Lie II theorem · 2010. 8. 12. · Introduction A General phenomenon we have is that when we have a...

Lie II theorem

Chenchang Zhu

Institut Fourier, Grenoble

Institut de mathematiquesEPFL, Lausanne

July 11, 2008Slides available at

http://www.crcg.de/wiki/index.php5?title=User:Zhu

Introduction

A General phenomenon we have isthat when we have a global object,say a group G, it has itscorresponding local object Gloc alocal group, and infinitesimal object g

a Lie algebra.

//

��

//oo

G

��

oo

global

∞simal local globalPoissonmanifold

local sym-plecticgroupoid

symplecticgroupoid

Lie alge-broid

local Liegroupoid

Liegroupoid

NQ-manifold

local Lie n-groupoid

Lie n-groupoid

......

...

Introduction


a Lie algebra.

g //

��

//oo

G

��

oo

∞simal global



symplecticgroupoid

Lie alge-broid

local Liegroupoid

Liegroupoid

NQ-manifold


Lie n-groupoid

......

...

Introduction


a Lie algebra.

g //

��

Gloc

��

//oo

G

��

oo

∞simal local global



symplecticgroupoid

Lie alge-broid

local Liegroupoid

Liegroupoid

NQ-manifold


Lie n-groupoid

......

...

Introduction


a Lie algebra.

gC.H. formula

//

��

Gloc

��

//differentiateoo

G

��

oo




symplecticgroupoid

Lie alge-broid

local Liegroupoid

Liegroupoid

NQ-manifold


Lie n-groupoid

......

...

Introduction


a Lie algebra.

gC.H. formula

//

��

Gloc

��

extension//

differentiateooG

��

restrictionoo




symplecticgroupoid

Lie alge-broid

local Liegroupoid

Liegroupoid

NQ-manifold


Lie n-groupoid

......

...

Introduction


a Lie algebra.

gC.H. formula

//

��

Gloc

��

extension//

Canonical

differentiateooG

��

restrictionoo




symplecticgroupoid

Lie alge-broid

local Liegroupoid

Liegroupoid

NQ-manifold


Lie n-groupoid

......

...

Introduction


a Lie algebra.

gC.H. formula

//

��

Gloc

��

extension//

Canonical

differentiateooG

��

restrictionoo




symplecticgroupoid

Lie alge-broid

local Liegroupoid

?−→(stacky)Liegroupoid

NQ-manifold


Lie n-groupoid

......

...

Groupoids


localsym-plecticgroupoid

symplecticgroupoid

Lie alge-broid

local Liegroupoid


NQ-manifold

localLie n-groupoid

Lie n-groupoid

......

...

A groupoid consists

G0 space of objects • •

G1 space of arrows • •vv

source and target s, t : G1 ⇒ G0

multiplication G1 ×G0G1

m−→ G1

• •vv

•vvjj

identity, inverse

satisfy usual coherenceconditions (e.g. associativity)

Groupoids


localsym-plecticgroupoid

symplecticgroupoid

Lie alge-broid

local Liegroupoid


NQ-manifold

localLie n-groupoid

Lie n-groupoid

......

...

A groupoid consists





m−→ G1

• •vv

•vvjj

identity, inverse


A Lie groupoid is a groupoid whoseG0, G1 are manifolds, all structuremaps are smooth, and t, s aresurjective submersions.

Groupoids

A groupoid consists





m−→ G1

• •vv

•vvjj

identity, inverse


A Lie groupoid is a groupoid whoseG0, G1 are manifolds, all structuremaps are smooth, and t, s aresurjective submersions.

A local Lie groupoid Gloc : Gloc1 ⇒

Gloc0 only has local multiplication:

∃ an neighborhood V of Gloc0 in

Gloc1 s.t. V ×G0

V m−→ Gloc

1 .

Nerve of a groupoid NG

X3 = G1 ×G0G1 ×G0

G1

�� X2 = G1 ×G0

G1

��

OOOO OO

• •ww

•ww

hh

X1 = G1��

OO OO

• •ww

X0 = G0

OO

• •

Similarly, NGloc.

Simplicial objects



∃ an neighborhood V of Gloc0

in Gloc1 s.t. V ×G0

V m−→ Gloc

1 .


G1 ×G0G1 ×G0

G1

�� G1 ×G0

G1

��

OOOO OO

• •ww

•ww

hh

G1��

OO OO

• •ww

G0

OO

• •

Similarly, NGloc.

