Morita equivalence of vector bundles

Cristian OrtizIME-USP

(Joint with M. del Hoyo - IMPA)

Workshop on Mathematical Physics - ICTP-IFT-UNESPJune 16-17, 2016

Outline

1. Lie groupoids + examples

2. VB-groupoids + examples

3. Morita equivalence of VB-groupoids

4. Applications

5. Ongoing research

Why Lie groupoids?

Useful to understand:

i) local symmetries, e.g. parallel transport of flat connections

ii) singular spaces, e.g. orbifolds

iii) global picture to generalized geometries, e.g. Poissonmanifolds, Dirac structures, moment maps, quantization

Remark

I local symmetries = representations of Lie groupoids

I singular spaces = Morita equivalence of Lie groupoids

I global picture = integration of Lie algebroids to Liegroupoids

Today

i), ii) and iii) in interaction vector bundles over singular spaces

Why Lie groupoids?

Useful to understand:

i) local symmetries, e.g. parallel transport of flat connections

ii) singular spaces, e.g. orbifolds

iii) global picture to generalized geometries, e.g. Poissonmanifolds, Dirac structures, moment maps, quantization

Remark

I local symmetries = representations of Lie groupoids

I singular spaces = Morita equivalence of Lie groupoids

I global picture = integration of Lie algebroids to Liegroupoids

Today

i), ii) and iii) in interaction vector bundles over singular spaces

Why Lie groupoids?

Useful to understand:

i) local symmetries, e.g. parallel transport of flat connections

ii) singular spaces, e.g. orbifolds

iii) global picture to generalized geometries, e.g. Poissonmanifolds, Dirac structures, moment maps, quantization

Remark

I local symmetries = representations of Lie groupoids

I singular spaces = Morita equivalence of Lie groupoids

I global picture = integration of Lie algebroids to Liegroupoids

Today

i), ii) and iii) in interaction vector bundles over singular spaces

Lie groupoids

A groupoid consists on two sets G and M , together withstructural maps s, t : G →M , m : G(2) → G, i : G → G and1 : M → G, satisfying the following conditions:

1. s(gh) = s(h), t(gh) = t(g).

2. (gh)k = g(hk), whenever s(g) = t(h) and s(h) = t(k).

3. 1t(g)g = g = g1s(g)

4. gg−1 = 1t(g), g−1g = 1s(g).

Equivalently, a groupoid is a small category in which everymorphism is invertible.Smooth data Lie groupoid

Lie groupoids

A groupoid consists on two sets G and M , together withstructural maps s, t : G →M , m : G(2) → G, i : G → G and1 : M → G, satisfying the following conditions:

1. s(gh) = s(h), t(gh) = t(g).

2. (gh)k = g(hk), whenever s(g) = t(h) and s(h) = t(k).

3. 1t(g)g = g = g1s(g)

4. gg−1 = 1t(g), g−1g = 1s(g).

Equivalently, a groupoid is a small category in which everymorphism is invertible.Smooth data Lie groupoid

Lie groupoids

A groupoid consists on two sets G and M , together withstructural maps s, t : G →M , m : G(2) → G, i : G → G and1 : M → G, satisfying the following conditions:

1. s(gh) = s(h), t(gh) = t(g).

2. (gh)k = g(hk), whenever s(g) = t(h) and s(h) = t(k).

3. 1t(g)g = g = g1s(g)

4. gg−1 = 1t(g), g−1g = 1s(g).

Equivalently, a groupoid is a small category in which everymorphism is invertible.Smooth data Lie groupoid

Examples

1. Lie groups: G ⇒M = {pt}2. Manifolds: M ⇒M

3. Actions: GnM ⇒M . Called the transformation groupoid.

4. Submersions: any submersion f : M → N determines a Liegroupoid M ×f M ⇒M (submersion groupoid)

5. Vector bundles: E →M determines GL(E)⇒M (generallinear groupoid)

6. other examples include: fundamental groupoid, holonomygroupoid, monodromy groupoid...

Morphisms

A Lie groupoid morphism is a smooth map ϕ : G → H which isalso a functor.

