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� � 2. First order processes ◮ Most basic: manifold M to TM , Lie group G to Lie algebra g . M × M has Lie algebroid TM ◮ ◮ Foliation F on M to tangent distribution. ◮ Holonomy/Monodromy groupoids of F have Lie algebroid T ( F ) ◮ Group action G × M → M to infinitesimal action g → X ( M ) . ◮ Action groupoid G < M ⇒ M has action Lie algebroid g < M ◮ Principal bundle P ( M , G ) to Atiyah sequence TP G ◮ Gauge groupoid P × P TP has Lie algebroid G G ◮ Parallel translation in vector bundle E on M to connection ∇ in E . ◮ Frame groupoid of all isomorphisms between fibres has Lie algebroid for which sections are all ∇ X for all ∇ and all X ∈ X ( M ) (and all ∇ X − ∇ ′ X ) ◮ . . . All are instances of the process Lie groupoid = ⇒ Lie algebroid G ⇒ M A G There are double and multiple versions of this.

� � 2. First order processes ◮ Most basic: manifold M to TM , Lie group G to Lie algebra g . M × M has Lie algebroid TM ◮ ◮ Foliation F on M to tangent distribution. ◮ Holonomy/Monodromy groupoids of F have Lie algebroid T ( F ) ◮ Group action G × M → M to infinitesimal action g → X ( M ) . ◮ Action groupoid G < M ⇒ M has action Lie algebroid g < M ◮ Principal bundle P ( M , G ) to Atiyah sequence TP G ◮ Gauge groupoid P × P TP has Lie algebroid G G ◮ Parallel translation in vector bundle E on M to connection ∇ in E . ◮ Frame groupoid of all isomorphisms between fibres has Lie algebroid for which sections are all ∇ X for all ∇ and all X ∈ X ( M ) (and all ∇ X − ∇ ′ X ) ◮ . . . All are instances of the process Lie groupoid = ⇒ Lie algebroid G ⇒ M A G There are double and multiple versions of this.

� � 2. First order processes ◮ Most basic: manifold M to TM , Lie group G to Lie algebra g . M × M has Lie algebroid TM ◮ ◮ Foliation F on M to tangent distribution. ◮ Holonomy/Monodromy groupoids of F have Lie algebroid T ( F ) ◮ Group action G × M → M to infinitesimal action g → X ( M ) . ◮ Action groupoid G < M ⇒ M has action Lie algebroid g < M ◮ Principal bundle P ( M , G ) to Atiyah sequence TP G ◮ Gauge groupoid P × P TP has Lie algebroid G G ◮ Parallel translation in vector bundle E on M to connection ∇ in E . ◮ Frame groupoid of all isomorphisms between fibres has Lie algebroid for which sections are all ∇ X for all ∇ and all X ∈ X ( M ) (and all ∇ X − ∇ ′ X ) ◮ . . . All are instances of the process Lie groupoid = ⇒ Lie algebroid G ⇒ M A G There are double and multiple versions of this.

� � 2. First order processes ◮ Most basic: manifold M to TM , Lie group G to Lie algebra g . M × M has Lie algebroid TM ◮ ◮ Foliation F on M to tangent distribution. ◮ Holonomy/Monodromy groupoids of F have Lie algebroid T ( F ) ◮ Group action G × M → M to infinitesimal action g → X ( M ) . ◮ Action groupoid G < M ⇒ M has action Lie algebroid g < M ◮ Principal bundle P ( M , G ) to Atiyah sequence TP G ◮ Gauge groupoid P × P TP has Lie algebroid G G ◮ Parallel translation in vector bundle E on M to connection ∇ in E . ◮ Frame groupoid of all isomorphisms between fibres has Lie algebroid for which sections are all ∇ X for all ∇ and all X ∈ X ( M ) (and all ∇ X − ∇ ′ X ) ◮ . . . All are instances of the process Lie groupoid = ⇒ Lie algebroid G ⇒ M A G There are double and multiple versions of this.

� � 2. First order processes ◮ Most basic: manifold M to TM , Lie group G to Lie algebra g . M × M has Lie algebroid TM ◮ ◮ Foliation F on M to tangent distribution. ◮ Holonomy/Monodromy groupoids of F have Lie algebroid T ( F ) ◮ Group action G × M → M to infinitesimal action g → X ( M ) . ◮ Action groupoid G < M ⇒ M has action Lie algebroid g < M ◮ Principal bundle P ( M , G ) to Atiyah sequence TP G ◮ Gauge groupoid P × P TP has Lie algebroid G G ◮ Parallel translation in vector bundle E on M to connection ∇ in E . ◮ Frame groupoid of all isomorphisms between fibres has Lie algebroid for which sections are all ∇ X for all ∇ and all X ∈ X ( M ) (and all ∇ X − ∇ ′ X ) ◮ . . . All are instances of the process Lie groupoid = ⇒ Lie algebroid G ⇒ M A G There are double and multiple versions of this.