Simplicial manifold X

X3

��

0

1

23

X2

��

OOOO OO

1

20

��

0 11’0

0’1

X1��

OOOO

0 1 0 1

OOOO

X0

OO

• •

Simplicial objects





V m−→ Gloc

1 .


G1 ×G0G1 ×G0

G1

�� G1 ×G0

G1

��

OOOO OO

• •ww

•ww

hh

G1��

OO OO

• •ww

G0

OO

• •

Similarly, NGloc.


X3

��

0

1

23

X2

��

OOOO OO

1

20

��

0 11’0

0’1

X1��

OOOO

0 1 0 1

OOOO

X0

OO

• •

Example: simplical simplex ∆[n]∆[n]k := {k-dim faces in an n-

dim simplex. }

Simplicial objects





V m−→ Gloc

1 .


G1 ×G0G1 ×G0

G1

�� G1 ×G0

G1

��

OOOO OO

• •ww

•ww

hh

G1��

OO OO

• •ww

G0

OO

• •

Similarly, NGloc.


X3

��

0

1

23

X2

��

OOOO OO

1

20

��

0 11’0

0’1

X1��

OOOO

0 1 0 1

OOOO

X0

OO

• •


dim simplex. }simplical horn Λ[n, l]Λ[n, l]k :={k-dim faces in an

(n, j)-horn. }

Simplicial objects


X3

��

0

1

23

X2

��

OOOO OO

1

20 0 11’0

0’1

X1��

OOOO

0 1 0 1

OOOO

X0

OO

• •



(n, j)-horn. }

Λ[1,1] Λ[1,0] Λ[2,2] Λ[2,1] Λ[2,0] Λ[3,3] Λ[3,2] ...

Simplicial objects


X3

��

0

1

23

X2

��

OOOO OO

1

20 0 11’0

0’1

X1��

OOOO

0 1 0 1

OOOO

X0

OO

• •



(n, j)-horn. }

Λ[1,1] Λ[1,0] Λ[2,2] Λ[2,1] Λ[2,0] Λ[3,3] Λ[3,2] ...

hom(∆[n], X ) = Xn

Simplicial objects


X3

��

0

1

23

X2

��

OOOO OO

1

20 0 11’0

0’1

X1��

OOOO

0 1 0 1

OOOO

X0

OO

• •



(n, j)-horn. }

Λ[1,1] Λ[1,0] Λ[2,2] Λ[2,1] Λ[2,0] Λ[3,3] Λ[3,2] ...

hom(∆[n], X ) = Xn

hom(Λ[2, 1], X ) = X1 ×X0 X1

Simplicial objects


X3

��

0

1

23

X2

��

OOOO OO

1

20 0 11’0

0’1

X1��

OOOO

0 1 0 1

OOOO

X0

OO

• •



(n, j)-horn. }

Λ[1,1] Λ[1,0] Λ[2,2] Λ[2,1] Λ[2,0] Λ[3,3] Λ[3,2] ...

hom(∆[n], X ) = Xn

hom(Λ[2, 1], X ) = X1 ×X0 X1

Definition

Kan condition: horn filling ∃ or ∃!Kan(m, j): hom(∆[m], X ) −→

hom(Λ[m, j], X ) is asurjective submersion.

Kan!(m, j): hom(∆[m], X ) −→hom(Λ[m, j], X ) is anisomorphism.

Kan condition

Λ[1,1] Λ[1,0] Λ[2,2] Λ[2,1] Λ[2,0] Λ[3,3] Λ[3,2] ...

hom(∆[n], X ) = Xn

hom(Λ[2, 1], X ) = X1 ×X0X1

Definition




a a b ab ba b−1 −1

Kan(2,2) Kan(2,0)

b ab a

Kan(2,1)

x

y

zx

y

z(ab)

a b

z

h

a bb

ab 1

(ab)

zab

Kan condition

Λ[1,1] Λ[1,0] Λ[2,2] Λ[2,1] Λ[2,0] Λ[3,3] Λ[3,2] ...

hom(∆[n], X ) = Xn

hom(Λ[2, 1], X ) = X1 ×X0X1

Definition





Kan(2,2) Kan(2,0)

b ab a

Kan(2,1)

x

y

zx

y

z(ab)

a b

z

h

a bb

ab 1

(ab)

zab

X = NG⇔ X satisfies all Kan(m, j)and Kan!(≥ 2, j).

