Exponentiability in Double Categories and the Glueing Construction - PowerPoint PPT Presentation

Exponentiability in Double Categories and the Glueing Construction Susan Niefield Union College Schenectady, NY July 2019

� Idea What are the “exponentiable” objects Y in a double category s ⊙ � D 1 � D 1 × D 0 D 1 � D 0 ? id • t For C at , P os , T op , L oc , and T opos , can show directly: Y is exponentiable in D ⇐ ⇒ Y is exponentiable D 0 Showed they satisfy D 1 ≃ D 0 / 2 , generalizing Artin-Wraith glueing. [N 2012; JPAA]

Goal To prove: Y is exponentiable in D ⇐ ⇒ Y is exponentiable in D 0 in a general theorem assuming D 1 ≃ D 0 / 2 plus ... Plan 1. Double categories and the examples 2. Glueing categories 3. Lax Functors and Adjoints 4. Exponentiability in double categories

� � � � Double Categories A double category D is a (pseudo) category object in CAT s ⊙ � D 1 � D 1 × D 0 D 1 � D 0 id • t Objects: objects of D 0 � Y of D 0 Horizontal morphisms: morphisms f : X Vertical morphism: objects of D 1 , denoted by v : X s X t • Cells: morphisms of D 1 , denoted by f s � Y s X s X s Y s ϕ v � • • w X t X t Y t Y t f t

� � � � � � � � � � Double Categories: Examples [N 2012; JPAA] O ( f s ) � O ( Y s ) O ( X s ) O ( X s ) O ( Y s ) � Y , • X s X t T op : top spaces X , X � O ( X t ) , • • v � ⊇ w O ( X s ) cont maps O ( X t ) O ( X t ) O ( Y t ) O ( Y t ) lex O ( f t ) f s � Y s X s X s Y s � Y , X s L oc : locales X , X X t , v � w • • ≥ • locale maps lex X t X t Y t Y t f t f s � Y s X s X s Y s � Y , X s T opos : S -toposes X , X X t , • v � • • w geom. morph. lex X t X t Y t Y t f t

� � � � � � � Double Categories: Examples (cont.) f s � Y s X s X s Y s � Y , X s C at : categories X , X X t , • v � • • w functors profunctors X t X t Y 1 Y 1 f t f s � Y s X s X s Y s � Y , X s P os : posets X , X X t , • v � • • w ≤ monotone order ideals X t X t Y 1 Y 1 f t

� � � � � � � � Glueing Categories (G1) D 0 has finite limits (G2) id • : D 0 � D 1 has a left adjoint Γ with unit i s � Γ v X s X s Γ v id • γ v v � • • “ cotabulator ” Γ v X t X t Γ v Γ v i t � D 0 / 2 is an equivalence, where 2 = Γ( id • (G3) Γ 2 : D 1 1 ), and the following are pullbacks in D 0 i s � Γ v i t � Γ v Γ v Γ v X s X s X t X t and Γ 2 v Γ 2 v 1 1 2 2 1 1 2 2 i s i t (G4) D is “horizontally invariant”

� Glueing Categories: Examples T op : Given v : O ( X s ) O ( X t ), define Γ v = X s ⊔ X t with • U = U s ⊔ U t open, if U s , U t are open and U t ⊆ v ( U s ) 2 is the Sierpinski space L oc : Γ v defined by “Artin-Wraith glueing” along v 2 is the Sierpinski locale O ( 2 ) T opos : Γ v defined by “Artin-Wraith glueing” along v 2 is the Sierpinski topos S 2

Glueing Categories: Examples, cont. C at : Γ v is the “collage” of the profunctor v | Γ v | = | X s | ⊔ | X t | , morphisms in X s , X t , and via v 2 is the arrow category P os : Γ v is the “collage” of the ideal v 2 is the non-discrete 2-point poset Note Companions and conjoints are used for Γ − 1 in the examples, 2 but not in general, so they are not part of glueing categories.

Lax Functors Definition � E consists of functors F 0 : D 0 � E 0 and A lax functor F : D � E 1 compatible with s and t , and cells F 1 : D 1 � F 1 ( id • � F 1 ( w ⊙ v ) id • X ) and F 1 w ⊙ F 1 v F 0 X satisfying naturality and coherence conditions. Oplax and pseudo functors are defined with the cells in the opposite direction and invertible, respectively. Get a 2-category LxDbl of double categories and lax functors. Note Why LxDbl ?

Adjoints in LxDbl Lemma (Grandis/Par´ e 2004) � E , and The following are equivalent for a lax functor F : D � D 0 and G 1 : E 1 � D 1 compatible with s, t. functors G 0 : E 0 (a) G is lax and F ⊣ G in LxDbl . (b) F 0 ⊣ G 0 , F 1 ⊣ G 1 , and G is lax. (c) F 0 ⊣ G 0 , F 1 ⊣ G 1 , and F is oplax. Definition (Aleiferi 2018) D is pre-cartesian (cartesian) if D ∆ � D × D and D ! � 1 have (pseudo) right adoints × and 1. Proposition Every glueing category is pre-cartesian. Proof. ∆, ! are pseudo, and D 1 ≃ D 0 / 2 has finite limits since D 0 does.

Exponentiability in Pre-cartesian Double Categories Definition An object Y is pre-exponentiable in D if the lax functor � D has a right adjoint in LxDbl , and D is − × Y : D pre-cartesian closed if every object is pre-exponentiable. Theorem If Y is pre-exponentiable in D , then − × Y is oplax and Y is exponentiable in D 0 . The converse holds, if D is a glueing category. Proof. By the Lemma, Y is pre-exp iff − × Y is oplax and Y , id • Y are exp � 2 ) via D 1 ≃ D 0 / 2 , which is in D 0 , D 1 , resp. But, id • Y �→ ( Y × 2 exp in D 1 when Y is exp in in D 0 , and so the result follows. Note For Proposition and Theorem, horizontal invariance of D is used to show compatibility with s , t required in the Lemma.

