i n t w i se Ka n f i b r a t i on . L e t &Is be t he & a l e s it e fo r a

lo ca lly N o e t h e r i a n s c h e m e S . I t i s e a s y t o s h o w t h a t a m a p p : X - - , Y o f s i m p l ic i a l

p re sheaves on & Is i s a l oca l f i b r a t i o n i f and on l y i f e ach s t a l k m ap P x: X x ~ Y x c o r -

r esponding t o each geo m e t r i c po i n t x o f S is a Kan f i b r a t i o n . T h i s i s t r ue f o r a l l

0 r o t h e n d ie c k s it es w i t h e n o u g h ' p o i n t s ' o r s t a lk s . I n p a r t ic u l a r , t h e c a n o n i c a l m a p

t / : X ~ X " is a l oc a l f i b r a t i o n f o r e a c h s i m p l i c i al p r e s h e a f X o n & I s . ( T h i s g en e r -

alizes t o a r b i t r a r y s i t e s . ) Bu t no w l e t F be an ab e l i an g r o up , a nd l e t F a l so de no t e

the c o r r e s p o n d i n g c o n s t a n t p r e s h e a f o f a b e l i a n g r o u p s o n & I s - T h e n , f o r e a c h & a l e

m ap U ~ S ,

F ( U ~ S ) = 1-I F ,

no(U)

where

no ( U )

is t h e s et o f c o n n e c t e d c o m p o n e n t s o f U . T h e c a n o n i c a l m a p r /: F - - , P

is g iv en at U ~ S b y t h e d i a g o n a l h o m o m o r p h i s m

A : F ~ I- F .

no(U)

It f ol lo w s t h a t A i n d u c e s t h e c a n o n i c a l m a p

17BF : B F--* B F

at U--* S, a n d so t l sF(U--* S) i s n o t a K a n f i b r a t i o n i f U i s d i s c o n n e c t e d . T h e s i m -

plicial p re sh ea f B F is c o n s t r u c t e d f r o m F b y p o i n tw i s e a p p l i c a t i o n o f t h e u s u a l n e r v e

functor.

L et S P r e ( C ) f C S P r e ( C ) b e t h e f u l l s u b c a t e g o r y o f l o c a ll y f i b r a n t p r e s h e a v e s . T h e

g0al o f t h i s s ec t i on i s t o sho w t ha t S Pr e ( C) f s a t is f i e s t he ax i om s [4 ] and [17 ] f o r a

c ate go ry o f f i b r a n t o b j e c t s f o r a h o m o t o p y t h e o r y . T h i s m e a n s t h a t t w o c la s se s o f

m aps i n S P r e ( C ) f a r e s p e c i f ie d , n a m e l y f i b r a t i o n s a n d w e a k e q u i v a l e n c e s , w h i c h

satis fy a l is t o f ax i om s . T h i s l i s t w i l l be w r i t t en d ow n l a t e r . T he f i b r a t i o ns f o r t h i s

theor y a r e t he l oca l f i b r a t i o ns , a s de f i ned abo ve .

T h e w e a k e q u i v a l e n c e s a r e h a r d e r t o d e f i n e , s i n c e t h e d e f i n i t i o n i s c o m b i n a t o r i a l

and l o c al . W e m u s t f i r s t a r r a n g e f o r a c a l c u lu s o f l o c a l f ib r a t i o n s , i n t h e s t y l e o f

[14]. I say tha t a c lass , .n t o f s im pl ic ia l se t mo no m or ph ism s i s l oca l l y sa tura ted i f i t

sat is fies the fo l lo win g ax iom s:

(1) A l l i s o m o r p h i s m s b e l o n g t o a .

(2) ~ i s d o s e d u n d e r c o b a s e c h a n g e w i t h r e s p e c t t o a r b i t r a r y m a p s .

( 3 ) ~ i s c l o sed unde r r e t r ac t s .

( 4 ) ~ i s c l o s e d u n d e r f i n i t e c o m p o s i t i o n a n d f i n i t e d i r e c t s u m .

Lemma

1 .1 .

The c l as s ~ p o f si m p l i c ia l s e t m on om or ph i s m s w h i c h has t he l oc al l e f t

~ fting p r op e r t y w i t h r e s pe c t t o a f i x e d s i m p l ic i a l p r e s he a f m ap p : X - ~ Y i s loc a ll y

Satura ted.


6/53

40 J.F. Jardine

Pr oo f . ~ tp i s t he co l l e c t ion o f a l l s imp l i c ia l se t i nc lus ions i " K o L such tha t , fo r

e a c h d i a g r a m

c t

K , X ( U )

i l l p

L , Y ( U )

#

t he re i s a cov e r ing s i eve R C C ( - , U) wh e re , fo r e ach tp: V- - , U in R , the re is a

d i a g r a m

a tp*

K , X ( U ) , X ( V )

L , Y ( U ) , Y ( V )

O b s e r v e t h a t t h e l i f ti n g s 0 , a r e n o t r e q u i r e d t o b e c o h e r e n t i n a n y w a y . T h e a xio ms

(1 ) , (2 ) , and (3) t r iv i a l . To ve r i fy (4) ( and to v e r i fy a lo t o f o the r th ing s) w e use a

s t a n d a r d r e f i n e m e n t p r i n c i p l e f o r c o v e ri n g s i ev e s. S u p p o s e t h a t R C C ( - , U ) i s

c o v e r i n g , a n d s u p p o s e t h a t S , C C ( - , F ) i s a c h o i c e o f c o v e r i n g s i e ve f o r each

q~: V- - , U in R . Le t R o S . be the co l l e c t ion o f a l l mo rph i sm s W -- , U o f C having a

f a c t o r i z a t i o n

W , U

\ /

V

wh ere (p e R and g ~S~ . Th en the re f ine m en t p r inc ip le , wh ich i s e as i ly p roved ,

asser ts th a t R o S~, i s co ve r ing .

N o w s u p p o s e t h a t t h e r e i s a d i a g r a m

K I

K3

#

, X ,

u)

P

Y ( U )


7/53


where i l a n d

i 2 a r e

i n M p . T h e n t h e r e i s a c o v e r i n g s ie v e R C C ( - , U ) s u c h t h a t , f o r

each ~ : V - ~ U i n R , t h e r e i s a d i a g r a m

K 1 , X ( U ) , X ( V )

K2 p

Ks , Y(U ) , Y(V)

There i s a cov e r ing s i eve S~ ,C C (- , V) such tha t , fo r e ach : W --, V in S~ ,, t he r e i s

a d i a g r a m

a ~* gJ*

K~ , X( U) 2 X( V) , X( W)

i l

Ks 13 ' Y(U) , , , Y(V) ~, , Y( W)

This g ives a ch oice of l i f t ing 0~, ~, fo r eac h fac tor iza t ion y = ~ o ~v of each y R o S , .

P ick ing on e l i f t i ng 0~ ,w fo r e ach y sho ws tha t i2oil l i f t s l oca l ly a long R o S . , and

so ~ tp i s c lo sed un de r f in i t e com po s i t io n . Mp i s c lo sed u nde r f in i t e d i rec t s um s ,

since t h e c o v e r i n g s ie v e s i n C ( - , U ) a r e c l o s e d u n d e r f i n i t e i n t e rs e c t io n . [ ]

T h e m e m b e r s o f t h e s m a l l e s t l o c a l l y s a t u r a t e d c l a s s o f m o n o m o r p h i s m s w h i c h

conta ins the inc lusions

ATcCA n, n

> 0 , a r e c a ll ed

strong anodyne extensions.

S t a n -

d ard n o n s e n s e [ 14 , p . 6 1 ] , t o g e t h e r w i t h L e m m a I . I , i m p l i e s t h a t a l l i n c l u s i o n s

(A IT )U( {e} S )CA I S, e=O, 1, (1.2)

which a re indu ced by inc lus ion s T C S o f finite s i m p l i c i a l s e t s , a r e s t r o n g a n o d y n e

ex tens ions . One shows , fo l lowing [14 ] aga in , t ha t i f T C S a r e f i n i te , t h e n t h e s e t o f

inclusions

K C L

o f s i m p l i c i a l s e t s s u c h t h a t t h e i n d u c e d m a p

(L x T) U ( K x S ) C L x S

is s t ro n g a n o d y n e , i s a l o c a l l y s a t u r a t e d c l as s w h i c h c o n t a i n s a l l m a p s o f t h e f o r m

(1 .2), a n d h e n c e a l l i n c l u s i o n s A ~ C A n , n > 0 , g i v in g


8/53


9/53

Simplicial presheaves

4 3

f a c t o r s t h r o u g h a d i a g r a m

oA , x( u)

A , X(U) R

fo r s o m e c o v e r i n g s ie v e R . B u t t h e n , f o r e a c h : V - -, U i n R , t h e r e i s a c o m m u t a t i v e

(p*

x ( u )

x ( u ) R

d iag ram

, x ( v )

' X ( V ) ~ o , R

L X ( U ) , L X ( V )

w h e re ~ * R i s t h e s e t o f a l l m o r p h i s m s ~u: W - , V o f C s u c h t h a t ~p o / i s i n R .

(o*R = C ( - , V ) , a n d s o t h e i n d i c a t e d m a p r is a n i s o m o r p h i s m . [ ]

Coro l lary 1 .7 .

I f p : X - ~ Y is a l o c a l f i b r a t io n , t h e n s o i s L p : L x - , L Y .

P r o o f . C o n s i d e r t h e d i a g r a m

~ X

x(u ) , LX( U) , A~

Y ( U ) , L Y ( U ) , A n

r6, 13

L e m m a 1 .6 a n d t h e r e m a r k p r e c e d i n g i t i m p l y th a t a l i ft s l o c a l l y t o X . L o c a l f i b r a -

tions a r e c lo s e d u n d e r c o m p o s i t i o n , b y a n a r g u m e n t d u a l t o t h a t g i ve n fo r L e m m a

1.1, a n d s o 1 / o p i s a l o c a l f i b r a t i o n . T h u s , b y r e f i n i n g f u r t h e r , o n e f i n d s li f t i n g s o f

to L X . [ ]

Coro l lary 1 .8 .