Kan condition

Λ[1,1] Λ[1,0] Λ[2,2] Λ[2,1] Λ[2,0] Λ[3,3] Λ[3,2] ...

hom(∆[n], X ) = Xn

hom(Λ[2, 1], X ) = X1 ×X0X1

Definition





Kan(2,2) Kan(2,0)

b ab a

Kan(2,1)

x

y

zx

y

z(ab)

a b

z

h

a bb

ab 1

(ab)

zab


Definition (Duskin, Henriques,Getzler)

Lie n-groupoid: all Kan(m, j) andKan!(≥ n + 1, j).

Kan condition

Λ[1,1] Λ[1,0] Λ[2,2] Λ[2,1] Λ[2,0] Λ[3,3] Λ[3,2] ...

hom(∆[n], X ) = Xn

hom(Λ[2, 1], X ) = X1 ×X0X1

Definition





Kan(2,2) Kan(2,0)

b ab a

Kan(2,1)

x

y

zx

y

z(ab)

a b

z

h

a bb

ab 1

(ab)

zab


Definition (Duskin, Henriques,Getzler)

Lie n-groupoid: all Kan(m, j) andKan!(≥ n + 1, j).

NGloc NG⇋ Kanification

Quillen’s small object argument


Kan(2,2) Kan(2,0)

b ab a

Kan(2,1)

x

y

zx

y

z(ab)

a b

z

h

a bb

ab 1

(ab)

zab

X = NG⇔ X satisfies all Kan(m, j)and Kan!(≥ 2, j).Lie n-groupoid satisfies allKan(m, j) and Kan!(≥ n + 1, j).NGloc

NG ⇋ Kanification

Kanification we fill all the horns

⊔Λ[k , j] //

��

X

��⊔∆[k ] // X 1 .



Kan(2,2) Kan(2,0)

b ab a

Kan(2,1)

x

y

zx

y

z(ab)

a b

z

h

a bb

ab 1

(ab)

zab


NG ⇋ Kanification


⊔Λ[k , j]× hom(Λ[k , j], X ) //

��

X

��⊔∆[k ]× hom(Λ[k , j], X ) // X 1 .



Kan(2,2) Kan(2,0)

b ab a

Kan(2,1)

x

y

zx

y

z(ab)

a b

z

h

a bb

ab 1

(ab)

zab


NG ⇋ Kanification


⊔Λ[k , j]× hom(Λ[k , j], X ) //

��

X

��⊔∆[k ]× hom(Λ[k , j], X ) // X 1 .

We obtain a sequence of simplicialobjects

X → X 1 → X 2 . . .

Then Kan(X ) := colim Xβ.



Kan(2,2) Kan(2,0)

b ab a

Kan(2,1)

x

y

zx

y

z(ab)

a b

z

h

a bb

ab 1

(ab)

zab


NG ⇋ Kanification


⊔Λ[k , j]× hom(Λ[k , j], Xβ) //

��

Xβ

��⊔∆[k ]× hom(Λ[k , j], Xβ) // X 1+β.


X → X 1 → X 2 . . .

Then Kan(X ) := colim Xβ.



Kan(2,2) Kan(2,0)

b ab a

Kan(2,1)

x

y

zx

y

z(ab)

a b

z

h

a bb

ab 1

(ab)

zab


NG ⇋ Kanification


⊔Λ[k , j]× hom(Λ[k , j], Xβ) //

��

Xβ

��⊔∆[k ]× hom(Λ[k , j], Xβ) // X 1+β.


X → X 1 → X 2 . . .

Then Kan(X ) := colim Xβ.But Kan(X )n is not necessarily amanifold!


Kanification we fill all the horns∐

Λ[k , j]× hom(Λ[k , j], Xβ) //

��

Xβ

��∐

∆[k ]× hom(Λ[k , j], Xβ) // Xβ+1.We obtain a sequence of simplicialobjects

X → X 1 → X 2 . . .

Then Kan(X ) := colim Xβ.Can’t have Kan(X )n necessarily amanifold!

Lemma

Kan(Gloc) is a Kan simplicialmanifold.

Lemma

If X is Kan, Kan(X ) ∼M.E. X areMorita equivalent.

Lemma

If X ∼M.E. Y are Morita equivalent,then Kan(X ) ∼M.E. Kan(Y ).