Examples

1. Lie groups: G ⇒M = {pt}2. Manifolds: M ⇒M

3. Actions: GnM ⇒M . Called the transformation groupoid.

4. Submersions: any submersion f : M → N determines a Liegroupoid M ×f M ⇒M (submersion groupoid)

5. Vector bundles: E →M determines GL(E)⇒M (generallinear groupoid)

6. other examples include: fundamental groupoid, holonomygroupoid, monodromy groupoid...

Morphisms

A Lie groupoid morphism is a smooth map ϕ : G → H which isalso a functor.

Equivalences

A morphism ϕ : G → H which is fully faithful and essentiallysurjective is called an equivalence.

Examples

1. manifolds diffeomorphisms

2. (GS ⇒ S) ↪→ (G ⇒M) is equivalence iff S is transversal toevery orbit

3. f : M → N surjective submersion. Then(M ×f M ⇒M)→ (N ⇒ N) is an equivalence.

RemarkEquivalences are also called Morita maps between Lie groupoids.

Equivalences

A morphism ϕ : G → H which is fully faithful and essentiallysurjective is called an equivalence.

Examples

1. manifolds diffeomorphisms

2. (GS ⇒ S) ↪→ (G ⇒M) is equivalence iff S is transversal toevery orbit

3. f : M → N surjective submersion. Then(M ×f M ⇒M)→ (N ⇒ N) is an equivalence.

RemarkEquivalences are also called Morita maps between Lie groupoids.

Transversal Geometry

Orbits and Isotropies

Given a Lie groupoid G ⇒M and x ∈M :

I isotropy group Gx = s−1 ∩ t−1(x) (Lie group)

I orbit Ox = {t(g) ∈M ; g ∈ s−1(x)} (immersed submanifold)

I orbit space M/G = {Ox;x ∈M} (topological space)

I normal representation Gx y νx

e.g. for GnM ⇒M , one gets the usual slice representation.

RemarkA morphism ϕ : G → H induces a Lie group morphismGx → Hϕ(x), a continuous map M/G → N/H and an intertwiningmap Gx y νx → H(x) y νϕ(x).

Morita map iff isomomorphism at the transversal level

Transversal Geometry

Orbits and Isotropies

Given a Lie groupoid G ⇒M and x ∈M :

I isotropy group Gx = s−1 ∩ t−1(x) (Lie group)

I orbit Ox = {t(g) ∈M ; g ∈ s−1(x)} (immersed submanifold)

I orbit space M/G = {Ox;x ∈M} (topological space)

I normal representation Gx y νx

e.g. for GnM ⇒M , one gets the usual slice representation.

RemarkA morphism ϕ : G → H induces a Lie group morphismGx → Hϕ(x), a continuous map M/G → N/H and an intertwiningmap Gx y νx → H(x) y νϕ(x).

Morita map iff isomomorphism at the transversal level

Transversal Geometry

Orbits and Isotropies

Given a Lie groupoid G ⇒M and x ∈M :

I isotropy group Gx = s−1 ∩ t−1(x) (Lie group)

I orbit Ox = {t(g) ∈M ; g ∈ s−1(x)} (immersed submanifold)

I orbit space M/G = {Ox;x ∈M} (topological space)

I normal representation Gx y νx

e.g. for GnM ⇒M , one gets the usual slice representation.

RemarkA morphism ϕ : G → H induces a Lie group morphismGx → Hϕ(x), a continuous map M/G → N/H and an intertwiningmap Gx y νx → H(x) y νϕ(x).

Morita map iff isomomorphism at the transversal level

Transversal Geometry

Orbits and Isotropies

Given a Lie groupoid G ⇒M and x ∈M :

I isotropy group Gx = s−1 ∩ t−1(x) (Lie group)

I orbit Ox = {t(g) ∈M ; g ∈ s−1(x)} (immersed submanifold)

I orbit space M/G = {Ox;x ∈M} (topological space)

I normal representation Gx y νx

e.g. for GnM ⇒M , one gets the usual slice representation.

RemarkA morphism ϕ : G → H induces a Lie group morphismGx → Hϕ(x), a continuous map M/G → N/H and an intertwiningmap Gx y νx → H(x) y νϕ(x).

