Orbi Mapping Spaces Dorette Pronk (with Vesta Coufal, Carmen Rovi, Laura Scull, Courtney Thatcher) Dalhousie University Union College, October 19, 2013
Outline Orbispaces Orbigroupoids Groupoid Maps Morita Equivalence Orbimaps Orbi Mapping Spaces Groupoid Mapping Spaces Examples The Orbi Mapping Groupoid - A Homotopy Colimit
� � � � Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps Orbispaces Definition ◮ An orbispace is a Morita equivalence class of orbigroupoids. ◮ An orbigroupoid G is a groupoid in the category of paracompact Hausdorff spaces π 1 s i � G 1 G 1 × s , G 0 , t G 1 � G 1 � G 0 m u π 2 t such that the source and target maps are étale, and the diagonal ( s , t ): G 1 → G 0 × G 0 is proper (closed with compact fibers). D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps Example 1: The Silvered Interval The circle S 1 with the Z / 2-action by reflection. reflection id morphisms objects The source map is defined by projection, the target by the action. D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps Example 2: The Order 3 Cone 2/3 1/3 morphisms id objects D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps Example 3: The Teardrop 2/3 morphisms 1/3 id ����������� ����������� ����������� ����������� id 3X ����������� ����������� ����������� ����������� ����������� ����������� �������� �������� ������� ������� objects �������� �������� ������� ������� �������� �������� ������� ������� �������� �������� ������� ������� �������� �������� ������� ������� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� ��� D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps Example 4: The Order 2 Corner V 4 ⋉ D D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps Example 5: The Order 3 Corner D 6 ⋉ D D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps Example 6: G -Points ∗ G D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps Example 7: G -Lines D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps Groupoid Maps Definition A morphism ϕ : G → H of topological groupoids is a pair of maps ϕ 0 : G 0 → H 0 and ϕ 1 : G 1 → H 1 , which commute with all the structure maps. D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps Z / 2-Points of the Order 2 Corner ◮ What are the groupoid maps ∗ Z / 2 → V 4 ⋉ D ? ◮ For any X ∈ D , ϕ X 0 ( P ) = X and ϕ X 1 ( 0 ) = ϕ X 1 ( 1 ) = ( X , id ) . ◮ For any X on the horizontal (vertical) axis of D , ψ X 0 ( P ) = X and ψ X 1 ( 1 ) = ( X , τ ) ( ψ X 1 ( 1 ) = ( X , σ ) ). ◮ χ ( P ) = O and χ ( 1 ) = ( O , ρ ) . D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps Z / 2-Points of the Order 2 Corner ◮ What are the groupoid maps ∗ Z / 2 → V 4 ⋉ D ? ◮ For any X ∈ D , ϕ X 0 ( P ) = X and ϕ X 1 ( 0 ) = ϕ X 1 ( 1 ) = ( X , id ) . ◮ For any X on the horizontal (vertical) axis of D , ψ X 0 ( P ) = X and ψ X 1 ( 1 ) = ( X , τ ) ( ψ X 1 ( 1 ) = ( X , σ ) ). ◮ χ ( P ) = O and χ ( 1 ) = ( O , ρ ) . D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps Z / 2-Points of the Order 2 Corner ◮ What are the groupoid maps ∗ Z / 2 → V 4 ⋉ D ? ◮ For any X ∈ D , ϕ X 0 ( P ) = X and ϕ X 1 ( 0 ) = ϕ X 1 ( 1 ) = ( X , id ) . ◮ For any X on the horizontal (vertical) axis of D , ψ X 0 ( P ) = X and ψ X 1 ( 1 ) = ( X , τ ) ( ψ X 1 ( 1 ) = ( X , σ ) ). ◮ χ ( P ) = O and χ ( 1 ) = ( O , ρ ) . D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps Z / 2-Points of the Order 2 Corner ◮ What are the groupoid maps ∗ Z / 2 → V 4 ⋉ D ? ◮ For any X ∈ D , ϕ X 0 ( P ) = X and ϕ X 1 ( 0 ) = ϕ X 1 ( 1 ) = ( X , id ) . ◮ For any X on the horizontal (vertical) axis of D , ψ X 0 ( P ) = X and ψ X 1 ( 1 ) = ( X , τ ) ( ψ X 1 ( 1 ) = ( X , σ ) ). ◮ χ ( P ) = O and χ ( 1 ) = ( O , ρ ) . D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps Paths D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
� � Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps 2-Cells Definition A 2-cell α : ϕ ⇒ ψ is a map α : G 0 → H 1 , such that s ◦ α = ϕ 0 , t ◦ α = ψ 0 , which satisfies the naturality condition, i.e., for each g ∈ G 1 , α ( sg ) � ψ 0 ( sg ) ϕ 0 ( sg ) ϕ 1 ( g ) ψ 1 ( g ) � ψ 0 ( tg ) ϕ 0 ( tg ) α ( tg ) commutes in H , m ( ψ 1 ( g ) , α ( sg )) = m ( α ( tg ) , ϕ 1 ( g )) . D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps 2-Cells Between Paths Here are two paths with a unique 2-cell between them. Note that these paths have the same image in the quotient space. D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps 2-Cells Between Paths These two paths do not have a 2-cell between them, although they have the same image in the quotient space. D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps 2-Cells Between Paths And these two paths have two 2-cells between them. D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
� � � Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps 2-Cells Between G -Points If ϕ ( P ) = ψ ( P ) , then 2-cells α : ϕ ⇒ ψ : ∗ G ⇒ H correspond to elements h ∈ H ψ ( P ) such that h ψ ( g ) h − 1 = ϕ ( g ) for all g ∈ G , ψ ( g ) � h h ϕ ( g ) D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps Z / 3-Points of the Order 3 Corner ◮ There are two Z / 3 points of the order-3-corner with a non-trivial map on groups: ϕ 0 ( P ) = O , ϕ 1 ( 1 ) = ρ and ψ 0 ( P ) = O , ψ 1 ( 1 ) = ρ 2 . ◮ There are three transformations from one to the other (corresponding to the three reflections) and three transformations from each point to itself (corresponding to the rotations). D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
Orbigroupoids Orbispaces Groupoid Maps Orbi Mapping Spaces Morita Equivalence Orbimaps Essential Equivalences, I An essential equivalence φ : G → H satisfies the following two properties: 1 ( Essentially surjective ) G 0 × H 0 H 1 − → H 0 is an open surjection, ������ ������ ������ ������ H ������� ������� Gobj ������ ������ ������� ������� obj ������ ������ ������� ������� ������ ������ ������� ������� ������ ������ φ may not be surjective on objects, but every object in H is isomorphic to an object in the image of G . D. Pronk (with V. Coufal, C. Rovi, L. Scull, C. Thatcher) Orbi Mapping Spaces
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