Properties of Kan(−)

Lemma

Kan(Gloc) is a Kan simplicialmanifold.

Lemma

If X is Kan, Kan(X ) ∼M.E. X areMorita equivalent.

Lemma

If X ∼M.E. Y are Morita equivalent,then Kan(X ) ∼M.E. Kan(Y ).

Calculation:

Kan(X )0 = X0

Kan(X )1 = X1 ⊔ X1 ×X0X1 ⊔

(

X1×X0

X1 ⊔ (X1 ×X0X1)×X0

X1 . . .)

. . . ,

Kan(X )2 = X2 ⊔ X1 ×X0 X1

⊔ hom(Λ[3, j], X ) . . . ,

... 7 of them

...

...

...4 of them

...

Kan(X)2

Kan(X)1

Truncation

Calculation:

Kan(X )0 = X0

Kan(X )1 = X1 ⊔ X1 ×X0 X1 ⊔(

X1×X0

X1 ⊔ (X1 ×X0X1)×X0

X1 . . .)

. . . ,

Kan(X )2 = X2 ⊔ X1 ×X0X1

⊔ hom(Λ[3, j], X ) . . . ,

(1)

... 7 of them

...

...

...4 of them

...

Kan(X)2

Kan(X)1

n-Truncation:

τn(X )k =

{

Xk , if k ≤ n;

Xk/Xk+1, if k > n.

τn(Kan simplicial manifold) is an-groupoid!

Truncation

Calculation:

Kan(X )0 = X0

Kan(X )1 = X1 ⊔ X1 ×X0 X1 ⊔(

X1×X0

X1 ⊔ (X1 ×X0X1)×X0

X1 . . .)

. . . ,

Kan(X )2 = X2 ⊔ X1 ×X0X1

⊔ hom(Λ[3, j], X ) . . . ,

(1)

... 7 of them

...

...

...4 of them

...

Kan(X)2

Kan(X)1

n-Truncation:

τn(X )k =

{

Xk , if k ≤ n;

Xk/Xk+1, if k > n.

τn(Kan simplicial manifold) is an-groupoid! Canonical Extensionof Gloc

G is given by

Gloc τ1(Kan(NGloc))

Truncation

Calculation:

Kan(X )0 = X0

Kan(X )1 = X1 ⊔ X1 ×X0 X1 ⊔(

X1×X0

X1 ⊔ (X1 ×X0X1)×X0

X1 . . .)

. . . ,

Kan(X )2 = X2 ⊔ X1 ×X0X1

⊔ hom(Λ[3, j], X ) . . . ,

(1)

... 7 of them

...

...

...4 of them

...

Kan(X)2

Kan(X)1

n-Truncation:

τn(X )k =

{

Xk , if k ≤ n;

Xk/Xk+1, if k > n.


G is given by


Problem: manifolds individually arevery nice, but they don’t form agood category. careful: Quotient,fibre product, limit, ...τ1(Kan(NGloc)) in general is only atopological groupoid.

Get Lie stuff

n-Truncation:

τn(X )k =

{

Xk , if k ≤ n;

Xk/Xk+1, if k > n.


G is given by


Problem: manifolds individually arevery nice, but they don’t form agood category. careful: Quotient,fibre product, limit, ...τ1(Kan(NGloc)) in general is only atopological groupoid.

To recover the local Lie structure,we need

either, take stacky quotient a stacky Lie groupoid

G +3 M ,

or, take τ2(Kan(NGloc)) aLie 2-groupoid

X2_*4 X1

+3 M .

They are equivalent under

Theorem (Duskin, [Z1])Stacky Lie groupoids 1−1

←→/∼

Lie

2-groupoids.

Universal groupoids

To recover the local Lie structure,we need

either, take stacky quotient a stacky Lie groupoid

G +3 M ,

or, take τ2(Kan(NGloc)) aLie 2-groupoid

X2_ *4 X1

+3 M .

They are equivalent under

Theorem (Duskin, [Z1])stacky Lie groupoids 1−1

←→/∼

Lie

2-groupoids.

bigon(X2)

��X1( //

//

stack��O�O�O

M)

top gpd G(moduli spaceoo o/ o/ o/ o/ o/ o/ //

// M)

What is G? G is the universalstacky groupoid of A—Liealgebroid of Gloc . τ1(Kan(NGloc))is the universal topologicalgroupoid of A (Cattaneo+Felder,Crainic+Fernandes).

simplicial set associated to A

bigon(X2)

��X1( //

//


M)


// M)

What is G? G is the universalstacky groupoid of A—Liealgebroid of Gloc. τ1(Kan(NGloc))is the universal topologicalgroupoid of A (Cattaneo+Felder,Crainic+Fernades).