Morita map iff isomomorphism at the transversal level

Transversal Geometry

Orbits and Isotropies

Given a Lie groupoid G ⇒M and x ∈M :

I isotropy group Gx = s−1 ∩ t−1(x) (Lie group)

I orbit Ox = {t(g) ∈M ; g ∈ s−1(x)} (immersed submanifold)

I orbit space M/G = {Ox;x ∈M} (topological space)

I normal representation Gx y νx

e.g. for GnM ⇒M , one gets the usual slice representation.

RemarkA morphism ϕ : G → H induces a Lie group morphismGx → Hϕ(x), a continuous map M/G → N/H and an intertwiningmap Gx y νx → H(x) y νϕ(x).

Morita map iff isomomorphism at the transversal level

Transversal Geometry

Orbits and Isotropies

Given a Lie groupoid G ⇒M and x ∈M :

I isotropy group Gx = s−1 ∩ t−1(x) (Lie group)

I orbit Ox = {t(g) ∈M ; g ∈ s−1(x)} (immersed submanifold)

I orbit space M/G = {Ox;x ∈M} (topological space)

I normal representation Gx y νx

e.g. for GnM ⇒M , one gets the usual slice representation.

RemarkA morphism ϕ : G → H induces a Lie group morphismGx → Hϕ(x), a continuous map M/G → N/H and an intertwiningmap Gx y νx → H(x) y νϕ(x).

Morita map iff isomomorphism at the transversal level

VB-groupoids

Definition (due to Pradines)

A VB-groupoid is given by

Γs //

t//

q��

E

qE��

Gs //

t//M

(1)

structure maps of defining Γ, i.e. s, t, m, i, 1, are vector bundlemorphisms covering the corresponding struture maps of G, i.e.s, t,m, i, 1.

Categorification

VB-groupoids are categorified vector bundles. V B(G) category ofVB-groupoids over G.

VB-groupoids

Definition (due to Pradines)

A VB-groupoid is given by

Γs //

t//

q��

E

qE��

Gs //

t//M

(1)

structure maps of defining Γ, i.e. s, t, m, i, 1, are vector bundlemorphisms covering the corresponding struture maps of G, i.e.s, t,m, i, 1.

Categorification

VB-groupoids are categorified vector bundles. V B(G) category ofVB-groupoids over G.

Examples of VB-groupoids

1. TG ⇒ TM tangent groupoid T ∗G ⇒ Lie(G)∗

2. For ∗⇒ ∗, recover 2-vector spaces (higher gauge theories)

3. Representations: vector bundle E −→M with a linearaction of G ⇒M , i.e. morphism ∆ : G ⇒ GL(E). Thetransformation VB-groupoid is G n E ⇒ E.

4. Important representations: G y TM for an etale groupoid;GO y νO normal representation; Π(M) y E is just a flatvector bundle!

5. more general notion of representation up to homotopy(AriasAbad-Crainic, Gracia Saz-Mehta)

Operations on VB-groupoids

Direct sum, dualization, pullback along groupoid morphisms

RemarkExtension of the usual constructions for representations

Examples of VB-groupoids

1. TG ⇒ TM tangent groupoid T ∗G ⇒ Lie(G)∗

2. For ∗⇒ ∗, recover 2-vector spaces (higher gauge theories)

3. Representations: vector bundle E −→M with a linearaction of G ⇒M , i.e. morphism ∆ : G ⇒ GL(E). Thetransformation VB-groupoid is G n E ⇒ E.

4. Important representations: G y TM for an etale groupoid;GO y νO normal representation; Π(M) y E is just a flatvector bundle!

5. more general notion of representation up to homotopy(AriasAbad-Crainic, Gracia Saz-Mehta)

Operations on VB-groupoids

Direct sum, dualization, pullback along groupoid morphisms

RemarkExtension of the usual constructions for representations

Examples of VB-groupoids

1. TG ⇒ TM tangent groupoid T ∗G ⇒ Lie(G)∗

2. For ∗⇒ ∗, recover 2-vector spaces (higher gauge theories)

3. Representations: vector bundle E −→M with a linearaction of G ⇒M , i.e. morphism ∆ : G ⇒ GL(E). Thetransformation VB-groupoid is G n E ⇒ E.

4. Important representations: G y TM for an etale groupoid;GO y νO normal representation; Π(M) y E is just a flatvector bundle!