� � 2. First order processes ◮ Most basic: manifold M to TM , Lie group G to Lie algebra g . M × M has Lie algebroid TM ◮ ◮ Foliation F on M to tangent distribution. ◮ Holonomy/Monodromy groupoids of F have Lie algebroid T ( F ) ◮ Group action G × M → M to infinitesimal action g → X ( M ) . ◮ Action groupoid G < M ⇒ M has action Lie algebroid g < M ◮ Principal bundle P ( M , G ) to Atiyah sequence TP G ◮ Gauge groupoid P × P TP has Lie algebroid G G ◮ Parallel translation in vector bundle E on M to connection ∇ in E . ◮ Frame groupoid of all isomorphisms between fibres has Lie algebroid for which sections are all ∇ X for all ∇ and all X ∈ X ( M ) (and all ∇ X − ∇ ′ X ) ◮ . . . All are instances of the process Lie groupoid = ⇒ Lie algebroid G ⇒ M A G There are double and multiple versions of this.

� � �� � � � � � � � � � � � � � � 3. Double Lie groupoids The elements of a double Lie groupoid S are ‘squares’ which have horizontal sides from a Lie groupoid H ⇒ M and vertical sides from a Lie groupoid V ⇒ M , with corner points from a manifold M . h 2 S V v 2 s v 1 � � � � �� M H h 1 Horizontal composition (when v ′ 1 = v 2 ) has vertical sources and targets as follows : h ′ h ′ 2 h 2 h 2 2 s ′ · H s v 2 s v 1 v ′ s ′ v ′ v ′ v 1 2 1 2 h 1 h ′ h ′ 1 h 1 1

� � �� � � � � � � � � � � � � � � 3. Double Lie groupoids The elements of a double Lie groupoid S are ‘squares’ which have horizontal sides from a Lie groupoid H ⇒ M and vertical sides from a Lie groupoid V ⇒ M , with corner points from a manifold M . h 2 S V v 2 s v 1 � � � � �� M H h 1 Horizontal composition (when v ′ 1 = v 2 ) has vertical sources and targets as follows : h ′ h ′ 2 h 2 h 2 2 s ′ · H s v 2 s v 1 v ′ s ′ v ′ v ′ v 1 2 1 2 h 1 h ′ h ′ 1 h 1 1

� � �� � � � � � � � � � � � � � � 3. Double Lie groupoids The elements of a double Lie groupoid S are ‘squares’ which have horizontal sides from a Lie groupoid H ⇒ M and vertical sides from a Lie groupoid V ⇒ M , with corner points from a manifold M . h 2 S V v 2 s v 1 � � � � �� M H h 1 Horizontal composition (when v ′ 1 = v 2 ) has vertical sources and targets as follows : h ′ h ′ 2 h 2 h 2 2 s ′ · H s v 2 s v 1 v ′ s ′ v ′ v ′ v 1 2 1 2 h 1 h ′ h ′ 1 h 1 1

� � �� � � � � � � � � � � � � � � 3. Double Lie groupoids The elements of a double Lie groupoid S are ‘squares’ which have horizontal sides from a Lie groupoid H ⇒ M and vertical sides from a Lie groupoid V ⇒ M , with corner points from a manifold M . h 2 S V v 2 s v 1 � � � � �� M H h 1 Horizontal composition (when v ′ 1 = v 2 ) has vertical sources and targets as follows : h ′ h ′ 2 h 2 h 2 2 s ′ · H s v 2 s v 1 v ′ s ′ v ′ v ′ v 1 2 1 2 h 1 h ′ h ′ 1 h 1 1

� � �� � � � � � � � � � � � � � � 3. Double Lie groupoids The elements of a double Lie groupoid S are ‘squares’ which have horizontal sides from a Lie groupoid H ⇒ M and vertical sides from a Lie groupoid V ⇒ M , with corner points from a manifold M . h 2 S V v 2 s v 1 � � � � �� M H h 1 Horizontal composition (when v ′ 1 = v 2 ) has vertical sources and targets as follows : h ′ h ′ 2 h 2 h 2 2 s ′ · H s v 2 s v 1 v ′ s ′ v ′ v ′ v 1 2 1 2 h 1 h ′ h ′ 1 h 1 1

� � �� � � � � � � � � � � � � � � 3. Double Lie groupoids The elements of a double Lie groupoid S are ‘squares’ which have horizontal sides from a Lie groupoid H ⇒ M and vertical sides from a Lie groupoid V ⇒ M , with corner points from a manifold M . h 2 S V v 2 s v 1 � � � � �� M H h 1 Horizontal composition (when v ′ 1 = v 2 ) has vertical sources and targets as follows : h ′ h ′ 2 h 2 h 2 2 s ′ · H s v 2 s v 1 v ′ s ′ v ′ v ′ v 1 2 1 2 h 1 h ′ h ′ 1 h 1 1

� � �� � � � � � � � � � � � � � � 3. Double Lie groupoids The elements of a double Lie groupoid S are ‘squares’ which have horizontal sides from a Lie groupoid H ⇒ M and vertical sides from a Lie groupoid V ⇒ M , with corner points from a manifold M . h 2 S V v 2 s v 1 � � � � �� M H h 1 Horizontal composition (when v ′ 1 = v 2 ) has vertical sources and targets as follows : h ′ h ′ 2 h 2 h 2 2 s ′ · H s v 2 s v 1 v ′ s ′ v ′ v ′ v 1 2 1 2 h 1 h ′ h ′ 1 h 1 1

� � � � � � � � � � � � � � � � 4. Double Lie groupoids, p2 The main compatibility condition between the two structures is that products of the form s 2 s 1 s 3 s 4 are well-defined: composing each row horizontally and then the results vertically and composing each column vertically and then the results horizontally give the same result.