I f p : X ~ Y i s a l o c a l f i b r a t i o n , t h e n s o is p : P ( ~ Y . I n p a r ti cu l a r , i f

X is a p r e s h e a f o f K a n c o m p l e x e s , t h e n X i s l o c al l y f i b r a n t .


10/53

4 4 J . F . J a r d i ne

L e t K b e a f i n i t e s i m p l i c i a l s e t , a n d l e t X b e a l o c a l l y f i b r a n t s i m p l i c i a l p r es h ea f

o n C . L e t

K

f

: X ( U )

g

be a pa i r o f s i m p l i c i a l s e t m aps . f i s s a i d t o be

l o c a l l y h o m o t o p i c

to g (write

f ' - l o c g ) i f t he r e i s a cove r i ng s i eve R C C ( - , U) such t h a t , f o r e ach (a : V- - , U in R ,

t h e r e i s a d i a g r a m

f

K , X ( U )

1 1

K x A 1 h~ , X ( V )

g

K , X ( U )

O n e s a y s t h a t f i s l o c a l l y h o m o t o p i c t o g ( re l L ) , w h e r e L C K , i f f ] ~ = g [t, a n d e ac h

h o m o t o p y h~, is c o n s t a n t o n L .

L e m m a

1.9 . L o c a l h o m o t o p y ( r el a n y s u b c o m p l e x ) o f m a p s K ~ X ( U ) is a n eq uiva-

l e n c e r e l a t i o n i f K i s f i n i t e a n d X i s l o c a l l y f i b r a n t .

P r o o f . I t s u ff ic e s t o s h o w t h a t lo c a l h o m o t o p y o f v e r t ex m a p s

A ~ Y ( U )

is an

e q u i v a l e n c e r e l a t io n i f Y is l o c a l ly f i b r a n t . T o s ee t h i s , o b s e r v e t h a t e a c h c om m a

c a t e g o r y C ~ U i n h e r it s a c a n o n i c a l t o p o l o g y f r o m t h e s it e C , a n d t h a t i f X Iu is the

c o m p o s i t e f u n c t o r

X

(C ,L U) P -- C p ,S , V ----~ ~-. V,

t h e n

X I v

i s l o c a l ly f i b r a n t i f X i s. N o w s u p p o s e t h a t t h e r e i s a d i a g r a m

L

K

t7

,x(u)

K

a n d f o r m t h e p u l l b a c k d i a g r a m


11/53


( X I v ) K / L , ( X

1

, X

f f

v ) x

v) L

of s impl ic ia l p resheaves o n E IU . Then f and g de te rmine ver t ices o f the loca l ly

f ibrant s implicial presheaf

( X [ u ) K / L ,

and f= l oc g ( re l L ) i f and on ly i f the co r res -

pond ing ver t ices a re loca l ly homotop ic .

Suppose tha t there i s a cover ing s ieve R C C ( - , U) such tha t , fo r each ~p: V ~ U

in R, there is a d ia gra m

X

A , X ( U )

A 1 w~, , X ( V )

A Y , X ( U )

so th at x '-- loc y a s v e r t i c e s o f X . T h e n t h e r e i s a c o v e r i n g s i ev e S~ C C ( - , V ) s u c h

tha t, fo r eac h y : W --* V in S~o th e re i s a d i ag ra m

(we , So*x; - )

A 2 , X( V)

A 2 , X( W)

Oy,

where

( w , s o tp * x , - )

i s the u n iqu e m ap on A2 which sends do tz to w( l l) and

d l t z

to

s#*x in

X ( V ) .

T h en th e r e i s a d i ag r am

A Y , X ( U )

d2

A I , A 2 vy ,

' X ( W )

/ , , ,

X

A " ' , X ( U )


12/53

46

J.F. Jardine

f o r e ach com po s i t e o y w i t h (0 e R and y e S , , an d so y = loc x . T he t r ans it iv i t y is

s i m i l a r . Re f l ex i v i t y i s t r i v ia l . [ ]

L e t

C t

be a sm a l l Gr o t hend i eck s i t e w i t h t e r m i na l ob j ec t t ( l i ke I t : i n C~U) . L e t

X b e a l o c a l l y f i b r a n t s i m p l i c i a l p r e s h e a f o n C t a n d t a k e a v e r t e x x ~ X ( t ) o . Let xU

b e t h e i m a g e o f x i n X ( U ) u n d e r t h e m a p X ( t ) - - - , X ( U ) w h i c h i s i n d u c e d b y U - t

i n C t. T h e s et o f l o ca l h o m o t o p y c la ss es o f m a p s ( A n, OA n ) - -' ( X ( U ) , x u ) has ele-

m e n t s d e n o t e d b y [ ( A n, O dn ), ( X ( U ) , x v ) h o c . L e t t i n g U v a r y g iv e s a p r e s h e a f

n.P(x, x) (U ) := { [(A",

OA"), X(U),xv)hoc}.

I t i s e a s i l y s een t ha t n ~ ( X , x ) i s sepa ra te d for a l l n_> 1 . I de f ine n , , ( X , x ) to be the

a s s o c i a te d s h e a f o f n P n ( X , x ) , n n ( X , x ) m a y b e i d e n t i f i e d w i t h L n P n ( X , x ) . Similar

c o n s i d e r a t i o n s a p p l y t o p a t h c o m p o n e n t s ; n P X i s t he p r e shea f o f l oca l hom ot opy

c l a s s e s o f v e r t i c e s , a n d n o X i s i t s a s soc i a t ed shea f .

A c o m b i n a t o r i a l p a i r i n g

m p " 7 tP n ( X , x )

X/tP(X, x) ~

r g n ( X , x )

m a y b e d e f i n e d a s fo l lo w s . L e t f a n d g b e m a p s ( An, O A n ) - - ' ( X ( U ) , x u ) which

r ep r e sen t l oc a l h om ot op y c l a s se s . T he r e i s a cove r i ng s i eve R C C ( - , U) such tha t,

f o r e ach ~0: V- - , U i n R , t h e r e i s a d i a g r a m

(xv . . . . . xu , f , - , g )

An +l ' X ( U )

A n + l , X ( V )

w ,

T h en { [dnw~,] loc}0 ,~n i s an R - co m pa t i b l e f am i l y , a nd hence de f i ne s an e lem en t

[{ [d , , w , hoc} , e n ] o f n n( X , x ) ( U ) w h i c h i s i n d e p e n d e n t o f t h e c h o i c e s t h a t h a v e b een

m a d e . I t f o l l o w s t h a t

([f]~oc, [g]loc) ~ [{ [amW~l lo~}~R]

d e f i n e s t h e U - s e c t io n c o m p o n e n t o f t h e p r e s h e a f m a p m p . A p p l y i n g t h e a s so cia te d

s h e a f f u n c t o r t o m p g iv e s a p a i r i n g

m : n , , (X , x ) nn(X , x ) "-* nn (X , x ) .

T h e c o n s t a n t m a p x v : ( A " ,a A n) ~ ( X ( U ) , x u ) de t e r m i nes a d i s t i ngu i shed e l em en t

e v e n , , ( X , x ) ( U )

i n t h e o b v i o u s w a y , a n d

~ * ( e v ) = e v

f o r e ach ( p: V ~ U .

P r o p o s i t i o n 1 .10 . n n ( X , x ) , a s d e f i n e d a b o v e , i s a s h e a f o f g r o u p s f o r n > . l w hich

i s a b e l ia n f o r n > 2 .


13/53

Sim plicial presheaves

47

l~roof. T h i s w i l l on l y be a ske t ch . T h e i dea i s t o t ak e t he a r g um en t s o f [21, p . 9 ]

and m a k e t h e m l o c a l. O n e g e ts a w a y w i t h th i s b e c a u s e o n l y f i n i t e l y m a n y c h o i c e s

o f l if ts ( an d hen ce r e f i ne m en t s ) a r e r eq u i r ed a t e ach s t age .

S uppose t ha t { [X~]loc}~eR i s an R- c om pa t i b l e f am i l y i n

z t P n ( X , x ) ( U )

f o r s o m e

cover ing s ieve R, an d tha t [Zhoc i s an e lem ent of

rcPn(X,x ) ( U ) .

S u p p o s e f u r t h e r t h a t ,

fo r e ach q : V ~ U i n R , t he r e i s a d i ag r am

A ~ + I ( X v ,. .. ,X v ,X ~ , -, ~ o *z ) ~ X ( V )

A n

T hen t he f am i l y

{ [ d n w ~ ] } ~ R

is R - c o m p a t i b l e a n d r e p r e se n t s t h e p r o d u c t

[ { [ x ~ l l o c } ~ ~ R l [ [ Z h o c l

in

~n(S , ) (U ) .

Now l e t

u , o , w : ( An , a A n ) ~ ( X ( U ) , x v )

r e p r e s e n t e l e m e n t s o f

n P ( X , x ) ( U ) .