5. more general notion of representation up to homotopy(AriasAbad-Crainic, Gracia Saz-Mehta)

Operations on VB-groupoids

Direct sum, dualization, pullback along groupoid morphisms

RemarkExtension of the usual constructions for representations

Examples of VB-groupoids

1. TG ⇒ TM tangent groupoid T ∗G ⇒ Lie(G)∗

2. For ∗⇒ ∗, recover 2-vector spaces (higher gauge theories)

3. Representations: vector bundle E −→M with a linearaction of G ⇒M , i.e. morphism ∆ : G ⇒ GL(E). Thetransformation VB-groupoid is G n E ⇒ E.

4. Important representations: G y TM for an etale groupoid;GO y νO normal representation; Π(M) y E is just a flatvector bundle!

5. more general notion of representation up to homotopy(AriasAbad-Crainic, Gracia Saz-Mehta)

Operations on VB-groupoids

Direct sum, dualization, pullback along groupoid morphisms

RemarkExtension of the usual constructions for representations

Examples of VB-groupoids

1. TG ⇒ TM tangent groupoid T ∗G ⇒ Lie(G)∗

2. For ∗⇒ ∗, recover 2-vector spaces (higher gauge theories)

3. Representations: vector bundle E −→M with a linearaction of G ⇒M , i.e. morphism ∆ : G ⇒ GL(E). Thetransformation VB-groupoid is G n E ⇒ E.

4. Important representations: G y TM for an etale groupoid;GO y νO normal representation; Π(M) y E is just a flatvector bundle!

5. more general notion of representation up to homotopy(AriasAbad-Crainic, Gracia Saz-Mehta)

Operations on VB-groupoids

Direct sum, dualization, pullback along groupoid morphisms

RemarkExtension of the usual constructions for representations

Examples of VB-groupoids

1. TG ⇒ TM tangent groupoid T ∗G ⇒ Lie(G)∗

2. For ∗⇒ ∗, recover 2-vector spaces (higher gauge theories)

3. Representations: vector bundle E −→M with a linearaction of G ⇒M , i.e. morphism ∆ : G ⇒ GL(E). Thetransformation VB-groupoid is G n E ⇒ E.

4. Important representations: G y TM for an etale groupoid;GO y νO normal representation; Π(M) y E is just a flatvector bundle!

5. more general notion of representation up to homotopy(AriasAbad-Crainic, Gracia Saz-Mehta)

Operations on VB-groupoids

Direct sum, dualization, pullback along groupoid morphisms

RemarkExtension of the usual constructions for representations

Morita invariance of representations

Known resultϕ : G −→ H Morita map, then ϕ∗ : Rep(H)→ Rep(G)equivalence of categories

Geometric application (quotient vector bundles)

f : M → N surjective submersion, then

Vect(N) ∼= Rep(M ×f M),

the correspondence is given by B 7→ f∗B

RemarkB = TN is not obtained as a quotient of TM . In general TM isnot isomorphic to f∗TN .

Morita invariance of representations

Known resultϕ : G −→ H Morita map, then ϕ∗ : Rep(H)→ Rep(G)equivalence of categories

Geometric application (quotient vector bundles)

f : M → N surjective submersion, then

Vect(N) ∼= Rep(M ×f M),

the correspondence is given by B 7→ f∗B

RemarkB = TN is not obtained as a quotient of TM . In general TM isnot isomorphic to f∗TN .

Morita invariance of representations

Known resultϕ : G −→ H Morita map, then ϕ∗ : Rep(H)→ Rep(G)equivalence of categories

Geometric application (quotient vector bundles)

f : M → N surjective submersion, then

Vect(N) ∼= Rep(M ×f M),

the correspondence is given by B 7→ f∗B

RemarkB = TN is not obtained as a quotient of TM . In general TM isnot isomorphic to f∗TN .

Morita invariance of V B(G)?

Questionϕ : G → H Morita map, then ϕ∗ : V B(H)→ (G) equivalence ofcategories?

Not necessarily.

Example

f : S1 −→ ∗ is a surjective submersion. The natural morphismϕ : (S1 × S1 ⇒ S1)→ (∗⇒ ∗) is Morita map. But

ϕ∗ : 2-Vect→ V B(S1 × S1),

is not essentialy surjective. The Mobius VB-groupoid is notisomorphic to a pullback!

