Topological Groupoids and Exponentiability Susan Niefield (joint - PowerPoint PPT Presentation

Topological Groupoids and Exponentiability Susan Niefield (joint with Dorette Pronk) July 2017

Overview Goal: Study exponentiability in categories of topological groupoid. Starting Point: Consider exponentiability in Gpd ( C ), where C is: ◮ finitely complete ◮ cartesian closed ◮ locally cartesian closed Application: Adapt to various categories of orbigroupoids.

Exponentiability in C Suppose C is finitely complete. An object Y is called exponentiable � C has a right adjoint, and C is called cartesian if − × Y : C closed if every object is exponentiable. � B is exponentiable in C if it is exponentiable in A morphism Y the slice category C / B , and C is called locally cartesian closed if every morphism is exponentiable. � B is exponentiable and r : Z � B , we follow Note that if q : Y the abuse of notation and write the exponential as r q : Z Y � B .

� � � � � � � � Properties of Exponentiability Composition of exponentiables is exponentiable, and pullback along any morphism preserves exponentiability. ∆ � B × B and Y are exponentiable, then every If the diagonal B q � B is exponentiable, since morphism Y q � B Y Y B ❄ ❄ ⑧⑧⑧⑧⑧⑧⑧ ❄ ❄ ❄ ❄ π 2 � id , q � ❄ Y × B Y × B and � id , q � � Y × B π 2 � B Y Y Y × B Y × B Y × B B q q × id π 1 B B B × B B × B Y Y 1 1 ∆

Exponentiable Spaces A space Y is exponentiable in Top iff O ( Y ) is a continuous lattice (Day/Kelly, 1970) A sober space is exponentiable in Top iff it is locally compact (Hoffmann/Lawson, 1978) A subspace inclusion is exponentiable in Top iff it is locally closed (N, 1978)

Cartesian Closed Coreflective Subcategories of Top Suppose M ⊆ Top . Given a space X , let ˆ X denote the set X with the topology generated by the set of continuous maps � X | M ∈ M} { f : M Say a X is M -generated if X = ˆ X , and let Top M denote the full subcategory of Top consisting of M -generated space. Proposition (N, 1978) If M is a class of exponentiable spaces s.t. M × N ∈ Top M , for all M , N ∈ M , then Top M is cartesian closed. . X × Y is the product and Z Y = � × Y = � Note X ˆ lim � Y Z M M is the exponential in Top M . .

Examples K = compact T 2 spaces; Top K = compactly generated spaces E = exponentiable spaces; Top E = exponentiably generated spaces Note In both cases, one can show locally closed inclusions are � B ˆ exponentiable in Top M . Thus, if ∆: B × B is locally closed, then the slice Top M / B is cartesian closed. .

� � � � � Groupoids in C An object G of Gpd ( C ) is a diagram in C of the form i s � G 0 c � G 1 G 2 G 1 G 1 G 1 G 1 G 0 G 0 e t making G a category in C in which every morphism is invertible, � G 0 is t in where G 2 = G 1 × G 0 G 1 . Unless otherwise stated, G 1 the pullback when G 1 is on the left and s when it is on the right. � H are pairs f i : G i � H i ( i = 0 , 1) compatible Morphisms f : G with the groupoid structure, i.e., homomorphisms . � H are morphisms ϕ : G 0 � H 1 such that 2-Cells f ⇒ g : G � ϕ s , g 1 � � H 2 G 1 G 1 H 2 � f 1 ,ϕ t � � c H 2 H 2 H 1 H 1 c i.e., natural transformations .

� � Exponentiable Objects in Gpd ( C ) Gpd ( C ) is cartesian closed whenever C is, and H G is defined by f 0 ( H G ) 0 � � H G 0 × H G 1 � X 0 0 1 g 0 f 1 � ( H G ) 0 × H G 0 ( H G ) 1 � × ( H G ) 0 � X 1 1 g 1 where ( f 0 , g 0 ) and ( f 1 , g 1 ) make H G the “groupoid of homomorph- � H . isms” G In particular, Gpd ( Top K ) and Gpd ( Top E ) are cartesian closed.

Exponentiable Objects in Gpd ( C ), cont. This construction of H G uses only the fact that G 0 , G 1 , and G 2 are exponentiable and not that C is cartesian closed, and so: Proposition If G 0 , G 1 , and G 2 are exponentiable in C , then G is exponentiable in Gpd ( C ). . Proposition If G is exponentiable in Gpd ( C ), then G 0 is exponent- iable in C . The converse holds if s or t is exponentiable in C . . Note If G is an ´ etale groupoid, then all structure morphisms are exponentiable in Top . We conjecture that, in this case, H G is ´ etale when H is also ´ etale and G 1 / G 0 is compact. .

� Example ∼ = � 1, which is If C also has finite coproducts, we can consider I I : 0 exponentiable in Gpd ( C ). � ¯ ( B I I ) 0 = B 1 , the “object of morphisms β : b b ” ( B I I ) 1 = B 2 × B 1 B 2 via c , the “object of squares” α � b t b s b s b t β s � β t ¯ ¯ ¯ ¯ b s b s α � b t b t ¯ α � α � with s ( β s α � β t ) = β s , t ( β s α � β t ) = β t , . . . ¯ ¯ ( B I I ) 2 is the “object of horizontally composable squares” I is an orbigroupoid when B is. Note B I .