� � � � � � � � � � � � � � � � 4. Double Lie groupoids, p2 The main compatibility condition between the two structures is that products of the form s 2 s 1 s 3 s 4 are well-defined: composing each row horizontally and then the results vertically and composing each column vertically and then the results horizontally give the same result.

� � � � � � � � � � � � � � � � 4. Double Lie groupoids, p2 The main compatibility condition between the two structures is that products of the form s 2 s 1 s 3 s 4 are well-defined: composing each row horizontally and then the results vertically and composing each column vertically and then the results horizontally give the same result.

� � � � � � � � � � � � � � � � 4. Double Lie groupoids, p2 The main compatibility condition between the two structures is that products of the form s 2 s 1 s 3 s 4 are well-defined: composing each row horizontally and then the results vertically and composing each column vertically and then the results horizontally give the same result.

� � � � � � � � � � � � � � � � 4. Double Lie groupoids, p2 The main compatibility condition between the two structures is that products of the form s 2 s 1 s 3 s 4 are well-defined: composing each row horizontally and then the results vertically and composing each column vertically and then the results horizontally give the same result.

� � �� � � � � � � � �� 5. Lie algebroids of a double Lie groupoid Given a double Lie groupoid, one S V can take the Lie algebroid of either groupoid structure on S . �� M H Take the Lie algebroid of the A V S AV vertical structure; the horizontal groupoid structure prolongs to the vertical Lie algebroid. �� M H Take the Lie algebroid of the A H ( A V S ) AV horizontal groupoid. � M AH A H ( A V S ) is a Lie algebroid over base AV . The vertical structure A H ( A V S ) → AH is at present just a vector bundle.

� � �� � � � � � � � �� 5. Lie algebroids of a double Lie groupoid Given a double Lie groupoid, one S V can take the Lie algebroid of either groupoid structure on S . �� M H Take the Lie algebroid of the A V S AV vertical structure; the horizontal groupoid structure prolongs to the vertical Lie algebroid. �� M H Take the Lie algebroid of the A H ( A V S ) AV horizontal groupoid. � M AH A H ( A V S ) is a Lie algebroid over base AV . The vertical structure A H ( A V S ) → AH is at present just a vector bundle.

� � �� � � � � � � � �� 5. Lie algebroids of a double Lie groupoid Given a double Lie groupoid, one S V can take the Lie algebroid of either groupoid structure on S . �� M H Take the Lie algebroid of the A V S AV vertical structure; the horizontal groupoid structure prolongs to the vertical Lie algebroid. �� M H Take the Lie algebroid of the A H ( A V S ) AV horizontal groupoid. � M AH A H ( A V S ) is a Lie algebroid over base AV . The vertical structure A H ( A V S ) → AH is at present just a vector bundle.

� � �� � � � � � � � �� 5. Lie algebroids of a double Lie groupoid Given a double Lie groupoid, one S V can take the Lie algebroid of either groupoid structure on S . �� M H Take the Lie algebroid of the A V S AV vertical structure; the horizontal groupoid structure prolongs to the vertical Lie algebroid. �� M H Take the Lie algebroid of the A H ( A V S ) AV horizontal groupoid. � M AH A H ( A V S ) is a Lie algebroid over base AV . The vertical structure A H ( A V S ) → AH is at present just a vector bundle.

� � �� � � � � � � � �� 5. Lie algebroids of a double Lie groupoid Given a double Lie groupoid, one S V can take the Lie algebroid of either groupoid structure on S . �� M H Take the Lie algebroid of the A V S AV vertical structure; the horizontal groupoid structure prolongs to the vertical Lie algebroid. �� M H Take the Lie algebroid of the A H ( A V S ) AV horizontal groupoid. � M AH A H ( A V S ) is a Lie algebroid over base AV . The vertical structure A H ( A V S ) → AH is at present just a vector bundle.

� � � � � � � �� � � � �� � � � � � �� � � � 6. Lie algebroids of a double Lie groupoid, p2 Recap from previous frame: Now do it the other way: S V S V �� M �� M H H � V A V S AV A H S � � �� M � M H AH A H ( A V S ) A V ( A H S ) AV AV � M � M AH AH

� � � � � � � �� � � � �� � � � � � �� � � � 6. Lie algebroids of a double Lie groupoid, p2 Recap from previous frame: Now do it the other way: S V S V �� M �� M H H � V A V S AV A H S � � �� M � M H AH A H ( A V S ) A V ( A H S ) AV AV � M � M AH AH

� � � � � � � �� � � � �� � � � � � �� � � � 6. Lie algebroids of a double Lie groupoid, p2 Recap from previous frame: Now do it the other way: S V S V �� M �� M H H � V A V S AV A H S � � �� M � M H AH A H ( A V S ) A V ( A H S ) AV AV � M � M AH AH

� � � � � � � �� � � � �� � � � � � �� � � � 6. Lie algebroids of a double Lie groupoid, p2 Recap from previous frame: Now do it the other way: S V S V �� M �� M H H � V A V S AV A H S � � �� M � M H AH A H ( A V S ) A V ( A H S ) AV AV � M � M AH AH

� � � � � � � �� � � � �� � � � � � �� � � � 6. Lie algebroids of a double Lie groupoid, p2 Recap from previous frame: Now do it the other way: S V S V �� M �� M H H � V A V S AV A H S � � �� M � M H AH A H ( A V S ) A V ( A H S ) AV AV � M � M AH AH Every manifold has a canonical involution T 2 S → T 2 S which ‘interchanges the order of differentiation’.