Morita invariance of V B(G)?

Questionϕ : G → H Morita map, then ϕ∗ : V B(H)→ (G) equivalence ofcategories?

Not necessarily.

Example

f : S1 −→ ∗ is a surjective submersion. The natural morphismϕ : (S1 × S1 ⇒ S1)→ (∗⇒ ∗) is Morita map. But

ϕ∗ : 2-Vect→ V B(S1 × S1),

is not essentialy surjective. The Mobius VB-groupoid is notisomorphic to a pullback!

Morita invariance of V B(G)?

Questionϕ : G → H Morita map, then ϕ∗ : V B(H)→ (G) equivalence ofcategories?

Not necessarily.

Example

f : S1 −→ ∗ is a surjective submersion. The natural morphismϕ : (S1 × S1 ⇒ S1)→ (∗⇒ ∗) is Morita map. But

ϕ∗ : 2-Vect→ V B(S1 × S1),

is not essentialy surjective. The Mobius VB-groupoid is notisomorphic to a pullback!

VB-Morita maps

DefinitionGiven Γ ∈ V B(G) and Γ′ ∈ (G′) a VB-Morita map Γ→ Γ′ is agroupoid morphism Φ : Γ→ Γ′ which is a Morita map ofgroupoids and also a vector bundle morphism.

Example

A 2-vect V1 ⇒ V0 is equivalent to a 2-term complex of vectorspaces ∂ : C → E. A VB-Morita map between 2-vectsΦ : (V1 ⇒ V0)→ (V ′0 ⇒ V ′1) corresponds to a quasi-isomorphismof complexes.

VB-Morita maps

DefinitionGiven Γ ∈ V B(G) and Γ′ ∈ (G′) a VB-Morita map Γ→ Γ′ is agroupoid morphism Φ : Γ→ Γ′ which is a Morita map ofgroupoids and also a vector bundle morphism.

Example

A 2-vect V1 ⇒ V0 is equivalent to a 2-term complex of vectorspaces ∂ : C → E. A VB-Morita map between 2-vectsΦ : (V1 ⇒ V0)→ (V ′0 ⇒ V ′1) corresponds to a quasi-isomorphismof complexes.

Characterization of VB-Morita maps

TheoremΦ : Γ→ Γ′ is VB-Morita iff the base map ϕ : G → G′ is Moritaand the induced chain map φ : (C → E)→ (C ′ → E′) is aquasi-isomorphism.

Examples

1. ϕ : G → G′ is a Morita map iff Tϕ : TG −→ TG′ is aVB-Morita map

2. duals of VB-Morita maps are VB-Morita maps

3. ϕ : G → G′ Morita map and Γ′ ∈ V B(G′), then the inducedmap ϕ∗Γ′ → Γ′ is VB-Morita

4. ϕ : G → H Morita map, then ϕ∗(T ∗H)→ T ∗G is VB-Morita

Characterization of VB-Morita maps

TheoremΦ : Γ→ Γ′ is VB-Morita iff the base map ϕ : G → G′ is Moritaand the induced chain map φ : (C → E)→ (C ′ → E′) is aquasi-isomorphism.

Examples

1. ϕ : G → G′ is a Morita map iff Tϕ : TG −→ TG′ is aVB-Morita map

2. duals of VB-Morita maps are VB-Morita maps

3. ϕ : G → G′ Morita map and Γ′ ∈ V B(G′), then the inducedmap ϕ∗Γ′ → Γ′ is VB-Morita

4. ϕ : G → H Morita map, then ϕ∗(T ∗H)→ T ∗G is VB-Morita

Characterization of VB-Morita maps

TheoremΦ : Γ→ Γ′ is VB-Morita iff the base map ϕ : G → G′ is Moritaand the induced chain map φ : (C → E)→ (C ′ → E′) is aquasi-isomorphism.

Examples

1. ϕ : G → G′ is a Morita map iff Tϕ : TG −→ TG′ is aVB-Morita map

2. duals of VB-Morita maps are VB-Morita maps

3. ϕ : G → G′ Morita map and Γ′ ∈ V B(G′), then the inducedmap ϕ∗Γ′ → Γ′ is VB-Morita

4. ϕ : G → H Morita map, then ϕ∗(T ∗H)→ T ∗G is VB-Morita

Characterization of VB-Morita maps

TheoremΦ : Γ→ Γ′ is VB-Morita iff the base map ϕ : G → G′ is Moritaand the induced chain map φ : (C → E)→ (C ′ → E′) is aquasi-isomorphism.