Exponentiability: Examples From [N, 2012; TAC]: − × Y is pseudo, if Y is exponentiable in D 0 , for D = C at , P os , T op , L oc , T opos , and so for these D : Corollary Y is pre-exponentiable in D ⇐ ⇒ Y is exponentiable in D 0 . In particular, C at and P os are pre-cartesian closed. Note In [N 2012; TAC], we assumed more, i.e., D is fibrant. What can we add to (G1) - (G4) so that − × Y will be oplax for all glueing categories? How can we deal with ⊙ ?

� � � � � � � � � � � � � � � � � � � � � Exponentiability: Examples, cont. Suppose D 0 has pushouts and consider the pushout 3 1 1 1 ❘ ❘ ❉ ❘ ❉ ❘ ❘ ❉ i 0 ❘ ❘ ❘ i s ❘ ❘ ❉ ❘ ❉ ❘ ❘ ❉ ❘ ❘ ❘ id • γ 2 2 2 3 3 3 i 01 • 1 ❧ ❧ ❧ ③ ❧ ③ ❧ ❧ i t ③ ❧ 1 1 1 ❧ ❧ ❘ ❧ i s ❘ ❧ ❉ � 2 ③ i 1 ❘ ❧ ❉ ❘ ③ 1 1 2 ❧ ❘ ❧ ❉ i 0 ③ ❘ ❧ ❘ 1 1 1 ❧ ❘ i s ❘ ❘ ❉ ❘ ❉ ❘ ❘ ❉ ❘ ❘ ❘ i t i 12 id • γ 2 2 2 3 3 3 i 02 • 1 ❧ ❧ ❧ ③ ❧ ③ ❧ ❧ i t ③ 2 2 3 3 1 1 1 ❧ ❧ ❘ ❧ ❧ ❘ ❉ ❧ i 01 ❘ ③ i 2 ❉ ❘ ❧ ❘ ③ ❧ ❉ i 1 ❧ ❘ ③ ❧ ❘ ❘ 1 1 1 ❧ i s ❘ ❘ ❉ ❘ ❉ ❘ ❘ ❉ ❘ ❘ ❘ id • γ 2 2 2 3 3 3 i 12 • 1 ❧ ❧ ❧ ③ ❧ ③ ❧ ❧ i t ③ ❧ ❧ ❧ ❧ ❧ ③ i 2 ❧ ③ ❧ ❧ ③ ❧ 1 1 1 ❧ where i 02 is induced by vertically pasting along i 1 = i 12 i s = i 01 i t .

� � � � � � � � � Exponentiability: Examples, cont. The diagram below induces a morphism j s.t. ( ⋆ ) is commutative. X s X s ▼ ▼ ▼ j � Γ w ⊔ X t Γ v � γ v Γ v Γ v Γ v v • Γ( w ⊙ v ) Γ( w ⊙ v ) Γ w ⊔ X t Γ v ❘ ❘ ❘ qqq ❘ Γ w ⊔ X t Γ v Γ w ⊔ X t Γ v X t X t X t X t ( ⋆ ) ▼ ▼ ▼ ❧ ❧ ❧ � γ w Γ w Γ w Γ w w • 2 2 3 3 q q q i 02 X u X u Definition We say D has the 02-pullback condition if D 0 has pushouts and v � w � ( ⋆ ) is a pullback, for all X s X u . X t • • Note C at , P os , T op , L oc , and T opos satisfy the 02-pullback condition.

� � � � � Exponentiability: Examples, cont. Corollary Suppose D is a glueing category with the 02-pullback condition. Y is pre-exponentiable in D ⇐ ⇒ Y is exponentiable in D 0 Proof. (Sketch) ϕ � ( w ⊙ v ) × Y . It suffice to show Γ ϕ is iso, for ( w × Y ) ⊙ ( v × Y ) � Γ( w × Y ) ⊔ X t × Y Γ( v × Y ) Γ(( w × Y ) ⊙ ( v × Y )) Γ(( w × Y ) ⊙ ( v × Y )) Γ(( w × Y ) ⊙ ( v × Y )) Γ( w × Y ) ⊔ X t × Y Γ( v × Y ) Γ ϕ � Γ(( w ⊙ v ) × Y ) Γ(( w ⊙ v ) × Y ) ∼ pb = Y exp in D 0 ∼ = � � (Γ w ⊔ X t Γ v ) × Y Γ( w ⊙ v ) × Y Γ( w ⊙ v ) × Y Γ( w ⊙ v ) × Y Γ( w ⊙ v ) × Y Γ( w ⊙ v ) × Y (Γ w ⊔ X t Γ v ) × Y (Γ w ⊔ X t Γ v ) × Y (Γ w ⊔ X t Γ v ) × Y pb 2 2 3 3 i 02

◮ E. Aleiferi, Cartesian Double Categories with an Emphasis on Characterizing Spans, Ph.D. Thesis, Dalhousie University, 2018 (https://arxiv.org/abs/1809.06940). ◮ M. Grandis and R. Par´ e, Adjoints for double categories, Cahiers de Top. et G´ eom. Diff. Cat´ eg. 45 (2004), 193–240. ◮ S. B. Niefield, The glueing construction and double categories, J. Pure Appl. Algebra 216 (2012), 1827–1836. ◮ S. B. Niefield, Exponentiability via double categories, Theory Appl. Categ. 27, (2012), 10–26.