� � � � � � � �� � � � �� � � � � � �� � � � 6. Lie algebroids of a double Lie groupoid, p2 Recap from previous frame: Now do it the other way: S V S V �� M �� M H H � V A V S AV A H S � � �� M � M H AH A H ( A V S ) A V ( A H S ) AV AV � M � M AH AH Every manifold has a canonical involution T 2 S → T 2 S which ‘interchanges the order of differentiation’. It restricts to a diffeomorphism A H ( A V S ) ∼ = A V ( A H S ) .

� � � � � � � �� � � � �� � � � � � �� � � � 6. Lie algebroids of a double Lie groupoid, p2 Recap from previous frame: Now do it the other way: S V S V �� M �� M H H � V A V S AV A H S � � �� M � M H AH A H ( A V S ) A V ( A H S ) AV AV � M � M AH AH Every manifold has a canonical involution T 2 S → T 2 S which ‘interchanges the order of differentiation’. It restricts to a diffeomorphism A H ( A V S ) ∼ = A V ( A H S ) . Use this to transfer one structure to the other.

� � � � � � � �� � � � �� � � � � � �� � � � 6. Lie algebroids of a double Lie groupoid, p2 Recap from previous frame: Now do it the other way: S V S V �� M �� M H H � V A V S AV A H S � � �� M � M H AH A H ( A V S ) AV A V ( A H S ) AV � M � M AH AH Every manifold has a canonical involution T 2 S → T 2 S which ‘interchanges the order of differentiation’. It restricts to a diffeomorphism A H ( A V S ) ∼ = A V ( A H S ) . Use this to transfer one structure to the other. The result is the double Lie algebroid of S .

� � � � � � � � � � �� � � � � � �� � � � �� 7. Basic example For G ⇒ M any Lie groupoid, take S = G × G G × G G G × G G �� M �� M M × M M × M � G AG × AG AG TG � � �� M � M TM M × M T ( AG ) AG A ( TG ) AG � M � M TM TM

� � � � � � � � � � �� � � � � � �� � � � �� 7. Basic example For G ⇒ M any Lie groupoid, take S = G × G G × G G G × G G �� M �� M M × M M × M � G AG × AG AG TG � � �� M � M TM M × M T ( AG ) AG A ( TG ) AG � M � M TM TM

� � � � � � � � � � �� � � � � � �� � � � �� 7. Basic example For G ⇒ M any Lie groupoid, take S = G × G G × G G G × G G �� M �� M M × M M × M � G AG × AG AG TG � � �� M � M TM M × M T ( AG ) AG A ( TG ) AG � M � M TM TM

� � � � � � � � � � �� � � � � � �� � � � �� 7. Basic example For G ⇒ M any Lie groupoid, take S = G × G G × G G G × G G �� M �� M M × M M × M � G AG × AG AG TG � � �� M � M TM M × M T ( AG ) AG A ( TG ) AG � M � M TM TM

� � � � � � � � � � �� � � � � � �� � � � �� 7. Basic example For G ⇒ M any Lie groupoid, take S = G × G G × G G G × G G �� M �� M M × M M × M � G AG × AG AG TG � � �� M � M TM M × M T ( AG ) AG A ( TG ) AG � M � M TM TM

� � � � � � � � � � �� � � � � � �� � � � �� 7. Basic example For G ⇒ M any Lie groupoid, take S = G × G G × G G G × G G �� M �� M M × M M × M � G AG × AG AG TG � � �� M � M TM M × M T ( AG ) AG A ( TG ) AG � M � M TM TM

� � � � � � � � � � �� � � � � � �� � � � �� 7. Basic example For G ⇒ M any Lie groupoid, take S = G × G G × G G G × G G �� M �� M M × M M × M � G AG × AG AG TG � � �� M � M TM M × M T ( AG ) AG A ( TG ) AG � M � M TM TM

� � � � � � � � �� � � �� � � � � � �� � � � 7. Basic example For G ⇒ M any Lie groupoid, take S = G × G G × G G G × G G �� M �� M M × M M × M � G AG × AG AG TG � � �� M � M TM M × M T ( AG ) AG A ( TG ) AG � M � M TM TM There is a canonical diffeomorphism T ( AG ) ∼ = A ( TG ) .

