Groupoid Representations for Generalized Quotients Part 2: Morita - PowerPoint PPT Presentation

Groupoid Representations for Generalized Quotients Part 2: Morita Equivalence, Foliations, and Orbifolds Dorette Pronk Dalhousie University May 3, 2010

� � � � � � Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations Effective Descent and Groupoids If we write Sh ( Y ) ≃ Sh ( X ) × E Sh ( X ) and t � Sh ( X ) Sh ( Y ) s π X � � E Sh ( X ) , π X and we write u for δ , the truncated simplicial topos becomes π 1 s Y × s , X , t Y � X , � Y u m π 2 t i.e. , a localic groupoid. [Butz-Moerdijk] If E has enough points, we can get a topological groupoid with the same properties..

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations Description of E in terms of G Let G be a topological groupoid with source and target maps open or proper surjections. Definition • A G -space is a space p : X → G 0 with a left action by G 1 , α : G 1 × s , G 0 , p X → X , α ( g , x ) = g · x , such that • p ( g · x ) = t ( g ) ; • g 1 · ( g 2 · x ) = ( g 1 g 2 ) · x (where g 1 g 2 is composition in G ) (cocycle condition); • u ( p ( x )) · x = x (unit condition). • An (equivariant) G -sheaf is a local homeomorphism p : X → G 0 with a left G 1 -action. Remark ∼ We could also have given α : G 1 × s , G 0 , p X → X × p , G 0 , t G 1 .

� Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations The Topos Sh ( G ) • A morphism ϕ : E → E ′ between G -sheaves is a morphism of spaces over G 0 , ϕ � E ′ E � � � � � � � � � � p � p ′ � � � � � G 0 that respects the G -action, ϕ ( g · x ) = g · ϕ ( x ) . • The category of G -sheaves forms a Grothendieck topos Sh ( G ) . We also write Sh G ( X ) for Sh ( G ⋉ X ) .

� � Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations Etale Complete Groupoids Which groupoids can be obtained in this fashion? • The source and target maps must be open or closed surjections. • There is geometric morphism π G 0 : Sh ( G 0 ) → Sh ( G ) with π ∗ G 0 : Sh G ( X ) → Sh ( G 0 ) the forgetful functor (that forgets the action). • The groupoid G is étale complete if the following square of toposes is a pullback: t � Sh ( G 0 ) Sh ( G 1 ) π G 0 s π G 0 � Sh ( G ) Sh ( G 0 )

� � � Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations Etale Groupoids • A groupoid is étale if both source and target maps are local homeomorphisms. • For an étale groupoid, t : G 1 → G 0 is an element of Sh ( G ) . t • We have that Sh ( G ) / ( G 1 → G 0 ) ≃ Sh ( G 0 ) . • Every étale groupoid is étale complete, that is, the following square is a weak pullback: t Sh ( G 0 ) /π ∗ G 0 ( G 1 ) ≃ Sh ( G 1 ) Sh ( G 0 ) ≃ Sh ( G ) / G 1 π G 0 s π G 0 � Sh ( G ) Sh ( G 0 ) .

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations A Site for Sh ( G ) • A bisection of G consists of an open subset U ⊆ G 0 with a section σ : U → G 1 of s such that V = t ◦ σ ( U ) ⊆ G 0 is ∼ open and t ◦ σ : U → V . • The objects of C ( G ) are the domains of all possible bisections of G . • An arrow ( U , σ ): U → U ′ is a bisection σ : U → G 1 such that t ◦ σ ( U ) ⊆ U ′ . • A family of arrows ( U i , σ i ): U i → U ) is a covering if � t ◦ σ i ( U i ) = U . i ∈ I

� Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations Etendues Definition A Grothendieck topos E is an étendue if it contains an epimorphism U � � 1, such that E / U is a topos of sheaves on a topological space, i.e. , E / U ≃ Sh ( X ) . Remarks • In our example above, we had t � � 1 ) � G 0 ) ( U ( G 1 . = � � � � � � � � � � � � � � � � � t � � � � � � � G 0 • We can also describe étendues as toposes with a site with monic maps.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations The 2-Category of Topological Groupoids • A homomorphism ϕ : G → H of topological groupoids is an internal functor in the category of topological spaces. It consists of continuous maps ϕ 0 : G 0 → H 0 , ϕ 1 : G 1 → H 1 , which commute with all the structure maps. • A 2-cell α : ϕ ⇒ ψ : G ⇒ H is represented by a continuous map α : G 0 → H 1 , such that s ◦ α = ϕ and t ◦ α = ψ and m ( ψ 1 ( g ) , α ( s ( g ))) = m ( α ( t ( g )) , ϕ 1 ( g )) .

� � � Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations Essential Equivalences of Groupoids Definition A morphism ϕ : G → H of topological groupoids is an essential equivalence when it satisfies the following two conditions: • ϕ is essentially surjective on objects in the sense that t ◦ π 2 is an open surjection: π 2 t � H 1 � H 0 G 0 × H 0 H 1 π 1 � s � H 0 G 0 ; ϕ 0 • F is fully faithful in the sense that the following diagram is a pullback: ϕ 1 G 1 H 1 ( s , t ) � ( s , t ) ϕ 0 × ϕ 0 � H 0 × H 0 G 0 × G 0

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations Equivalences of Toposes Proposition An essential equivalence ϕ : G → H of groupoids gives rise to an equivalence of categories ∼ ϕ : Sh ( G ) → Sh ( H ) . Remark This gives us one of the implications for our proposed Morita equivalence theorem.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations Fibered Products The 2-category of topological groupoids has both strong and weak fibered products. Let ϕ : G → K and ψ : H → K . • The strong fibered product G × K H has space of objects G 0 × K 0 H 0 and space of arrows G 1 × K 1 H 1 . • The weak fibered product G × K H has space of objects k G 0 × K 0 K 1 × K 0 H 0 = { ( x , k , y ) | ϕ ( x ) → ψ ( y ) } and the elements of the space of arrows with source ( x , k , y ) and target ( x ′ , k ′ , y ′ ) are determined by pairs of arrows g ∈ G 1 and h ∈ H 1 such that k ′ ϕ ( g ) = ψ ( h ) k in K .

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations Properties of Essential Equivalences • The weak pullback of an essential equivalence along an arbitrary morphism of topological groupoids is again an essential equivalence. • If θ is an essential equivalence and there is a 2-cell α : θ ◦ ϕ ⇒ θ ◦ ψ then there is a 2-cell α ′ : ϕ ⇒ ψ such that θ ◦ α ′ = α . • The class of essential equivalences is closed under 2-isomorphisms. • The class of essential equivalences admits a right calculus of fractions.

� � � � Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations Localic Groupoid Representations • (Moerdijk, 1988) For open étale complete localic groupoids, there is an equivalence of categories with isomorphism classes of maps: [ OEC-LocGrpds ][ W − 1 ] ≃ [ Toposes ] • Essential step in the proof: show that for each geometric morphism ϕ : Sh ( G ) → Sh ( H ) there exist an essential equivalence w : K → G and a groupoid homomorphism f : G → H such that Sh ( f ) � ϕ ◦ Sh ( w ) . w 1 � Sh ( H 0 ) � G 1 Sh ( K 0 ) K 1 π H 0 w 0 ( s , t ) ( s , t ) � Sh ( G ) � Sh ( H ) � G 0 × G 0 Sh ( G 0 ) K 0 × K 0 w 0 × w 0 π G 0 ϕ

� � Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations Comments • Morphisms between open étale complete localic groupoids should be equivalence classes of spans w f � H G K where w is an essential equivalence. • For open étale complete localic groupoids G and H , Sh ( G ) ≃ Sh ( H ) if and only if there exists a localic groupoid K with essential equivalences ϕ ψ � H G K • In this case we call the two groupoids G and H Morita equivalent.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations We have some issues left: • This result is about categories, not about 2-categories. • Spacial groupoids and spacial toposes?

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations Etendues Theorem There is an equivalence of bicategories EtaleGrpd [ W − 1 ] , SpEtendues 2-iso ≃ But we can do a bit better...

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations • Proposition An étale complete open topological groupoid is Morita equivalent to an étale groupoid if and only if all its isotropy groups are discrete. • We will call such groupoids topological foliation groupoids, and denote their category by TopFolGrpd. • Theorem There is an equivalence of bicategories TopFolGrpd [ W − 1 ] . SpEtendues 2-iso ≃

� � Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations Consequences • For any two topological foliation groupoids G and H , Sh ( G ) ≃ Sh ( H ) if and only if there is a third such groupoid K with essential equivalences ϕ ψ � H . G K • In this case we will call G and H Morita equivalent . • A geometric morphism G → H corresponds to a span of groupoid homomorphisms ψ ϕ G K � H , where ψ is an essential equivalence. We call such a span a generalized morphism . • Generalized morphisms are composed using chosen weak pullbacks and the identities are represented by spans of identity arrows.