B y

success ive re f inement , there i s a cover ing s ieve

R CCt (- , U)

s u c h t h a t , f o r e a c h

~: V ~ U i n R , t h e r e ar e c o m m u t a t i v e d i a gr a m s

(Xv . . . . X v , ~o*u, - , *v )

A n + l ~" X ( V )

A n + l

(xv . . . , x v , d , w~ _ l, - , q ,* w)

A '~ +] ~ X ( V )

z l n+ l

A n + l ( x v . . . , x v, ( o* u , -, ~ o * w )

,, ~ X ( V )

/1 n+ l


14/53

48

J.F. Jardine

( X v , . . . , X v ' w ~ n _ l , ~o W~O

- , W n + l ,

n+2)

Ann+2 , X ( V )

zln+ 2

B u t t h e n t h e d i a g r a m

OA~ +1

N

An+l

(X V . . . . . X V , (O 'U, d n W~n + 1 , dn W~n +

2)

, x ( v )

c o m m u t e s f o r e a c h (0: V - ~ U i n R , a n d s o

[ [ w h o ~ ] ( [ [ o ] i o d [ [ U h o c ] )

= [ [ W h o c ] ' [ { I d ~ w ~ - l h o c } ~ R ]

= [ { I d n w ~ n + l ] l o c } ~ eR ]

= [ { [ d n w n ~ + 2 ] l o c } ~ R ] [ [U ] l o c ]

= ( [ [ wh o c l [ [O l lo c ] ) [ [u h o c l

in

r t , ( X , x ) ( u ) .

I t f o l l ows t h a t t he m u l t i p l i c a t i on m ap i s a s soc i a t i ve . S i m i l a r a rgu -

m e n t s g iv e t h e r e s t o f t h e r e s u lt . [ ]

E a c h o f t h e s i t es C ~ U o f L e m m a 1 .8 h a s a t e r m i n a l o b j e c t , n a m e l y th e id en tity

m a p 1 v : U - - , U , a n d s o x ~ X ] v ( l v ) o d e t e r m i n e s a s h e a f o f h o m o t o p y g r o u p s

ztn(X [ u , x ) i f X i s l o c a l ly f i b r a n t . A m a p f : X ~ Y o f l o c a l l y f i b r a n t s im p l i ci a l p re -

sheaves i s s a i d t o be a combinator ia l weak equ iva lence i f e a c h o f t h e i n d u c e d m ap s

f , : n 0 ( X ) ~

no(Y),

f.:n .(X lu, x)-~n.(YIu,fx),

U ~ C , x ~ X ( U ) o

a r e i s o m o r p h i s m s o f s h e a v e s . R e c a l l t h a t , i f C is a s it e s u c h t h a t t h e s h e a f ca te go ry

S h y ( C ) h a s e n o u g h p o i n t s , o r s t a l k s , t h e n X is l o c a l ly f i b r a n t i f a n d o n l y i f each

o f t h e s t a l k s X y i s a K a n c o m p l e x . I n t h i s c a se , a m a p f : X ~ Y is a c o m b in a to r ia l

w e a k e q u i v a le n c e i f a n d o n l y i f e a c h o f t h e i n d u ce d s t al k m a p s fy : X y --, Yy is a weak

e q u i v a l e n c e o f K a n c o m p l e x e s i n t h e s i m p l i c i a l s e t c a t e g o r y S .


15/53

Simplicial presheaves

49

p r o p o s i t i o n 1 .11 . S u p p o s e g i v e n a c o m m u t a t i v e d i a g r a m

g

X , Y

\ / ;

Z

o f m o r p h i s m s o f l o c a l l y f i b r a n t s i m p l i c i a l p r e sh e a v es . I f a n y t w o o f f , g , o r h ar e

c o m b i n a t o r i a l w e a k e q u i v a l e n c e s , t h e n s o i s t h e t h i r d .

P r o o f . S u p p o s e t h a t g a n d h a r e c o m b i n a t o r i a l w e a k e q u i v a l e n c e s . T h e n

g ,: n 0 ( X ) - ' 7 to (Y ) i s a n i s o m o r p h i s m , s o t h a t , f o r e a c h y ~ Y ( U ) o , U e C , t h e r e i s a

cove ring s i eve R such t ha t , f o r e ach to : V-- * U in R , t he r e i s a d i a g r a m

OA 1 (tp*y,g( x) ) Y ( V )

A 1

where x e X ( V ) o . E a c h s u c h w ,, i n d u c e s a n i s o m o r p h i s m o f sh e a v e s

( w , ) . : ~ t n (Y I v , * Y ) -= - nn (Y I v , g ( x , ) ) .

Mor e gene r a l l y , i f

Y "

i s a l o c a l l y f i b r a n t s i m p l i c i a l p r e s h e a f o n C t a n d t h e r e i s a

OA ]

N

diagram

A l

(x ,x ' )

, r ' ( t )

th en t h e r e is a n i n d u c e d i s o m o r p h i s m

w . :

r c n ( Y ; x ) - - r c n ( Y ; x ' )

w h ic h i s n a t u r a l i n t h e o b v i o u s s e n s e . I n e f f e c t, i f U i s a n o b j e c t o f C t a n d

~:(zln, aAn) -~ (Y '(U ),x u) r e p r e s e n t s a n e l e m e n t o f

n P ( Y ' x ) ( U ) ,

t hen t he r e i s a

c ov erin g s ie v e S C C t ( - , U ) s u c h t h a t , f o r e a c h (p: V - -, U i n S t h e r e i s a c o m m u t a t i v e

diagram

(or, w u) *

( , d n x d ) U ( O d n x A 1) Y ' ( U ) Y ' ( V )

A n x A l


16/53

5 0

J.F. Jardine

w h e r e w U i s t h e c o m p o s i t e

O A n x A I pr Z I 1 w

, , Y ' ( t ) , Y ' ( U ) .

T h e n { [ w , d l ] lo c } , ~ R i s a n S - c o m p a t i b l e f a m i l y i n P '

n n ( Y , x ) ( U ) ,

and so deter.

m i n e s a n e l e m e n t w , [ a ] l o c o f

1 t n ( Y ' , x ' ) ( U ) ;

w,[a ] lo c i s i nd epe nd en t o f the rep re sen.

t a t i v e o f [ a ] l o c . T h e i n d u c e d p r e s h e a f m a p

w . : z t P ( Y ; x ) - * Z t n ( Y ' , x ' )

i s m o n i c a n d l o c a l ly e p i m o r p h i c . F i n a l l y , o b s e r v e t h a t , f o r e a c h (o: V ~ U in th e

o r ig in a l cov e r ing s i eve R , the s i te i som orp h i sm (C~ U)~(q~:

V --* U ) = - C $ V

induces an

i s o m o r p h i s m

r t n( Y ] u , y ) ( t p : V ~ U ) = z t n( Y l v , t p * y) ( l v )

which i s na tu ra l i n Y.

P u t t i n g a l l o f t h e a b o v e t o g e t h e r g i v e s c o m m u t a t i v e d i a g r a m s

7t, ,(YI u, y )( lv )

l t . ( Z l v , fY ) ( 1 u )

tp*

(o*

, ~ t . (Ylv, y)(~o: v ~ u )

1

, n , ,( Z [ v , f y ) ( : V ~ U )

r t n (Y [u , y )(~p : V ~ U ) =

r tn (Y l v , tp*y ) (1v )

, J l ' ,

z t ,, ( Z [ v , fy ) (q~ : V ~ U ) --- r t ,, ( Z l v , f~o*y ) ( 1 v )

7t, ,(Yl v , q~*y)( l v)

t

Tt, ,(Z l v , f tp*y ) (1 v )

(w~o),

( fw, , ) .

r t n (X ]v ,X~o) (1v )

'

r t . (Ylv ,

gxe)(1 v) ~

, n . ( Z I e , f g x D ( 1 v )

fo r e ach ~p: V- - , U in th e cove r ing s i eve R . Th us , a l l o f t he m ap s f , a re isomor-

p h i s m s , a n d s o f i s a c o m b i n a t o r i a l w e a k e q u i v a l en c e i f g a n d h a r e . T h e o t h e r cases

a re t r iv i a l . [ ]

L o c a l f i b r a t i o n s b e t w e e n l o c a l l y f i b r a n t s i m p l i ci a l p r e s h e a v e s a r e c h a r a ct e ri z ed by

hav ing the loca l r ig h t l i f t i n g p r op e r t y w i th re spec t t o a l l s imp l i c i a l se t inc lus ions o f


17/53


the form A ~ C A n, n > O . T h e f o l l o w i n g r e s u l t, w h i c h i s t h e k e y t o th i s t h e o r y , i m -

p lie s t ha t a m ap o f loca l l y f i b r an t s i m p l i c i a l p r e sh eave s i s a loca l f i b r a t i on an d a

c o m b i n a t o ri a l w e a k e q u i v a l e n c e i f a n d o n l y i f i t h a s t h e l o c a l ri g h t l i ft i n g p r o p e r t y

with respec t to a l l i nc l us i ons o f f i n i t e s i m p l i c i a l s e t s ; such m aps wi l l be ca l l ed t r i v i a l

l ocal f i bra t i o ns .

T h e o r e m

1 .12 . A m a p p : X ~ Y b e t w e e n l o c a l l y f i b r a n t s i m p l i c ia l p r e s h e a v e s i s a

l oca l f i b r a t i o n a n d a c o m b i n a t o r i a l w e a k e q u i v a l e n c e i f a n d o n l y i f it h a s t h e l o c a l

rig ht l i ft in g p r o p e r t y w i t h r e s p ec t t o a l l i n cl u s i o n s o f t h e f o r m a A n C A n, n > O .

0_/1n

+1

An+l

p r o o f . A m a p ( A 0, OA n) ~ ( X ( U ) , x ) r e p r e s e n ts t h e t r iv i a l e le m e n t o f r c n ( X { u , x) ( 1 v )

i f and on l y i f t he r e i s a cov e r i ng s i eve R C C ( - , U) such t h a t , f o r e ach (0 : V- - , U i n

R, there i s a d iagram

((o*a,x v , . . . , X v )

, x ( v )

It f o ll o w s t h a t a m a p p : X ~ Y w h i c h h a s t h e l o c a l ri g h t l if t in g p r o p e r t y w i t h r e s p e c t

to all OA" C A n, n >_O, i s a c o m b i n a t o r i a l w e a k e q u i v a l e n c e . S u c h a m a p i s c l e a r l y

a l oc al f i b r a t i o n , b y t h e o b s e r v a t i o n p r e c e d i n g t h e p r o o f o f L e m m a 1 . 6.