� � � � � � � � 8. In particular . . . Put G = M × M . Then the preceding example is S = M 4 and the two forms of the double Lie algebroid are T ( p ) � p TM � T ( TM ) TM T ( TM ) TM T ( p ) � p p TM � � M � M TM TM p and the canonical diffeomorphism T 2 M → T 2 M is the standard ‘interchange of order of differentiation’ J which also interchanges the bundle structures on T 2 M . T ( p ) � T 2 M TM p TM J p TM � M T 2 M TM TM T ( p ) � M TM

� � � � � � � � 8. In particular . . . Put G = M × M . Then the preceding example is S = M 4 and the two forms of the double Lie algebroid are T ( p ) � p TM � T ( TM ) TM T ( TM ) TM T ( p ) � p p TM � � M � M TM TM p and the canonical diffeomorphism T 2 M → T 2 M is the standard ‘interchange of order of differentiation’ J which also interchanges the bundle structures on T 2 M . T ( p ) � T 2 M TM p TM J p TM � M T 2 M TM TM T ( p ) � M TM

� � � � � � � � 8. In particular . . . Put G = M × M . Then the preceding example is S = M 4 and the two forms of the double Lie algebroid are T ( p ) � p TM � T ( TM ) TM T ( TM ) TM T ( p ) � p p TM � � M � M TM TM p and the canonical diffeomorphism T 2 M → T 2 M is the standard ‘interchange of order of differentiation’ J which also interchanges the bundle structures on T 2 M . T ( p ) � T 2 M TM p TM J p TM � M T 2 M TM TM T ( p ) � M TM

� � � � � � � � 8. In particular . . . Put G = M × M . Then the preceding example is S = M 4 and the two forms of the double Lie algebroid are T ( p ) � p TM � T ( TM ) TM T ( TM ) TM T ( p ) � p p TM � � M � M TM TM p and the canonical diffeomorphism T 2 M → T 2 M is the standard ‘interchange of order of differentiation’ J which also interchanges the bundle structures on T 2 M . T ( p ) � T 2 M TM p TM J p TM � M T 2 M TM TM T ( p ) � M TM

� � � � � � � � 8. In particular . . . Put G = M × M . Then the preceding example is S = M 4 and the two forms of the double Lie algebroid are T ( p ) � p TM � T ( TM ) TM T ( TM ) TM T ( p ) � p p TM � � M � M TM TM p and the canonical diffeomorphism T 2 M → T 2 M is the standard ‘interchange of order of differentiation’ J which also interchanges the bundle structures on T 2 M . T ( p ) � T 2 M TM p TM J p TM � M T 2 M TM TM T ( p ) � M TM

� � � � � � � � 8. In particular . . . Put G = M × M . Then the preceding example is S = M 4 and the two forms of the double Lie algebroid are T ( p ) � p TM � T ( TM ) TM T ( TM ) TM T ( p ) � p p TM � � M � M TM TM p and the canonical diffeomorphism T 2 M → T 2 M is the standard ‘interchange of order of differentiation’ J which also interchanges the bundle structures on T 2 M . T ( p ) � T 2 M TM p TM J p TM � M T 2 M TM TM T ( p ) � M TM

� 9. Local representation p TM � Take ξ ∈ T 2 M with projections ξ Y T ( p ) � p p � m X If X = 0 then ξ is vertical and if Y = 0 then ξ is at a zero. So if X = Y = 0 then ξ can be identified with an element Z of TM . Represent elements of T 2 M ‘locally’ as ( X , Y , Z ) where the Z is called a core element . Write T 2 M ‘locally’ as TM ∗ TM ∗ TM . Then J : T 2 M → T 2 M is ‘locally’, J ( X , Y , Z ) = ( Y , X , Z ) .

� 9. Local representation p TM � Take ξ ∈ T 2 M with projections ξ Y T ( p ) � p p � m X If X = 0 then ξ is vertical and if Y = 0 then ξ is at a zero. So if X = Y = 0 then ξ can be identified with an element Z of TM . Represent elements of T 2 M ‘locally’ as ( X , Y , Z ) where the Z is called a core element . Write T 2 M ‘locally’ as TM ∗ TM ∗ TM . Then J : T 2 M → T 2 M is ‘locally’, J ( X , Y , Z ) = ( Y , X , Z ) .

� 9. Local representation p TM � Take ξ ∈ T 2 M with projections ξ Y T ( p ) � p p � m X If X = 0 then ξ is vertical and if Y = 0 then ξ is at a zero. So if X = Y = 0 then ξ can be identified with an element Z of TM . Represent elements of T 2 M ‘locally’ as ( X , Y , Z ) where the Z is called a core element . Write T 2 M ‘locally’ as TM ∗ TM ∗ TM . Then J : T 2 M → T 2 M is ‘locally’, J ( X , Y , Z ) = ( Y , X , Z ) .

� 9. Local representation p TM � Take ξ ∈ T 2 M with projections ξ Y T ( p ) � p p � m X If X = 0 then ξ is vertical and if Y = 0 then ξ is at a zero. So if X = Y = 0 then ξ can be identified with an element Z of TM . Represent elements of T 2 M ‘locally’ as ( X , Y , Z ) where the Z is called a core element . Write T 2 M ‘locally’ as TM ∗ TM ∗ TM . Then J : T 2 M → T 2 M is ‘locally’, J ( X , Y , Z ) = ( Y , X , Z ) .