Examples

1. ϕ : G → G′ is a Morita map iff Tϕ : TG −→ TG′ is aVB-Morita map

2. duals of VB-Morita maps are VB-Morita maps

3. ϕ : G → G′ Morita map and Γ′ ∈ V B(G′), then the inducedmap ϕ∗Γ′ → Γ′ is VB-Morita

4. ϕ : G → H Morita map, then ϕ∗(T ∗H)→ T ∗G is VB-Morita

Characterization of VB-Morita maps

TheoremΦ : Γ→ Γ′ is VB-Morita iff the base map ϕ : G → G′ is Moritaand the induced chain map φ : (C → E)→ (C ′ → E′) is aquasi-isomorphism.

Examples

1. ϕ : G → G′ is a Morita map iff Tϕ : TG −→ TG′ is aVB-Morita map

2. duals of VB-Morita maps are VB-Morita maps

3. ϕ : G → G′ Morita map and Γ′ ∈ V B(G′), then the inducedmap ϕ∗Γ′ → Γ′ is VB-Morita

4. ϕ : G → H Morita map, then ϕ∗(T ∗H)→ T ∗G is VB-Morita

More examples

Quotient bundles revisitedϕ : (M ×f M ⇒M)→ (N ⇒ N) is a Morita map. ThenTϕ : (TM ×Tf TM ⇒ TM)→ (TN ⇒ TN) is VB-Morita.Hence,

T (M ×f M) ∼=V B ϕ∗(TN)

Mobius VB-groupoid

Actually, E × E ⇒ E is VB-Morita equivalent to the pullback of a2-vect (V1 ⇒ V0).

More examples

Quotient bundles revisitedϕ : (M ×f M ⇒M)→ (N ⇒ N) is a Morita map. ThenTϕ : (TM ×Tf TM ⇒ TM)→ (TN ⇒ TN) is VB-Morita.Hence,

T (M ×f M) ∼=V B ϕ∗(TN)

Mobius VB-groupoid

Actually, E × E ⇒ E is VB-Morita equivalent to the pullback of a2-vect (V1 ⇒ V0).

More examples

Quotient bundles revisitedϕ : (M ×f M ⇒M)→ (N ⇒ N) is a Morita map. ThenTϕ : (TM ×Tf TM ⇒ TM)→ (TN ⇒ TN) is VB-Morita.Hence,

T (M ×f M) ∼=V B ϕ∗(TN)

Mobius VB-groupoid

Actually, E × E ⇒ E is VB-Morita equivalent to the pullback of a2-vect (V1 ⇒ V0).

Applications

Cohomology

Proposition

H•V B(Γ) is invariant by VB-Morita maps

Homotopy representations as coefficients for cohomologies

Gracia Saz-Mehta: H•V B(Γ) ∼= H•(G, EΓ)e.g. H•V B(TG) ∼= H•(G, AdG)

Corollary

H•(G, AdG) is Morita invariant

Since H•(G, AdG) ∼= H•def (G), the above gives a geometric proofof results on deformation theory (Crainic-Mestre-Struchiner)

Applications

Cohomology

Proposition

H•V B(Γ) is invariant by VB-Morita maps

Homotopy representations as coefficients for cohomologies

Gracia Saz-Mehta: H•V B(Γ) ∼= H•(G, EΓ)e.g. H•V B(TG) ∼= H•(G, AdG)

Corollary

H•(G, AdG) is Morita invariant

Since H•(G, AdG) ∼= H•def (G), the above gives a geometric proofof results on deformation theory (Crainic-Mestre-Struchiner)

Ongoing project: Vector bundles over stacks

Previous workGinot-Noohi-Xu, vector bundles over stack. But TX→ X is not anexample!

Another viewpoint

V B(G) :=V B(G)

[VB-Morita maps]

Example

XG the stack associated to G ⇒M . Then the VB-Morita class ofTG ⇒ TM represents the tangent stack TX→ X

Obrigado!