Representing - GitHub Pages

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 .