� � � Exponentiable Morphisms in Gpd ( C ) q � B is a fibration if G 1 � s , q 1 � � G 0 × B 0 B 1 has a right inverse in C . G q � B is If C is locally cartesian closed, then every fibration G exponentiable in Gpd ( C ) and r q : H G � B is defined by f 0 ( H G ) 0 � � H G 0 × B 0 ( B 0 × B 1 H G 1 1 ) � X 0 0 g 0 f 1 � ( H G ) 0 × B 0 H G 1 ( H G ) 1 � × B 0 ( H G ) 0 � X 1 1 g 1 σ � H b ,” and making ( H G ) 0 “the object of homomorphisms G b ( H G ) 1 “the object of 2-cells between distributers” σ � H b G b G b H b Σ � • • G β � H β σ � G ¯ G ¯ H ¯ H ¯ b b b b ¯ Note Composition is defined using the fibration assumption. .

Exponentiable Morphisms in Gpd ( C ), cont. q � B is a fibration with The construction of H G uses only that G the q i exponentiable, and not that C is locally cartesian closed. � B is a fibration with G i � B i ( i = 0 , 1 , 2) Proposition If G � B is exponentiable in Gpd ( C ). . exponentiable in C , then G ∆ i � B i × B i ( i = 0 , 1) are Corollary Suppose the diagonals B i exponentiable in a cartesian closed category C . Then every � B is exponentiable in Gpd ( C ) . fibration G � B are exponentiable in Note This implies that fibrations G ∆ i � B i ˆ Gpd ( Top M ), for M = K or E , if B i × B i are locally closed. .

� � � Example Revisited s � π 2 � B 2 π 1 � B 1 , and Define B I I t � B by s 0 = s , t 0 = t , s 1 : B 2 × B 1 B 2 α � α � π 1 � B 2 π 2 � B 1 , i.e., s 1 ( β s t 1 : B 2 × B 1 B 2 α � β t ) = α , t 1 ( β s α � β t ) = ¯ α . ¯ ¯ s � Proposition B I I t � B are fibrations in Gpd ( C ). . � ( B I Proof. � s , s 1 � : ( B I I ) 1 I ) 0 × B 0 B 1 is given by α � b t α � α � α � b t b s b s b t b s b s b t b s b s b t b s b s b t β s � β t �→ β s � and so β s � �→ β s � id ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ b s b s α � b t b t b s b s b s b s b t b t ¯ αβ − 1 s is a right inverse. The proof for t is similar. . I × B G s π 1 � B is a fibration, for all G � B . Note Can also show B I .

Example Revisited, cont. I × B G s π 1 � B is exponentiable in Gpd ( C ), if the Proposition B I components are exponentiable in C . . ∆ i � B i × B i ( i = 0 , 1) are Corollary Suppose the diagonals B i exponentiable in a cartesian closed category C . Then s π 1 � B is exponentiable, for all G I × B G � B in Gpd ( C ). . B I s π 1 � B is exponentiable in Gpd ( Top M ), for I × B G Note So, B I ∆ i � B i ˆ M = K or E , if the diagonals B i × B i are locally closed.

� � � � � � � The Pseudo-Slice 2-Category Gpd ( C ) / / B s π 1 � B ) is part of a 2-monad I × B G � B ) �→ ( B I The functor ( G on the 2-category Gpd ( C ) / B induced by the internal groupoid i s � B I × B B I c � B I B I I B I B I B I B I I I I I I B B e t and the 2-Kleisli category is the pseudo-slice Gpd ( C ) / / B whose � B , morphisms are triangles objects are homomorphism q : G � ˆ ϕ, f � � B I f I × B H I × B H � H B I G G H G G ✹ ❂ ϕ � ❂ ✹ � ✡✡✡✡✡ � ✁✁✁✁✁ ❂ ✹ ❂ ✹ q ❂ r ✹ q or ❂ s π 1 B B B B � ( g , ψ ) are 2-cells θ : f � g such that and 2-cells θ : ( f , ϕ ) ϕ � ✂✂ q q ❁ ψ ❁ ❁ rf rf rg rg r θ

Pseudo-Exponentiability in 2-Kleisli Categories An object Y is pseudo-exponentiable in a 2-category K if, for every Z , there is an object Z Y and natural equivalences K ( X × Y , Z ) ≃ K ( X , Z Y ) Theorem (N, 2007) Suppose that K is a 2-category with finite products and T , η, µ is a 2-monad on K such that η T ∼ = T η and � TX × TY is an isomorphism in K , for all X , Y . If T ( X × TY ) TY is 2-exponentiable in K , then T ( TZ TY ) � TZ TY is an equiv- alence and Y is pseudo-exponentiable in the Kleisli 2-category K T . . Remarks 1. |K T | = |K| , K T ( X , Y ) = K ( X , TY ), with composition via µ . 2. One can show that the 2-monad on Gpd ( C ) / B related to B I I satisfies the hypotheses of the theorem.

Pseudo-Exponentiability in Gpd ( C ) / / B � B is pseudo-exponentiable in Gpd ( C ) / Corollary G / B , if the s π 1 � B are exponentiable in C . . I × B G components of B I Corollary Gpd ( C ) / / B is pseudo-cartesian closed, if the diagonals ∆ i � B i × B i are exponentiable in a cartesian closed category C . . B i Note So, Gpd ( Top M ) / / B is pseudo-cartesian closed, for ∆ i � B i ˆ M = K or E , if the diagonals B i × B i are locally closed. . Corollary If C is locally cartesian closed category, then Gpd ( C ) is pseudo-locally cartesian closed. . Note A proof that Gpd ( Sets ) is pseudo-locally cartesian closed appeared in a 2003 paper by Palmgren posted on the arxiv. .