representing - github pages
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k¥4 - Representing spacesFix a profits group r cud ots f : P → Glen UF)
Thy Csohessrnysr's Criteria)Let F : CNL → SETS be continuous Puncher set.FGF) - a singleton.
For a: A→ C and p : B → C in As , consider¢ : F(Axis) → FCAT Xfce , FCB )
The F is representable e> the following one satisfiedHb
.If o is small
,the ¢ is surjective.
.
H2.
If A- He and GIF, the ¢ is bijective .
H3.dim FGFGI) ex
htt . If A- B and a- B is small,then ¢ is bijective .
If T saffrons Ep and
The Cmaow) End,pen, CE) - IF , th DE is representable.
Prod H1 : Tale lifts p. and p, of f- to A aid B such that
does ad fo fB are It Mn Cmo ) - conj .Tato g e bt Mn Cmo )
set. g loyal g-
'e es .
Since o is surjective, we can lift gto he ft Mn Im. )
.
Then (hp. hi'
, fo) defines a lift to AxeB,and
¢ Chp. hi'
, post - Cpa , eel .
H2:F#tT H2s shipped.
143 : Labor.
Ht : Furst , we need a lemma.
Kenny Assume that End#er, let - IF. Then A- any CE CNL
and arp left p : Ms Glen CC) of E, Ender, le)- C
.
Prod Exercise.
(Reduce to Astoria case ad induct an length CC) .)
Paok ko htt : Tahr o : A → C small m Ar.
We want to show0 e. Dq CAZA) → DECA) xq.co, Dq CA)
is injectme .Take for e Df (AxeA) such that del - OG) as deformations
.
Write f ↳ Cpi , pal ED CA) xseek, DECA) and similarly
T ↳ ( ri , ra) E" "
Bx assumption , I g :E It Mn (mo ) such that
Pi = girigi'
Nets oop ,-
oof. and Ooh -- ooh as litts ko C.
oop ,=oocg.r.gl/--oCq1oor,aCg.T '- ag. look 0cg . )
' '
= ocg.gitoop.olg.gs' ) "
= ocqgi' ) o -e, Hg,gi't
'
⇒ ocg.gs ) commutes with a- pie.Be the lemma , dg.gs ' ) E 1 t me , who
'
we can lift toa.a 2- ' ma . Multiplying g, by a-3 we can assume
a Cgi ) - algal .The g
- Cgs , g.) E 1 ' Mn lmaoo) and -
g - g-'.
o
Ganay let the CNL. Assure End (f) - IF and let R represent
Dq : CNL→SETS.
Then the restriction of Dq to Ahn is representedby Riwae, h .
Similarly Pa Dog without condition an f .
Rmd Sf R? represents Dog , then we have a natural isoDoi ⇐ Honour CR? -)
In port if f'e Df ( RM corr te id E Henan CR? RD
,
th for any AE CNL ad e E DHA) ,there is a unique
¢ : RD→A m CNL sit-
erse= oops p → Glen CR ')
→ toE Glen CA)
- X-
ThedagntspoeeLet adf = Mn CIF) with adjoint P- action , i.a Tor, XE odp,
o . X- e-G) Xptot '
RI adf = glue .
Ifwe replace Glen by some other
groupscheme
,
Mn od@ would be the Lie algebra AF.
Tabaliftf : M
→ Glen CIFCET) of f . Fer every NEM,write
p G)= (ft Ecco)) Eco) fer dot c- Mn (IM
.
The f- AT EM
plot) = pale CT) ⇐ Ate CGTDEGTI-ftteeGHEGVH.ecGD ECT)⇐s c Cox) f GT) = calf G) e-G) t EG) chipCT)⇒ cent) - cent e- (a) CGI EAT
'
Es Ce Z'fr,od E)--
space of 1- cocyel AMwith Coffs in adj
Goers Cheek that the F- week-space
structures a DEAFGT)ad 27M
,odf) agree.
If R ' represents D8 , show this also
agrees with Home (Mrs Hmi. .pl , F)
Two lofts e ,- G. tea.IE , pi- Atea) e- c- DEAFGI) define the
sane defamation Ed7. Xo Mn CIF) sit . p ,
- fttex) f. ft- ex)
⇐ a. tea) f- Cttexlfttea ) f ft - ex)
↳ c. e- = Xp tap - EX⇒ GCN - c.Colt X - e- Ca) X e- GT
' H ref↳ c,
and c,define the sous class in HIM
, cdf ).
Drops we have be op IF - ureter spaceDECKED ⇐ 24M
, adj ) Defaced ⇐ A' CM, ode)
↳ SP M satisfies Gop Cios.
f-ay op Its P, Hema, CH, Kp) is hurts),
then De CE CED is Awk dm IIF.
feed Let It =L- let . By inflation - restriction , we have0→ H' CMH
,odg) → H
' Cr, ode) → HYH , ode ) Elton CH
,IF")
-- -Duke since Mtt is ofin group fruits by Ep assumption D
RI Assn Endorses (E) - IF ad let R represent Dq .
Can sharethtr ± Whp) Ex ..-,x5Nlf. .-ft
whog- dima. Http, odes)r - dim
#It' (r
, ode)
Caf (Mazar) Say P - Gf, s where F- * Phd ad S is a Piuteset of prowess of f
> Svlx3u5vlp3.Assurer f is abs ironed and let R represent DE .Th dim R - I + h, - hawhen. hi - dump Hi (GF, s , adel .
Finery Show that n--1 case of this Cag es Leopoldt 's Coy .