� 9. Local representation p TM � Take ξ ∈ T 2 M with projections ξ Y T ( p ) � p p � m X If X = 0 then ξ is vertical and if Y = 0 then ξ is at a zero. So if X = Y = 0 then ξ can be identified with an element Z of TM . Represent elements of T 2 M ‘locally’ as ( X , Y , Z ) where the Z is called a core element . Write T 2 M ‘locally’ as TM ∗ TM ∗ TM . Then J : T 2 M → T 2 M is ‘locally’, J ( X , Y , Z ) = ( Y , X , Z ) .

� 9. Local representation p TM � Take ξ ∈ T 2 M with projections ξ Y T ( p ) � p p � m X If X = 0 then ξ is vertical and if Y = 0 then ξ is at a zero. So if X = Y = 0 then ξ can be identified with an element Z of TM . Represent elements of T 2 M ‘locally’ as ( X , Y , Z ) where the Z is called a core element . Write T 2 M ‘locally’ as TM ∗ TM ∗ TM . Then J : T 2 M → T 2 M is ‘locally’, J ( X , Y , Z ) = ( Y , X , Z ) .

� � � � 10. Local representation, p2 More generally, for any vector bundle E on M , there is a double vector bundle p E � TE E T ( q ) � q � M TM p p TM � Write elements as e ξ T ( q ) � q p � m X If X = 0 and e = 0 then ξ can be identified with an element of E . Write TE ‘locally’ as TM ∗ E ∗ E and elements as ( X , e 1 , e 2 ) . The e 2 is the core element. p E written locally as E ∗ E ∗ ∗ T ∗ M . Now dualize TE over E and we get T ∗ E E T ( q ) � q � M E ∗ p The core is now T ∗ M ,

� � � � 10. Local representation, p2 More generally, for any vector bundle E on M , there is a double vector bundle p E � TE E T ( q ) � q � M TM p p TM � Write elements as e ξ T ( q ) � q p � m X If X = 0 and e = 0 then ξ can be identified with an element of E . Write TE ‘locally’ as TM ∗ E ∗ E and elements as ( X , e 1 , e 2 ) . The e 2 is the core element. p E written locally as E ∗ E ∗ ∗ T ∗ M . Now dualize TE over E and we get T ∗ E E T ( q ) � q � M E ∗ p The core is now T ∗ M ,

� � � � 10. Local representation, p2 More generally, for any vector bundle E on M , there is a double vector bundle p E � TE E T ( q ) � q � M TM p p TM � Write elements as e ξ T ( q ) � q p � m X If X = 0 and e = 0 then ξ can be identified with an element of E . Write TE ‘locally’ as TM ∗ E ∗ E and elements as ( X , e 1 , e 2 ) . The e 2 is the core element. p E written locally as E ∗ E ∗ ∗ T ∗ M . Now dualize TE over E and we get T ∗ E E T ( q ) � q � M E ∗ p The core is now T ∗ M ,

� � � � 10. Local representation, p2 More generally, for any vector bundle E on M , there is a double vector bundle p E � TE E T ( q ) � q � M TM p p TM � Write elements as e ξ T ( q ) � q p � m X If X = 0 and e = 0 then ξ can be identified with an element of E . Write TE ‘locally’ as TM ∗ E ∗ E and elements as ( X , e 1 , e 2 ) . The e 2 is the core element. p E written locally as E ∗ E ∗ ∗ T ∗ M . Now dualize TE over E and we get T ∗ E E T ( q ) � q � M E ∗ p The core is now T ∗ M ,

� � � � 10. Local representation, p2 More generally, for any vector bundle E on M , there is a double vector bundle p E � TE E T ( q ) � q � M TM p p TM � Write elements as e ξ T ( q ) � q p � m X If X = 0 and e = 0 then ξ can be identified with an element of E . Write TE ‘locally’ as TM ∗ E ∗ E and elements as ( X , e 1 , e 2 ) . The e 2 is the core element. p E written locally as E ∗ E ∗ ∗ T ∗ M . Now dualize TE over E and we get T ∗ E E T ( q ) � q � M E ∗ p The core is now T ∗ M ,

� � � � 10. Local representation, p2 More generally, for any vector bundle E on M , there is a double vector bundle p E � TE E T ( q ) � q � M TM p p TM � Write elements as e ξ T ( q ) � q p � m X If X = 0 and e = 0 then ξ can be identified with an element of E . Write TE ‘locally’ as TM ∗ E ∗ E and elements as ( X , e 1 , e 2 ) . The e 2 is the core element. p E written locally as E ∗ E ∗ ∗ T ∗ M . Now dualize TE over E and we get T ∗ E E T ( q ) � q � M E ∗ p The core is now T ∗ M ,

� � � � 10. Local representation, p2 More generally, for any vector bundle E on M , there is a double vector bundle p E � TE E T ( q ) � q � M TM p p TM � Write elements as e ξ T ( q ) � q p � m X If X = 0 and e = 0 then ξ can be identified with an element of E . Write TE ‘locally’ as TM ∗ E ∗ E and elements as ( X , e 1 , e 2 ) . The e 2 is the core element. p E written locally as E ∗ E ∗ ∗ T ∗ M . Now dualize TE over E and we get T ∗ E E T ( q ) � q � M E ∗ p The core is now T ∗ M ,