F o r t h e c o n v e r s e , s a y t h a t a d i a g r a m

Od n

A

A, ,

Ot

#

, x ( u )

, Y ( U )

has a loc a l l i f t in g i f t he r e i s a cov e r i ng s i eve R C C ( - , U) such t h a t , f o r e ach tp : V--> U

in R , t h e r e i s a c o m m u t a t i v e d i a g r a m

O*o ~

OA ' X( V)

A n ~ Y ( V )

~o,o#


18/53

52

J .F . Jard ine

The idea i s to show tha t , i f p : X - -* Y i s a com bina to r ia l wea k equ iva lence and a loca l

f i b r a t io n , t h en ev e ry d i ag r am o f t h e f o r m D h as a l o ca l l if ti n g .

F i r s t o f a l l , i f D i s loca l ly hom oto p ic to d iag ram s De h av ing loca l l i f tings , then

D has a lo cal l i f t ing . In ef fec t , th ere is a cover ing s ieve R C C (- , U) such that , for

each ~" V-- -, U in R, the re i s a com m uta t iv e d iag ram

OA

["] OA n

X A 1

A o

*#

A n d I

, x ( v )

l p

, Y ( V )

h~

Fu r the rm ore , one i s assum ing t ha t , fo r each (0 in R , there i s a cover ing sieve

R v C C ( - , V ) s uch th a t , f o r e ach ~ : W ~ V in R ~ ,, t h e r e is a d iag r am

q/*oh~odl

, x ( w )

A n

A n

, Y ( W )

~ * o h ~ o d I

T h en th e r e i s a co v er in g s iev e R ~ , , v C C ( - , W ) s u ch th a t , f o r e ach y :

W - * W

in

Rv , , , there i s a d iag ra m

d o (y*q/*h, y*O~,,)

Of f' , (aA xA~) U(A xO) , X (W' )

A n dO , A nx A l , Y ( W' )

C o m p o s in g t h e r e f in em en t s g iv e s t h e c l a im .

N o w c o n s i d e r t h e d i a g r a m

OA" ~ X( U)

D ( ] / p n > _ l .

4,

A , Y ( U )

#

D i s l o ca l l y h o m o to p ic t o d i ag r am s o f t h e f o r m


19/53

S i m p l i c i a l p r e s h e a v e s

53

( f f ~ , X V , . . .

,Xv)

OA , X( V)

I

[ p ~p V- ~ U,

A n , Y ( V )

w h ere x i s th e i m a g e o f t h e v e r t e x 0 o f

OA n.

I n e f f e c t , t h e s u b c o m p l e x A ~ o f

OA n

c on tr ac ts o n t o t h e v e r te x O, a n d t h e h o m o t o p y e x t en d s l o c a l ly t o a h o m o t o p y o f

d i a g r a m s . B u t t h e n a ~ r e p r e s e n t s t h e t r i v i a l e l e m e n t o f n ,, _ l ( X [ v , X v ) ( l v ) =

~ , , _ l ( Y l v, P X v ) ( l v ) a n d s o ea c h D ~ i s l o ca l ly h o m o t o p i c t o d i a g r a m s o f t h e f o r m

XW

OA , x( w)

I

9 /, o ~ ' ] [ p q / : W - -* V .

4 ,

a , Y ( W )

Fina l ly , p . : n n ( X l w , X w ) - - * n n ( Y l w , P ( X w )) is a shea f ep i , an d so each D~,,~ ha s a

lo ca l l i ft i n g . T h u s D h a s a l o c a l l i f ti n g . T h e s h e a f i s o m o r p h i s m p , :

r to X = ~ n o Y

g ives t he r equ i r ed l oca l l i ft i ngs f o r eve r y ve r t ex o f Y . [ ]

Recal l [4] , [17] that a c a t e g o r y o f f i b r a n t o b j e c t s ( f o r a h o m o t o p y t h e o r y ) i s a

c ate go ry f~ w i t h p u l l b a c k s a n d a t e r m i n a l o b j e c t . , e q u i p p e d w i t h t w o c l a s se s o f

m a ps , c a l le d f i b r a t i o n s a n d w e a k e q u i v a l e n c e s , s u c h t h a t t h e f o l lo w i n g a x i o m s a r e

satisfied:

(A ) G iv en m a p s f : X ~ Y a n d g : Y ~ Z in ~ , i f a n y t w o o f f , g , o r g o f a r e w e a k

equiva lences , then so i s the th i rd .

(B ) T h e c o m p o s i t e o f tw o f i b r a t i o n s is a f i b r a ti o n . A n y i s o m o r p h i s m is a

fibration.

( C ) F i b r a t i o n s a n d t r i v i a l f i b r a t i o n s ( i . e . , m a p s w h i c h a r e f i b r a t i o n s a n d w e a k

equ i va l ences ) a r e c l o sed unde r pu l l back .

(D ) F o r a n y o b j e c t X o f ~ , t h e r e i s a c o m m u t a t iv e d i a g r a m

X x

s

/

X , X x X

A

w here A i s t h e d i a g o n a l m a p , s i s a w e a k e q u i v a l e n c e , a n d ( d o , d l ) i s a f i b r a t i o n .

(E ) F o r e a c h o b j e c t X o f f#, t h e m a p X - , i s a f i b r a t i o n .

T h e p o i n t o f w h a t w e h a v e d o n e s o f a r i n t h is s e c t i o n h a s b e e n t o p r o v e :


20/53

54 J.F. Jardine

T h e o r e m

1 .13 . T h e c a t e g o r y S P r e ( C ) f o f l o c a l l y f i b r a n t s i mp l i c ia l p r e sh e a v e s on

a n a r b i t r a r y G r o t h e n d i e c k s i t e C , t o g e t h e r w i t h t h e c l a s s e s o f c o mb i n a t o r i a l we a k

e q u i v a l e n c e s a n d l o c a l f i b r a t i o n s , s a t i s fi e s t h e a x i o m s f o r a c a t e g o r y o f f ib r an t

o b j e c ts f o r a h o m o t o p y t h eo r y .

P r o o f .

A x i o m ( A ) is P r o p o s i t i o n 1 . 1 1. T h e n o n - t r i v i a l p a r t o f A x i o m (B ) Was

o b s e r v e d i n t h e p r o o f o f C o r o l l a r y 1 . 7. L o c a l f i b r a t i o n s a n d t r i v ia l lo c a l fib ra tio ns

a r e d e f i n e d b y l o c a l l i f t i n g p r o p e r t i e s , b y T h e o r e m 1 . 1 2 , a n d a r e t h e r e f o r e c l o s e d

u n d e r b a s e c h a n g e , g i v i n g A x i o m ( C ) . T h e r e i s a c o m m u t a t i v e d i a g r a m

f o r e a c h l o c a l l y f i b r a n t s i m p l i c ia l p r e s h e a f X , w h i c h i s i n d u c e d b y t h e d i a g ra m o f

f in i t e s imp l i c i a l se t s

A 1

Od 1

A 0

f

i * i s a lo c a l f i b r a t i o n b y C o r o l l a r y 1 .5 . O n t h e o t h e r h a n d , d o : X "jl ~ X m ay be

iden t i f i ed wi th the m ap (dO)*: X A ' - - ,X ~. Th i s m ap (dO)* has the loca l le f t l if ting

p r o p e r t y w i t h r e s p e c t t o a l l OA n C A n , n > _O, by adjo in tness , and so do is a t r iv ia l

l o c a l f i b r a t i o n b y T h e o r e m 1 .1 2 . B u t t h e n s is a w e a k e q u i v a l e n c e b y P r op o s it io n

1 . 11 , a n d s o A x i o m ( D ) i s v e r i f ie d . A x i o m ( E ) i s a n a s s u m p t i o n . [ ]

T h e o b j e c t X I o f A x i o m ( D ) is c a ll e d a p a t h o b j e c t fo r X . I t i s im po r t an t t o no te

t h a t t h e p a t h o b j e c t c o n s t r u c t i o n o f t h e p r o o f o f T h e o r e m 1 .1 3 is fu n c t o r ia l an d

c l as s if i es n a t u r a l s i m p l i c i a l h o m o t o p y , b y a d j o i n t n e s s . M o r e p r e c is e l y , t h e r e is a

d i a g r a m

h

/ /

X

y,a I

(ao, d~)

' Y x Y

( f , g)


21/53

S im p l i c ia l p re sh e a v es

55

i f and o n ly i f there i s a d iag ram

X

X X A I h . , Y

X

where

X dl

i s the p resheaf wh ich i s def ined by

X x A I ( U ) = X ( U ) A 1.

Let

n (X , Y) be the se t hom (X, Y) in SPre (C) f , co l lapsed by the smal les t equ iva lence

re la t ion which i s genera ted by the s impl ic ia l homotopy re la t ion . The na tu : r a l i ty o f

ya ' imp l ies tha t r e (X , Y) i s the se t o f m orph ism s f rom X to Y o f a ca te gory

nSPre(C) f , whose ob jec ts a re those o f the o r ig ina l ca tegory SPre(C) f . Th is ca te -

g ory ap p r o x im a te s t h e a s s o c ia t ed h o m o to p y ca t eg o ry H o ( S P r e ( C ) f ) i n t h e s ens e

tha t there i s an i somorph ism

[ X , Y ] - - l i m 7 t( Z , Y ) ,

- . . ,

[rO: Z -- ' X ~ T riv J,X

where Tr iv ~X is the fu l l (f i lt e r ed ) sub ca teg ory o f the co m m a ca tego ry 7 tSPre(C) f~X

whose ob jec ts cons is t o f ma ps w hich a re r ep resen ted by t riv ia l f ib ra t ions , and [X, Y]

d en ote s m o r p h i s m s f r o m X to Y in H o ( S P r e ( C ) f ) . T h e n a tu r a l i t y o f 7 a ' im p l i es

th at t h e h o m o to p y ca t eg o r y m a y b e ap p r o x im a ted b y 7 tS P re ( C )f v ia a c a l cu lu s o f

f ractions (see also [4 , p . 425]) . The co rresp ond ing point for s im plicial sheaves is cen-

tral to the cup pro du ct cons truct ions of [17] .