Simplicial set S(A) associated to A

Sn(A) = Homalgd(T∆n, A),

Don’t know whether it’s a simplicialmanifold.


bigon(X2)

��X1( //

//


M)


// M)




S0(A) = M



bigon(X2)

��X1( //

//


M)


// M)




S0(A) = M

S1(A) = { a(t) ∈ A��

γ(t) ∈ M

: ρ(a(t)) = γ(t)}

= mfd



bigon(X2)

��X1( //

//


M)


// M)




S0(A) = M

S1(A) = { a(t) ∈ A��

γ(t) ∈ M

: ρ(a(t)) = γ(t)}

= mfd

S2(A) = {a(ǫ, t) : ∃b(ǫ, t), ∂ǫa− ∂tb

T∇(α, β), b(0, t) = b(1, t) = 0}

?? Banach manifold

S3(A) = ...


Different versions of integration



S0(A) = M

S1(A) = { a(t) ∈ A��

γ(t) ∈ M

: ρ(a(t)) = γ(t)}

= mfd

S2(A) = {a(ǫ, t) : ∃b(ǫ, t), ∂ǫa− ∂tb

T∇(α, β), b(0, t) = b(1, t) = 0}

?? Banach manifold

S3(A) = ...





S0(A) = M

S1(A) = { a(t) ∈ A��

γ(t) ∈ M

: ρ(a(t)) = γ(t)}

= mfd

S2(A) = {a(ǫ, t) : ∃b(ǫ, t), ∂ǫa− ∂tb

T∇(α, β), b(0, t) = b(1, t) = 0}

?? Banach manifold

S3(A) = ...


Theorem

τ2(S(A)) ∼M.E. τ2(Kan(NGloc)) isthe universal Lie 2-groupoid of A.




S0(A) = M

S1(A) = { a(t) ∈ A��

γ(t) ∈ M

: ρ(a(t)) = γ(t)}

= mfd

S2(A) = {a(ǫ, t) : ∃b(ǫ, t), ∂ǫa− ∂tb

T∇(α, β), b(0, t) = b(1, t) = 0}

?? Banach manifold

S3(A) = ...


Theorem

τ1(S(A)) ∼M.E. τ1(Kan(NGloc)) isthe universal topological groupoid.

Theorem





S0(A) = M

S1(A) = { a(t) ∈ A��

γ(t) ∈ M

: ρ(a(t)) = γ(t)}

= mfd

S2(A) = {a(ǫ, t) : ∃b(ǫ, t), ∂ǫa− ∂tb

T∇(α, β), b(0, t) = b(1, t) = 0}

?? Banach manifold

S3(A) = ...


Theorem


Theorem


Kan(NGloc) (1) comparing to S(A)

1 piecewise finite dimensional,

2 still has strict multiplication:m :Kan(NGloc)×M Kan(NGloc)→Kan(NGloc). (True also for itstruncations).

3 shows differentiation onlyneed Gloc.


Theorem


Theorem


Kan(NGloc) comparing to S(A) is

1 piecewise finite dimensional,

2 still has strict multiplication:m :Kan(NGloc)×M Kan(NGloc)→Kan(NGloc). (True also for itstruncations).

3 shows differentiation onlyneed Gloc .

Summary:

1 Universal Lie 2-groupoid of A,

G(A) := τ2(S(A))

∼M.E. τ2(Kan(NGloc))

Equivalent to

2 G(A) the universal stackygroupoid of A.Enrichment of

3 τ1(S(A)) ∼M.E.

τ1(Kan(NGloc)) the universaltopological groupoid of A.

Lie II

Summary:


G(A) := τ2(S(A))


Equivalent to


3 τ1(S(A)) ∼M.E.


Theorem (Lie II, [Z2])A Lie algebroid morphism A→ Band any groupoid GB of B, give

A→ B G(A)→ GB

a morphism of Lie 2-groupoids.

Proof.

A⇓

// B

Gloc(A) // 11Gloc(B)� � //

// //

Lie II

Summary:


G(A) := τ2(S(A))


Equivalent to


3 τ1(S(A)) ∼M.E.