� � � � � � 11. Canonical diffeomorphism R For any vector bundle E there is an isomorphism of double vector bundles T ∗ ( E ∗ ) E ∗ R � � M T ∗ ( E ) E ∗ E � M E where ϕ ∈ E ∗ , e ∈ E , θ ∈ T ∗ M . Locally this is ( ϕ, e , θ ) �→ ( e , ϕ, − θ ) Apply this to E = TM and we get R : T ∗ ( T ∗ M ) → T ∗ ( TM ) ,

� � � � � � 11. Canonical diffeomorphism R For any vector bundle E there is an isomorphism of double vector bundles T ∗ ( E ∗ ) E ∗ R � � M T ∗ ( E ) E ∗ E � M E where ϕ ∈ E ∗ , e ∈ E , θ ∈ T ∗ M . Locally this is ( ϕ, e , θ ) �→ ( e , ϕ, − θ ) Apply this to E = TM and we get R : T ∗ ( T ∗ M ) → T ∗ ( TM ) ,

� � � � � � � � � � � � � 12. Canonical diffeomorphism ♯ The canonical symplectic structure d λ on T ∗ M induces an isomorphism ♯ : T ∗ ( T ∗ M ) → T ( T ∗ M ) . Locally this is ( ϕ 1 , X , ϕ 2 ) �→ ( ϕ 1 , X , − ϕ 2 ) . T ∗ ( T ∗ M ) T ∗ M T ∗ ( E ∗ ) E ∗ R � ♯ � M E T ∗ ( E ) E ∗ � M TM T ( T ∗ M ) T ∗ M � M E � M TM

� � � � � � � � � � � � � 12. Canonical diffeomorphism ♯ The canonical symplectic structure d λ on T ∗ M induces an isomorphism ♯ : T ∗ ( T ∗ M ) → T ( T ∗ M ) . Locally this is ( ϕ 1 , X , ϕ 2 ) �→ ( ϕ 1 , X , − ϕ 2 ) . T ∗ ( T ∗ M ) T ∗ M T ∗ ( E ∗ ) E ∗ R � ♯ � M E T ∗ ( E ) E ∗ � M TM T ( T ∗ M ) T ∗ M � M E � M TM

� � � � � � � � � � � � � 12. Canonical diffeomorphism ♯ The canonical symplectic structure d λ on T ∗ M induces an isomorphism ♯ : T ∗ ( T ∗ M ) → T ( T ∗ M ) . Locally this is ( ϕ 1 , X , ϕ 2 ) �→ ( ϕ 1 , X , − ϕ 2 ) . T ∗ ( T ∗ M ) T ∗ M T ∗ ( E ∗ ) E ∗ R � ♯ � M E T ∗ ( E ) E ∗ � M TM T ( T ∗ M ) T ∗ M � M E � M TM

� � � � � � � � � � � � � 12. Canonical diffeomorphism ♯ The canonical symplectic structure d λ on T ∗ M induces an isomorphism ♯ : T ∗ ( T ∗ M ) → T ( T ∗ M ) . Locally this is ( ϕ 1 , X , ϕ 2 ) �→ ( ϕ 1 , X , − ϕ 2 ) . T ∗ ( T ∗ M ) T ∗ M T ∗ ( E ∗ ) E ∗ R � ♯ � M E T ∗ ( E ) E ∗ � M TM T ( T ∗ M ) T ∗ M � M E � M TM

� � � � � � 13. Canonical diffeomorphism Θ R R T ∗ T ∗ M T ∗ TM ( ϕ 1 , X , ϕ 2 ) ( X , ϕ 1 , − ϕ 2 ) ♯ ♯ Θ Θ TT ∗ M ( ϕ 1 , X , − ϕ 2 )

� � � � � � 13. Canonical diffeomorphism Θ R R T ∗ T ∗ M T ∗ TM ( ϕ 1 , X , ϕ 2 ) ( X , ϕ 1 , − ϕ 2 ) ♯ ♯ Θ Θ TT ∗ M ( ϕ 1 , X , − ϕ 2 )

� � � � � � 13. Canonical diffeomorphism Θ R R T ∗ T ∗ M T ∗ TM ( ϕ 1 , X , ϕ 2 ) ( X , ϕ 1 , − ϕ 2 ) ♯ ♯ Θ Θ TT ∗ M ( ϕ 1 , X , − ϕ 2 ) So Θ is locally ( ϕ 1 , X , ϕ 2 ) �→ ( X , ϕ 1 , ϕ 2 ) and involves no minus signs.

� � � � � � 13. Canonical diffeomorphism Θ R R T ∗ T ∗ M T ∗ TM ( ϕ 1 , X , ϕ 2 ) ( X , ϕ 1 , − ϕ 2 ) ♯ ♯ Θ Θ TT ∗ M ( ϕ 1 , X , − ϕ 2 ) So Θ is locally ( ϕ 1 , X , ϕ 2 ) �→ ( X , ϕ 1 , ϕ 2 ) and involves no minus signs. Θ may be regarded as the dual of J : T 2 M → T 2 M .