Theorem 1 .13 impl ies the a na logous r esu l t [17 ] fo r s impl ic ia l sheaves.

Corollary 1.14. T h e c a t e g o r y SShv(C)f of l o c a l l y f i b r a n t s i mp l i c i a l s h e a v e s o n a n

a rb itra ry G r o t h e n d i e c k s i te C , t o g e t h e r w i t h t h e c l a ss es o f l o c a l f i b r a t i o n s a n d c o m -

b in ato ria l we a k e q u i v a l e n c e s a s d e f i n e d a b o v e , s a t is f ie s t h e a x i o m s f o r a c a t e g o r y

o f i b r a n t o b j e c t s f o r a h o m o t o p y t h eo r y . M o r e o v e r ,

H o ( S S h v ( C ) f )

i s equ iva l en t to

Ho(SPre(C)f) .

Proof.

All f in i te l imi t s in SShv(C) a re fo rm ed as they a re in SPre(C ) , and so X r

is a simpl ic ia l sheaf i f X i s . Th is im pl ies the pa th ob jec t A xio m (D) fo r s im pl ic ia l

sheaves . The res t of the axioms are t r iv ial .

A m ap p : X- - , Y o f s impl ic ia l p resheaves i s a tr iv ia l loca l f ib ra t io n i f and o n ly i f

the ma ps

X A ~ ( i ' p , ) , X O a ~ y z P

ya4"


22/53

56

J . F . J a r d i n e

a r e l o c a l e p i m o r p h i s m s i n d e g r e e 0 ; th i s f o l lo w s f r o m T h e o r e m 1 .1 2 . B y ad jo in tn es s

an d Co r o l l a r i e s 1 .3 and 1 .4 , t h i s is equ i va l en t t o ( i * , p , ) be i ng a loca l ep im or ph isnl

i n a ll d e g r e es . O n t h e o t h e r h a n d , t h e c a n o n i c a l m a p

r / , :

X K-~ ~ x

i n d u c e s a n i s o m o r p h i s m o f t h e s h e a f a s s o c i a t e d t o X K w i t h ) ? K i f K is f in i te , since

t h e a s s o c i a t e d s h e a f f u n c t o r c o m m u t e s w i t h f i n it e li m i t s . I t f o ll o w s t h a t th e m ap

~ ( ~ - * * ) ,

~ x

1

~Sd R

i s a d egree -wise loca l ep i i f ( i* , p , ) i s , an d so # i s a t r iv ia l loca l f ib ra t io n i f p i s. We

h a v e a l r e a d y s e e n in C o r o l l a r y 1 .7 th a t t h e a s s o c i a t e d s h e a f f u n c t o r p r e se r ve s local

f i b r a t i on s . E v e r y m ap g : X- -- , Y o f S P r e ( C ) f ha s a f a c t o r i z a t i o n g = q o i , whe re q is

a l oca l f i b r a t i on and i is r i gh t i nv e r se t o a t r i v i a l l oca l f i b r a t i o n ; t h i s is t he f ac to riza.

t i o n l e m m a o f [ 4 ]. T h e r e f o r e , i f g is a c o m b i n a t o r i a l w e a k e q u i v a l e n c e, t h e n s o are

t~ a n d r a n d h e n c e g . T h u s , t h e a s s o c i a t e d s h e a f f u n c t o r p r e s e rv e s c o m b in a to ri al

w e a k e q u i v a le n c e s, a n d s o t h e r e a r e i n d u c e d f u n c t o r s

H o ( S P r e ( C ) f ) ,

A

' H o ( S S h v ( C ) f ) .

L e m m a 1 . 6 i m p l ie s t h a t t h i s i s a n e q u i v a l e n c e o f c a t e g o r i e s . [ ]

A n y o f t h e c l a s s i c a l c o n s t r u c t i o n s o f s i m p l i c i a l h o m o t o p y t h e o r y w h i c h i n v o l v e

o n l y f i n i t e l y m a n y s o l u t i o n s o f t h e K a n e x t e n s i o n c o n d i t i o n c a r r y o v e r t o t h e lo cally

f i b r a n t s i m p l ic i a l p r e s h e a f s e t ti n g . T h e l o n g e x a c t s e q u e n c e o f a f i b r a t i o n is an

e x a m p l e ; o n e c o n s t r u c t s t h e b o u n d a r y h o m o m o r p h i s m l o c a ll y b y a n a l o g y w ith the

c o n s t r u c t i o n o f [ 2 0 ] , g i v i n g

L e m m a

1 .15 . S u p p o s e t h a t

C t

i s a G r o t h e n d i e c k s i t e w i t h t e r m i n a l o b j e c t t . S u p -

p o s e t h a t p : X ~ Y i s a l o c a l f i b r a t i o n o f l o c a l l y f i b r a n t s i m p l i c i a l p r e s h e a ~ e s o n Ct

a n d t h a t x ~ X ( t ) o i s a g l o b a l c h o i c e o f b a s e p o i n t f o r X . L e t F x b e d e f i n e d b y the

C a r t e s i a n s q u a r e

i

Fx ,X

[ p

, ~ y

p x

Then there is a sequence o f pointed sheaves

8 i , p .

" '" ' 7 h (Fx , x )

' l t l ( X , x )

O i , p ,

' ~ o F ~ ' n o X ; ~ o Y


23/53

S impliciai presheaves 57

w hich is e x a c t a s a s e q u e n c e o f s h e a ve s , a n d c o n s i s ts o f g r o u p h o m o m o r p h i s m s in

the us u a l r a n g e .

F o r

C t , X

a n d x a s i n L e m m a 1 . 1 5 , o b s e r v e t h a t t h e r e a r e p u l l b a c k s q u a r e s

pr d~

P X ~ X a~ ' X

X

do

~ X

f 2 X

, P X

l

l p r

x

, X

w hich d e f i n e t h e p a t h s p a c e

P X

a n d t h e l o o p s p a c e ~ X r e s p e c t i v e ly , re l a t i v e t o t h e

c ho ic e o f x . T h e n P X i s t r i v i a l l y l o c a l l y f i b r a n t , a n d s o th e r e a r e i s o m o r p h i s m s o f

sheaves

(1.16)

7 [ i (X , x ) ~-. 7 t i _ (~ ' ~X , x ) ,

nl(X,x) -- ~o(~X)

i_>2,

by T h e o r e m 1 .1 2 a n d L e m m a 1 .1 5. O n e c a n s h o w t h a t

7 t i ( X , x )

i s a b e l i a n f o r i _> 2

by u s i n g t h i s f a c t , f o r t h e n 7ti_ l (~ '~X,x ) h a s tw o g r o u p m u l t i p l i c a t i o n s w h i c h h a v e

a c o m m o n i d e n t i t y a n d s a t i s fy a n i n t e r c h a n g e l a w .

N o w l e t C b e a r b i t r a r y a n d s u p p o s e t h a t Y i s a l o c a l l y f i b r a n t s i m p l i c i a l p r e s h e a f

on C . K a n ' s E x f u n c t o r [1 9] m a y b e u s e d t o c o n s t r u c t a p r e s h e a f o f K a n c o m -

plexes E x Y a n d a c a n o n i c a l m a p v : Y - , E x ~ Y.

Proposition

1 . 1 7 .

T h e m a p v : Y ~

E x ~

Y i s a c o m b i n a t o r i a l w e a k e q u i v a l en c e i f Y

a l o ca l ly f i b r a n t s i m p l ic i a l p r e s h e a f o n a n a r b i tr a r y G r o t h e n d i e c k s i te C .

Proof . R ec al l [19] tha t v :

Y - , E x ~ Y

i s a f i l t e r e d c o l i m i t o f m a p s o f t h e f o r m

[Y -',E xY , w h e r e E x i s r i g h t a d j o i n t t o t h e s u b d i v i s i o n f u n c t o r a n d Y ~ E x Y is

~ d u c e d p o i n t w i s e b y t h e l a s t v e r t e x m a p s s d d n - ~ A n . T h e i d e a i s t o s h o w t h a t

Ex Y is lo c a l l y f i b r a n t a n d t h a t t h e m a p Y - , E x Y i s a c o m b i n a t o r i a l w e a k e q u i v a -

lence. T h e r e s u l t t h e n f o l l o w s f r o m t h e f a c t t h a t , i f

.1"1 f2 f3

X o , x 2 , - - .

a f i lt e r ed s y s t e m i n S P r e ( C ) f s u c h t h a t e a c h J~ i s a c o m b i n a t o r i a l w e a k e q u i v a -


24/53

58 J.F. Jardine

l e nc e , t h e n l i m X i i s l o c a l ly f i b r a n t , a n d t h e c a n o n i c a l m a p

X o - ~ l i m X /

- - +

i s a c o m b i n a t o r i a l w e a k e q u i v a l e n c e .

T h e i n c l u s i o n s d A ~ c c s d d n i s a s t r o n g a n o d y n e e x t e n s i o n ; t h i s i s i m p l i c it in

K a n ' s L e m m a 3 . 4 [1 9], b u t a l es s c o m b i n a t o r i a l p r o o f m a y b e g i v e n w h i c h ta ke s

a d v a n t a g e o f t h e f a c t t h a t s d d " i s a c o n e o n s d 0 A n . I t f o l l o w s t h a t E x p re se rv es

l o c a l f i b r e s e q u e n c e s , a n d t h a t E x Y i s l o c a l l y f i b r a n t i n p a r t i c u l a r .