A→ B G(A)→ GB


Proof.

A⇓

// B

Gloc(A) // 11Gloc(B)� � // GB

// //

Lie II

Summary:


G(A) := τ2(S(A))


Equivalent to


3 τ1(S(A)) ∼M.E.



A→ B G(A)→ GB


Proof.

A⇓

// B

Gloc(A) //

⇓

11Gloc(B)� � // GB

Kan(Gloc(A)) // Kan(GB) //

Lie II

Summary:


G(A) := τ2(S(A))


Equivalent to


3 τ1(S(A)) ∼M.E.



A→ B G(A)→ GB


Proof.

A⇓

// B

Gloc(A) //

⇓

11Gloc(B)� � // GB

Kan(Gloc(A)) // Kan(GB)∼M.E. // GB

Lie II

Summary:


G(A) := τ2(S(A))


Equivalent to


3 τ1(S(A)) ∼M.E.



A→ B G(A)→ GB


Proof.

A⇓

// B

Gloc(A) //

⇓

11Gloc(B)� � // GB

τ2Kan(Gloc(A)) // τ2Kan(GB)∼M.E. // τ2GB

Lie II

Summary:


G(A) := τ2(S(A))


Equivalent to


3 τ1(S(A)) ∼M.E.



A→ B G(A)→ GB


Proof.

A⇓

// B

Gloc(A) //

⇓

11Gloc(B)� � // GB


G(A)

Lie II

Summary:


G(A) := τ2(S(A))


Equivalent to


3 τ1(S(A)) ∼M.E.



A→ B G(A)→ GB


Proof.

A⇓

// B

Gloc(A) //

⇓

11Gloc(B)� � // GB


G(A) GB

Connectedness

Theorem (Lie II, [Z2] )A Lie algebroid morphism A→ Band any groupoid GB of B, give

A→ B G(A)→ GB


Proof.

A⇓

// B

Gloc(A) //

⇓

11Gloc(B)� � // GB


G(A) GB

Theorem

The universal Lie 2-groupoid X (A)has 2-connected source fibre.

i.e. the Lie groupoid s−1(x) = {

x } ⇒ { x } has π0 = π1 =π2 = 1.

Connectedness


A→ B G(A)→ GB


Proof.

A⇓

// B

Gloc(A) //

⇓

11Gloc(B)� � // GB


G(A) GB

Theorem



x } ⇒ { x } has π0 = π1 =π2 = 1.Reason why it’s true Let G be theuniversal Lie group of g, then

1conn cover

G

OOO�O�O�

//G

⇑universal

oo

� � //

_?oo

Connectedness


A→ B G(A)→ GB


Proof.

A⇓

// B

Gloc(A) //

⇓

11Gloc(B)� � // GB


G(A) GB

Theorem




1conn cover

G

OOO�O�O�

//G

⇑universal

oo

univ. prop. of G ⇒ � � //= id

_?oo

Connectedness


A→ B G(A)→ GB


Proof.

A⇓

// B

Gloc(A) //

⇓

11Gloc(B)� � // GB


G(A) GB

Theorem




1conn cover

G

OOO�O�O�

//G

⇑universal

oo


univ. prop. of 1-c.c.⇒ = id_?oo

Connectedness

Theorem




1conn cover

G

OOO�O�O�

//G

⇑universal

oo



NowThe source fibre is a 1-groupoid(=differential stack, rather than a0-groupoid=manifold). We shouldexpect higher covers! E.g. S2 inthis world

BZ→ S2 → S2

S2 = S3 ×R⇒ S3 with π≤2S2 = 0.:(

Connectedness

Theorem




1conn cover

G

OOO�O�O�

//G

⇑universal

oo



NowThe source fibre is a 1-groupoid(=differential stack, rather than a0-groupoid=manifold). We shouldexpect higher covers! E.g. S2 inthis world

BZ→ S2 → S2

S2 = S3 ×R⇒ S3 with π≤2S2 = 0.:( No homotopy theory of gpds ):Hence can’t use it as a proof. Theproof is a direct verification.

References

Chenchang Zhu.n-groupoids and stacky groupoids,arxiv:math.DG/0801.2057.

C. Zhu.Lie II theorem for Lie algebroids via stacky Lie groupoids,arXiv:math/0701024v2 [math.DG].

Lie II theorem · 2010. 8. 12. · Introduction A General phenomenon we have is that when we have a...

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