� � � � � � 13. Canonical diffeomorphism Θ R R T ∗ T ∗ M T ∗ TM ( ϕ 1 , X , ϕ 2 ) ( X , ϕ 1 , − ϕ 2 ) ♯ ♯ Θ Θ TT ∗ M ( ϕ 1 , X , − ϕ 2 ) So Θ is locally ( ϕ 1 , X , ϕ 2 ) �→ ( X , ϕ 1 , ϕ 2 ) and involves no minus signs. Θ may be regarded as the dual of J : T 2 M → T 2 M . J is locally ( X , Y , Z ) �→ ( Y , X , Z ) .

� � � � � � 13. Canonical diffeomorphism Θ R R T ∗ T ∗ M T ∗ TM ( ϕ 1 , X , ϕ 2 ) ( X , ϕ 1 , − ϕ 2 ) ♯ ♯ Θ Θ TT ∗ M ( ϕ 1 , X , − ϕ 2 ) So Θ is locally ( ϕ 1 , X , ϕ 2 ) �→ ( X , ϕ 1 , ϕ 2 ) and involves no minus signs. Θ may be regarded as the dual of J : T 2 M → T 2 M . J is locally ( X , Y , Z ) �→ ( Y , X , Z ) . Dualizing over X gives ( X , ϕ 1 , ϕ 2 ) → ( ϕ 1 , X , ϕ 2 ) .

� � � � � � 13. Canonical diffeomorphism Θ R R T ∗ T ∗ M T ∗ TM ( ϕ 1 , X , ϕ 2 ) ( X , ϕ 1 , − ϕ 2 ) ♯ ♯ Θ Θ TT ∗ M ( ϕ 1 , X , − ϕ 2 ) So Θ is locally ( ϕ 1 , X , ϕ 2 ) �→ ( X , ϕ 1 , ϕ 2 ) and involves no minus signs. Θ may be regarded as the dual of J : T 2 M → T 2 M . J is locally ( X , Y , Z ) �→ ( Y , X , Z ) . Dualizing over X gives ( X , ϕ 1 , ϕ 2 ) → ( ϕ 1 , X , ϕ 2 ) . Then take the inverse.

� � � � � � 13. Canonical diffeomorphism Θ R R T ∗ T ∗ M T ∗ TM ( ϕ 1 , X , ϕ 2 ) ( X , ϕ 1 , − ϕ 2 ) ♯ ♯ Θ Θ TT ∗ M ( ϕ 1 , X , − ϕ 2 ) So Θ is locally ( ϕ 1 , X , ϕ 2 ) �→ ( X , ϕ 1 , ϕ 2 ) and involves no minus signs. Θ may be regarded as the dual of J : T 2 M → T 2 M . J is locally ( X , Y , Z ) �→ ( Y , X , Z ) . Dualizing over X gives ( X , ϕ 1 , ϕ 2 ) → ( ϕ 1 , X , ϕ 2 ) . Then take the inverse. This all extends to double Lie groupoids. The question is, why do we want to ?

� �� � � �� � � � � � � � � � 14. Double Lie groupoids again Take the Lie algebroids of a double Lie groupoid S : � V A V S AV A H S � � �� M � M H AH In each case take the dual. We get � V A ∗ V S A ∗ K A ∗ H S � � �� M � M H A ∗ K The groupoid K ⇒ M here is the ‘core groupoid’ of S . The elements of K are the s ∈ S for which both sources are identity elements. h s v 1 1

� �� � � �� � � � � � � � � � 14. Double Lie groupoids again Take the Lie algebroids of a double Lie groupoid S : � V A V S AV A H S � � �� M � M H AH In each case take the dual. We get � V A ∗ V S A ∗ K A ∗ H S � � �� M � M H A ∗ K The groupoid K ⇒ M here is the ‘core groupoid’ of S . The elements of K are the s ∈ S for which both sources are identity elements. h s v 1 1

� �� � � �� � � � � � � � � � 14. Double Lie groupoids again Take the Lie algebroids of a double Lie groupoid S : � V A V S AV A H S � � �� M � M H AH In each case take the dual. We get � V A ∗ V S A ∗ K A ∗ H S � � �� M � M H A ∗ K The groupoid K ⇒ M here is the ‘core groupoid’ of S . The elements of K are the s ∈ S for which both sources are identity elements. h s v 1 1

� �� � � �� � � � � � � � � � 14. Double Lie groupoids again Take the Lie algebroids of a double Lie groupoid S : � V A V S AV A H S � � �� M � M H AH In each case take the dual. We get � V A ∗ V S A ∗ K A ∗ H S � � �� M � M H A ∗ K The groupoid K ⇒ M here is the ‘core groupoid’ of S . The elements of K are the s ∈ S for which both sources are identity elements. h s v 1 1

� �� � � �� � � � � � � � � � 14. Double Lie groupoids again Take the Lie algebroids of a double Lie groupoid S : � V A V S AV A H S � � �� M � M H AH In each case take the dual. We get � V A ∗ V S A ∗ K A ∗ H S � � �� M � M H A ∗ K The groupoid K ⇒ M here is the ‘core groupoid’ of S . The elements of K are the s ∈ S for which both sources are identity elements. h s v 1 1