I t i s t r i v i a l to s h o w t h a t , i f X i s l o c a l l y f ib r a n t , t h e n X ~ E x X i n d u c e s a n iso mo r.

p h i s m i n n o . S u p p o s e t h a t

x e X ( U ) o

f o r a n o b j e c t U o f C . E x c o m m u t e s W ith

r e s t r i c t io n t o C ~ U , a n d s o w e m a y s u p p o s e t h a t C = C t , X an d x a re a s in th e s ta te .

m e n t o f L e m m a 1 .1 5. T h e n t h e r e i s a c o m m u t a t i v e d i a g r a m o f s h e a f h om o m o r.

p h i s m s

nl(X,x)

no( X)

, n l ( E x X , x )

, n 0 ( E x ~ X )

B u t E x

P X - ; .

i s a t r i v i a l l o c a l f i b r a t i o n , b y a d j o i n t n e s s a n d T h e o r e m 1 .1 2, a n d so

a is a n i s o m o r p h i s m . I t e r a t i n g t h i s p r o c e d u r e s h o w s t h a t a l l o f t h e i n d u c e d m ap s

l t i (X , x ) - -* I t i (Ex X , x ) , i>_ l ,

a r e i s o m o r p h i s m s . [ ]

O b s e r v e t h a t P r o p o s i t i o n 1 .1 7 is a t r iv i a l i t y i f S h v ( C ) h a s e n o u g h p o i n t s , f o r the n

E x ' c o m m u t e s w i t h a ll s t a lk c o n s t r u c t io n s , a n d K a n ' s t h e o r e m t h a t X ~ E x ' X is

a w e a k e q u i v a l e n c e f o r a ll K a n c o m p l e x e s X m a y b e j u s t q u o t e d . T h i s t h e o r e m had

t o b e r e p r o v e d i n t h e c o n t e x t a b o v e . I t w i l l b e c o m e i m p o r t a n t w h e n v a r io u s ho mo-

t o p y c a t e g o r i e s a r e c o m p a r e d i n t h e n e x t s e c ti o n . T h e s h e a v e s o f h o m o t o p y groups

w h i c h a r e a s s o c i a t e d t o p r e s h e a v e s o f K a n c o m p l e x e s a r e a l s o v e r y e a s y to d e sc ribe .

P r o p o s i t i o n 1 .1 8 . L e t X b e a p r e s h e a f o f K a n c o m p l e x e s o n a si te C t w i t h term in al

o b j e c t t , a n d t a k e x ~ X ( t ) o. L e t

7 t n s i r n P ( X ,

x ) b e t h e p r e s h e a f o f s i m p l i c i a l h om o to py

g r o u p s o f X , b a s e d a t x . T h e n t h e s h e a f a s so c ia t ed t o

7 t n S i r n p ( x ~

) is can on ical ly iso-

m o r p h i c t o n n ( X , x ) .

P r o o f . C o n s i d e r t h e m a p

11" 7tsnimP(X,X) -* LTlSnimP(X,x).

T h e s i m p l i c e s a , B : (An , a An ) ~ ( X ( U ) , x v ) r e p r e s e n t t h e s a m e e l e m e n t of


25/53

Simp licial presheaves 59

Lg ~i r~p (X , x ) (U)

i f a n d o n l y i f t h e y a r e l o c a l l y h o m o t o p i c ( r e l O A ' ) . T h u s , t h e r e i s

a f a c t o r iz a t i o n o f p r e s h e a f m a p s

7[nsimp X) X) r / , L ~ z s i m p ( x , x )

rc (X,x)

A ll o f th e m a p s i n t h is d i a g r a m b e c o m e i s o m o r p h i s m s w h e n t h e a s s o c i a t e d s h e a f

func tor i s ap pl ie d . [ ]

2 . G l o b a l t h e o r y

L e t C b e a n a r b i t r a r y s m a l l G r o t h e n d i e c k s i t e . T h e g l o b a l h o m o t o p y t h e o r y f o r

the f u ll c a t e g o r y S P r e ( C ) o f s i m p l i c i a l p r e s h e a v e s i s e s se n t i a l l y a t h e o r y o f c o f i b r a -

tions. Th ese a re e asy to def in e ; a

c o f i b r a t i o n

i s a m a p o f s i m p l i c i a l p r e s h e a v e s w h i c h

is a p o i n t w i s e m o n o m o r p h i s m . A s s o c i a t e d t o a n y si m p l i c ia l p r e s h e a f X o n C a n d

x e X ( U ) o

i s a shea f

zttnP(XIv, x )

o n C ~ U .

rttnP(Xlu, x )

i s t h e s h e a f a s s o c i a t e d t o

the p r e sh ea f wh i ch i s de f i ned by

where IX( V) [ i s t he r ea l i za t i o n o f t h e s i m p l i c i a l s et

X ( V ) ,

a n d

~ Z n ( [ X ( V ) [ , X v )

i s t he

usu al n t h h o m o t o p y g r o u p o f t h e s p a c e [ X ( V ) [ , b a s e d a t

X v = ( o* ( x) . n t p ( X [ u , X )

is a shea f o f g r o up s wh i ch i s abe l i an i f n > 2 . T h e sh ea f z t~P( X) o f t o po l o g i ca l pa t h

com ponen ts is de f i ned s i m i l a r l y . A m ap f : X- - , Y o f s i m p l i c i a l p r e shea ves i s s a i d t o

be a

t o p o l o g i c a l w e a k e q u i v a l e n c e

i f i t i n d u c e s i s o m o r p h i s m s o f

sheaves

f , : t p ( X l v , x ) --- ) l ttn P (X [ u , f X ) , U ~ C , x ~ X ( U ) o ,

f* : 7[~P(X) -~ ) ~ z ~ P ( y ) .

T h e re i s a c a n o n i c a l i s o m o r p h i s m

v ,x) - - . . ( s IXl Iu , x ) ,

w here S is t h e s i n g u l a r f u n c t o r , i n v ie w o f P r o p o s i t i o n 1 . 1 8 a n d t h e u s u a l a d j o i n t -

hess t r ic k s . I t f o l l o w s t h a t f : X ~ Y i s a t o p o l o g i c a l w e a k e q u i v a l e n c e i f a n d o n l y

if the as soc ia ted m ap S if I: S [X [ --* S [ Y [ i s a co m bi n a t o r i a l weak equ i v a l ence .

T hus, P r o po s i t i o n 1 .11 i m p l i e s

Lemma 2 .1 .

G i v e n s im p l ic ia l p r e s h e a f m a p s f : X ~ Y a n d g : Y - ~ Z , i f a n y t w o o f

f, g , or g o are t opo l og i ca l w eak eq u i va lences , t hen so i s t he t h i rd .

A t r iv ia l co f ibra t ion i : A - -, B

o f s i m p l i c i a l p r e s h e a v e s i s a m a p w h i c h i s b o t h a

f ib r a ti o n a n d a t o p o l o g i c a l w e a k e q u i v a l e n c e . W e a r e n o w w o r k i n g t o w a r d s a


26/53

60 J . F . J a rd in e

c l o s e d m o d e l s t r u c t u r e o n S P r e ( C ) w h i c h i s b a s e d o n c o f i b r a t i o n s a n d t o p o l o g i c a l

w e a k e q u i v a l e n c es . W e b e g i n b y p r o v i n g

P r o p o s i t i o n 2 .2 . Trivial cofibrations are closed under pushou t.

P r o o f . P o i n t w i s e w e a k e q u i v a l e n c e s a r e t o p o l o g i c a l w e a k e q u i v a l e n c e s , s o it i s

e n o u g h t o c o n s i d e r a n y p u s h o u t d i a g r a m o f t h e fo r m

J

A ,C

1

B ,D

with j a

cofibration,

a n d s h o w t h a t

i'

i s a t r ivial cof ibrat ion i f i i s . i ' i s a t r ivial

c o f i b r a t i o n i f a n d o n l y i f f o r e v e r y d i a g r a m o f t o p o l o g i c a l s p a c e s o f th e f or m

l a ~ l , I C ( U ) l

N

I/ ' I

I A I # , I D ( U ) I

t h e r e i s a c o v e r i n g s i e v e R C C ( - , U ) s u c h t h a t , f o r e a c h ~p: V - ~ U i n

R,

there are

d i a g r a m s

l a A ' , I

I ~ 1

~ o ~ ~

l a A l X I . ~ ' 1

I z r ' l x l A ' l

I c ( v ) l

cl~*la

l i ' l

h e

, I D ( V ) I

/

l a ~ n t

](p*] a

, I c ( v ) l

l i ' l

, I D ( V ) I

h ~ d I


27/53


where Cl~o*la i s the co ns t an t ho m o t o py on [(p*] a . T h i s i s sho wn by app l y i ng

T heor em 1 .12 , t oge t he r w i t h a l oca l s i m p l ex cho i ce a r gum en t , t o t he l oca l f i b r a t i on

as soc i a t ed t o t he m ap S

[ i ' t : S

IcI -- ' s

I D [

o f p r e s h e a v e s o f K a n c o m p l e x e s .

N o w t a k e a d i a g r a m o f th e f o r m E . T h e n t h e r e is a h o m o t o p y

l a A l

IOA I x I A ' I ' I C ( u ) l

1 J , i ' ,

hD

I A I x I I [ ' I D ( U ) I

I A I

and a subd i v i s i o n [L I - - IAn l ( i n t he c l a s s i ca l s ense [25 ] ) such t ha t , i n t he d i a g r am

I K I ~ I a A t

n F1

ILl-= I A I

h D d I

, I c u ) l

1i '1

, I D ( U ) I

the i m age o f t he r ea l i za t i on [ a [ o f e ach s i m p l ex a o f L i s co n t a i ne d e i t he r in [C( U) [

or

I B ( U ) I ,

w h e r e

I g [

i s t h e i n d u c e d s u b d i v i s i o n o f

l a A I .

I t f o l l o w s t h a t t h e h o m o -

t0 py l i ft i n g p r o p e r t y f o r E m a y b e r e p l a c e d b y t h e c o r r e s p o n d i n g p r o b l e m f o r

d i ag r am s o f t he f o r m

IKI , I c ( u ) l

E,

I i ' l

ILl B'

' I D ( U ) I

s uc h t h a t f l ' m a p s e a c h [ a [ i n t o e i t h e r [ C ( U ) [ o r [ B ( U ) [ .

T h e r e i s a s e q u e n c e o f s u b c o m p l e x e s

K = K o C K ~ C . . . C K n = L

of L , wh ere K i + 1 is o b t a i n e d f r o m K i b y a d j o i n i n g a s i m p l ex . S u p p o s e t h a t , f o r t h e

induced d iagram


28/53

62 J.F. Jardine

~t

IKI , Ic (u )l

E,

Ii'1

IK+t #,, ' ]D (U )]

t he r e i s a cove r i ng s i eve R c C ( - , U) such t ha t , f o r e ach ~p: V- * U in R , t he r e a re

d i a g r a m s

I K I [ A a l ] C ( V ) l

IKil x ]h 1 ] ,

[D(V)[

/ o

I1 a'

I g [ I C ( V ) l

li 'l

Igil h~dl ' ID(V)I

s u c h t h a t

( 1 ) i f / ~ ; l a l c I B ( U ) I , t h e n h o , ( l a l I A l l ) C I n ( v ) l ,

(2) i f #/ ' lalc IC(U)l , t hen ho , i s cons t an t on lal.

T h e s e c o n d i t i o n s a r e c o m p a t i b l e , s i n c e IA(U)I--IB(U)I n IC(U)I. Suppose tha t

Ki+l i s o b t a i n e d f r o m K + b y t h e p u s h o u t

OAn ' K i

A n , Ki + 1

a

I f P / t a l c l c ( u ) l , t h e n ho, m a y b e e x t e n d e d b y a h o m o t o p y h i : [K i+ I[ [ A a l - '

IO(V)l w h i c h i s c o n s t a n t o n tcrl. I f # ~ l a [ is c o n t a i n e d i n IB(U)I b u t n o t i n Ic(U)l,

t h e n ho, m a y b e e x t e n d e d t o a h o m o t o p y g o , : I g + + l l IAll- ~ ID(V)I such that

go,([a I

IAII)clB(V)I.

T h u s , s i n c e

i : A + B

i s a t r i v i a l c o f i b r a t i o n , t he r e is a

cove r i ng s i eve S O, CC ( - , V) such t ha t , f o r e ach q t: W + V i n So ,, t he r e a r e com m ut a-

t i v e d i a g r a m s


29/53


63

I K i l x l A ' l

1

I K i + I I [ A I [

, I c w ) l

Cl~*lo~

Ii'l

h~

ID(W)I

~ d o / J ' ' ~

/ ~

I~*1g~ dl

l~u*l 0

[ i [ ' I C ( W ) I

IK~+~I

h w d ' '

ID(W)I

Com pos i ng t h e ho m ot o p i e s h~, an d [V*[ g~ a l ong t he cove r i ng s i eve R o S ~ so l ve s

the l oca l l i f t i ng p r o b l e m f o r t h e i nc l us i on K --- ,K i+ 1 . [ ]

L e m m a 2 .1 a n d P r o p o s i t i o n 2 . 2 a l r e a d y i m p l y t h a t t h e c a t eg o r y S P r e ( C ) , t o g e t h e r

w ith t h e c l a s se s o f c o f i b r a t i o n s a n d t o p o l o g i c a l w e a k e q u i v a le n c e s a s d e f i n e d a b o v e ,

s atis fie s a l i s t o f ax i o m s w h i ch a r e d ua l t o t he ax i om s ( A ) - ( E ) o f t he l a s t s ec t i on ,

m ak in g S P r e ( C ) a c a t e g o r y o f c o f i b r a n t o b j e ct s f o r a h o m o t o p y t h e o r y . B u t m o r e

is tr ue . S ay t h a t a m ap p : X - - , Y is a

g l o b a l f i b r a t i o n

i f p h a s th e r i g h t l i ft i n g p r o p e r -

ty w i th r e spec t t o a l l t r i v i a l co f i b r a t i o ns . W e sha l l p r o ve

T h e o r e m 2 . 3 . S P r e ( C ) , wi th the c las ses o f co f ibra t ions , topo log ica l weak equ iva-

lences an d g l ob a l f i b r a t i ons a s de f i ne d ab ov e , s a ti sf ie s t he a x i om s f o r a c l o s e d m o de l

category.

Recal l tha t a

c l o se d m o d e l c a t eg o r y

i s a c a t e g o r y d / , t o g e t h e r w i t h t h r e e c l a s s e s

o f m a p s , c a l l e d c o f i b r a t i o n s , f i b r a t i o n s a n d w e a k e q u i v a l e n c e s, s u c h t h a t t h e f o l-

lowing ax i om s ho l d :

CM1. ~g i s c l o sed unde r f i n i t e d i r ec t and i nve r se l i m i t s .

C M2. G i v e n f : X ~ Y a n d g : Y ~ Z i n ~ , i f a n y t w o o f f , g o r g o f a r e w e a k e q u i v a-

lences , th en so i s the th i r d .

C M 3. I f f i s a r e t r a c t o f g i n th e c a t e g o r y o f a r r o w s o f d r ', a n d g i s a c o f i b r a t i o n ,

f ib ra tio n o r w e a k e q u i v a l e n c e , t h e n s o i s f .


30/53

64 J.F. Jardine

C M 4 . G i v e n a n y s o li d a r r o w d i a g r a m

U , X

I

V , Y

o f ~ g, w h e r e i i s a c o f i b r a t i o n a n d p i s a f i b r a t i o n , t h e n t h e d o t t e d a r r o w e xists

m a k i n g t h e d i a g r a m c o m m u t e i f e i t h e r i o r p i s a w e a k e q u i v a l e n c e .

C M S . A n y m a p f o f Ji g m a y be f a c t o r e d a s

(1) f = p o i , w h e r e p i s a f i b r a t i o n a n d i i s a c o f i b r a t i o n a n d a w e a k e q u iv a l en c e ,

( 2 ) f = q o j , w h e r e q is a f i b r a t i o n a n d a w e a k e q u i v a l e n c e a n d j i s a c o f i b r a ti o n .

C M 1 a n d C M 3 a r e t r iv i a l f o r S P r e ( C ) . T h e p a r t o f C M 4 t h a t i s n o t t h e de fi ni ti on

o f g l o b a l f i b r a t i o n i s p r o v e d w i t h a t r i c k o f J o y a l [ 1 8 ]. I n e f f e c t , g i v e n a d i ag r am

ot

U , X

V , Y

#

w h e r e p i s a t r i v i a l g l o b a l f i b r a t i o n ( i . e . , p i s a g l o b a l f i b r a t i o n a n d a t o po l o gi c al

w e a k e q u i v a l e n c e , a s u s u a l ) a n d i i s a c o f i b r a t i o n , f o r m t h e d i a g r a m

U

~t

I x

' , X , X

1

' Z p

/o' 1

V U X ' Y

w h e r e 0 i s t h e c a n o n i c a l m a p , q is a t r i v i a l f i b r a t i o n , a n d j i s a c o f i b r a t i o n . T he~

i ' i s a c o f i b r a t i o n , s o j i ' i s a t r i v i a l c o f i b r a t i o n . T h u s , t h e d o t t e d a r r o w e xists,


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Simplicktl presheaves 65

rn ak in g t h e d i a g r a m c o m m u t e , a n d C M 4 i s p r o v e d ( m o d u l o C M 5 ) . T h e p r o o f o f

T he ore m 2 . 3 is t h e r e f o r e r e d u c e d t o p r o v i n g t h e f a c t o r i z a t i o n a x i o m C M S .

T he s i te C is ' sm a l l ' , s o t ha t t he r e i s a c a r d i na l num be r a such t ha t a i s l a r ge r

th an t he c a r d i n a l i ty o f t h e s et o f s u b s e t s P M o r ( C ) o f t h e s et o f m o r p h i s m s M o r ( C )

o f C . A s i m p l i c i a l p r e sh ea f i s s a i d t o be a - b o u n d e d i f t h e c a r d i n a l i t y o f e a c h

x n (U ) , U ~ C , n _ > 0 , is s m a l l er t h a n a . O b s e r v e t h a t i f X i s a - b o u n d e d , t h e n s o i s

the a s s o c ia t e d s h e a f 27 . T h e k e y p o i n t i n t h e p r o o f o f C M 5 ( 1 ) a n d T h e o r e m 2 . 3 i s

L em m a 2 .4 . A map p : X - - , Y & a g lob a l f i b ra t ion i f and on ly i f i t has the r i gh t

l ifting pro per ty w ith respect to aH triv ial cofi 'brations i : U ~ V such that V is

a-bounded.

Proof. F i r s t o f a l l , l e t j : A -- , C be a t r i v i a l co f i b r a t i on , and supp ose t ha t B i s an

a -b ou nd ed s u b o b j e c t o f C . I c la i m t h a t t h e r e i s a n a - b o u n d e d s u b o b j e c t B ,o o f C

such th at

B c B ~ , c C

a n d s u c h t h a t

B ~ O A ~ B ~ ,

i s a t r i v i a l co f i b r a t i on . I n e f f ec t ,

given ~ , ~ r t i ( B ( U ) , B A A ( U ) , x ) , t h e r e is a co v e r in g s ie ve R C C ( - , U ) s u c h t h a t

q*j ,y is t r ivial in

r t i ( C ( V ) , A ( V ) , x v )

f o r e ach ~0: V- - , U i n R . ( T he r e l a t ive h om o-

t0 py g r o u p s a r e t o p o l o g i c a l ; t h e r e a l i z a t i o n n o t a t i o n 1 -[ h a s b e e n d r o p p e d f o r n o t a -

tiona l conv en i ence . I n a dd i t i o n , i c an be 0 . F o r exam pl e , z r 0 (B( U) , B G A ( U ) , x ) i s

def ined t o be t h e qu o t i e n t r to ( B ( U ) ) / r t o ( B O A ( U ) ) . ) C i s a f i l t e r ed co l i m i t o f i t s

a -b ou nd ed s u b o b j e c t s a n d R is a - b o u n d e d , s o th a t t h e r e is a s u b o b j e c t B y o f C

w hich c o n t a i n s B , s u c h t h a t t h e i m a g e y ' o f ~ i n

r ti(By(U) , By O A ( U ) , x )

v a n i s h e s

i n 7[ (By (V ) , B~, n A (V ) , X v) f o r e ach ( 0: V ~ U i n R . L e t B 1 = By , whe r e t he un i on

is t aken ov er a l l 7 ~

r t i ( B ( U ) , B A A ( U ) , x ) ,

U ~ C ,

x ~ B A A ( U ) o , i > _ O .

T h en B l i s

g -b ou nd ed . I t e r a t e t h e p r o c e d u r e t o p r o d u c e a - b o u n d e d o b j e c t s

and let

BC B ICB2C... ,

B = U 8 .

_>1

T hen B ,o i s a n a - b o u n d e d s u b o b j e c t o f C , a n d a n y e l e m e n t o f

z t i ( B t o ( U ) ,

B~NA(U) , y )

v a n i s h e s a l o n g s o m e c o v e r i n g s i e v e , s o

B ~ o A A - ~ B ~

i s a t r iv ia l cof i -

bration.

N o w s u p p o s e t h a t p : X - , Y h a s t h e r i g h t l i f t i n g p r o p e r t y w i t h r e s p e c t t o a l l

~ - b o u n d e d t r i v i a l c o f i b r a t i o n s , a n d c o n s i d e r t h e d i a g r a m

C ~ X

A , Y

where i is a tr i v i a l co f i b r a t i o n . C on s i d e r t he s e t o f pa r t i a l l i f t s


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6 6

J . F . J a r d i ne

C

A

, X

P

, Y

s u c h t h a t i ' i s a t r i v i a l c o f i b r a t i o n a n d B ~ C . T h i s s e t is i n d u c t i v e l y o r d e r e d . T o see

t h a t i t i s n o n - e m p t y , o b s e r v e t h a t A i s a f i l t e r e d c o l i m i t o f i ts a - b o u n d e d s u b.

c o m p l e x e s , a n d s o t h e r e is a p u s h o u t d i a g r a m

B ' N C , C

B ' , B

w h e r e B ' is a n a - b o u n d e d s u b o b j e c t o f A w h i c h i s n o t i n C , i " i s a t r i v ia l c o fi br a ti o n

b y t h e a b o v e , a n d i ' i s a t r iv i a l c o f i b r a t i o n b y P r o p o s i t i o n 2 . 2 . B u t t h e n t h e sa me

a r g u m e n t i m p l i e s t h a t t h e m a x i m a l p a r t i a l l if ts h av e th e f o r m

C ~ A

/ 1 1

A ~ Y

T h e s e e x is t, b y Z o , n ' s L e m m a . [ ]

L e m m a

2 .5 . E v e ry s im p l ic ia l p r e s h e a f m a p f : X ~ Y m a y b e f a c t o r e d

f

X , Y

\ /

Z

w h e r e i i s a t r i v ia l c o f i b r a t i o n a n d p i s a g l o b a l f i b r a t i o n .

P r o o f . T h i s p r o o f i s a t r a n s f i n i te s m a l l o b j e c t a r g u m e n t . C h o o s e a c a r d i n a l f l > 2a,

a n d d e f i n e a f u n c t o r F : f l - ~ S P r e ( C ) ~ Y o n t h e p a r t i a l l y o r d e r e d s et fl b y s e t ti n g

F ( O ) = f : X - -- , Y , X = X ( O ) ,

x ( o = l im X ( 7 )

~ , < (

f o r l i m i t o r d i n a l s ( ,


33/53


67

and b y r e q u ir in g t h a t th e m a p X ( y ) ~ X ( y + 1) b e d ef in e d b y t h e p u s h o u t d i ag r a m

H io

D

i iu , l lv

D D

oo t

l

X ( y ) , X ( y + 1)

such t h a t t h e i n d e x s et c o n s i s t s o f a l l d i a g r a m s o f t h e f o r m

uo , x ( r )

l

m , Y

w here t h e i o a r e c h o s e n f r o m a l is t o f r e p r e s e n t a t i v e s o f i s o m o r p h i s m c l as s e s o f a -

bounded t r iv i a l co f ib ra t ions . Le t

X ( f l ) = l i m X ( y ) ,

---I.

and c o n s i d e r t h e i n d u c e d f a c t o r i z a t i o n

i (]~)

x , x(/D

Y

o f f . T h e n

i ( f l )

i s a t r iv i a l co f ib ra t ion , s ince i t i s a f i l t e red co l imi t o f such . A l so ,

for any d ia g r am

u , x ( #)

~1 I ( )

V , Y

such t h a t V is a - b o u n d e d a n d i is a t r iv i a l c o f i b r a t i o n , t h e m a p

U - , X ( p )

m u s t f a c -

tor th ro ug h som e

X ( y ) - ~ X ( f l ) , y < f l ,

f o r o t h e r w i s e U h a s t o o m a n y s u b o b j e c t s .

The resul t fo l low s. [ ]

T he p r o o f o f C M 5 ( 2 ) i s r e l a t i v e l y e a s y b y c o m p a r i s o n . F i r s t o f a l l, o b s e r v e t h a t ,


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6 8 J.F. Jardine

f o r e a c h o b j e c t U o f C , t h e U - s e c ti o n s f u n c t o r X ~ X ( U ) has a lef t adjoint

?u" S ~ S Pr e ( C ) wh i ch s ends t he s i m p l i c ia l s e t Y t o t he s i m p l i c i a l p r e sh ea f Yu,

w h i c h i s d e f i n e d b y

Y . ( V ) : I I Y .

(a: V-~ U

A m ap q : Z ~ X has t he r i gh t l if t i ng p r op e r t y w i t h r e spec t t o a ll co f i b r a t i on s i f and

o n l y i f i t h a s t h e r i g h t l if t in g p r o p e r t y w i t h r e s p e c t t o a l l i n c l u s io n s S C A ~ of

s u b o b j e c t s o f th e d ~ , U ~ C , n _> 0 . O n e u s es a Z o r n ' s L e m m a a r g u m e n t o n a n in-

d u c t i v e l y o r d e r e d s e t o f p a r t i a l l i f ts to s e e t h i s , a s i n t h e p r o o f o f L e m m a 2 . 4. A

t r a n s f i n i t e s m a l l o b j e c t a r g u m e n t , a s i n L e m m a 2 . 5 , s h o w s t h a t e v e r y m a p f : Y --,X

h a s a f a c t o r i z a t i o n

J

Y ~ Z

\ S

X

wh er e j i s a co f i b r a t i on and q ha s t h e r i gh t l i f t ing p r op e r t y w i t h r e spec t t o all

c o f i b r a t i o n s . I n o t h e r w o r d s , q i s a n i n je c t iv e r e s o l u t i o n o f f i n S P r e ( C ) ~ X (s ee als0

[15 ]; t h i s i s r e a l l y j u s t t he s am e a r g um en t ) . Bu t t h en q i s a wea k eq u i va l ence a s well

a s a g l o ba l f i b r a t i o n . I n e f f ec t , q ha s t he r i gh t l i f t i ng p r o pe r t y w i t h r e spec t t o all

i nc l us i ons n n

d u C A v , U ~ C, n >_ O,

s o t h a t e a c h m a p o f s e c t i o n s q :

Z ( U ) --* X ( U ) is

a t r i v i a l f i b r a t i on o f s i m p l i c i a l s e t s . T hus , q i s a po i n t wi se , hence t opo l og i ca l , weak

e q u i v a l e n c e , a n d C M 5 ( 2 ) is p r o v e d .

T h e p r o o f o f T h e o r e m 2 . 3 i s a l s o c o m p l e t e . I t s a r g u m e n t is r o u g h l y p a r al le l to

t h a t g i v e n b y J o y a l f o r t h e c o r r e s p o n d i n g r e s u lt a b o u t s i m p l ic i a l s h e a v e s . M o r e ex-

p l i c i tl y , a co f i b r a t i o n ( r e sp . t op o l og i ca l w eak equ i va l ence ) o f s i m p l i c i a l sheaves is

j u s t a co f i b r a t i on ( r e sp . t opo l og i ca l weak equ i va l ence ) i n t he s i m p l i c i a l p r e shea f

ca t e go r y . A g l ob a l f i b r a t i on p : X ~ Y o f s i m p l i c i a l sheaves i s a m a p wh i ch ha s the

r i gh t l i f t i ng p r ope r t y w i t h r e spec t t o a l l t r i v i a l co f i b r a t i ons o f s i m p l i c i a l sheaves .

T h i s i s equ i va l en t t o s ay i ng t ha t p i s a g l oba l f i b r a t i on o f s i m p l i c i a l p r e sheaves by

t h e f o l l o w i n g :

Simplicial Presheaves by J.F. Jardine

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