Restriction Monads Category Theory 2016 Dalhousie and St. Marys - PowerPoint PPT Presentation

Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Restriction Monads Category Theory 2016 Dalhousie and St. Mary’s Universities Halifax, N.S., Canada Darien DeWolf Dalhousie University August 11, 2016 Restriction Monads

Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Restriction Categories (Cockett and Lack, 2002) A category X is called a restriction category when it can be equipped with an assignment ( f : A → B ) �→ ( f A : A → A ) of all arrows f in X to an endomorphism f satisfying: 1. For all maps f , f f A = f . 2. For all maps f : A → B and g : A → B ′ , f A g A = g A f A . 3. For all maps f : A → B and g : A → B ′ , g A f A = g A f A . 4. For all maps f : B → A and g : A → B ′ , g A f = f ( gf ) B . Restriction Monads

� � � � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Restriction Category Objects ◮ Obvious data needed: s � X 1 C = X 1 t × s X 1 X 0 u c t r ◮ What additional data is needed to allow us to diagrammatically express (R.1) - (R.4)? ◮ Also, want to keep an eye out and avoid using the fact that restriction categories are internal to Set . Restriction Monads

Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Restriction Monads In a bicategory, a restriction monad consists of a 0-cell x , 1-cells T , D , E : x → x and 2-cells ◮ η : 1 T ⇒ T , ◮ ι : E ⇒ T (monic), ◮ µ : T 2 ⇒ T , ◮ ∆ : T ⇒ TD , ◮ τ : D 2 ⇒ D 2 and ◮ [ µ | ∗ DE ] : DE ⇒ D , ◮ ρ : D ⇒ E (epic), ◮ ψ : DT ⇒ TD satisfying conditions corresponding to (R.1) through (R.4) plus the usual monad laws. Restriction Monads

Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Example: Par : Set → Set in Cat Define a functor Par : Set → Set by Par ( A ) = A � { ⋆ } and � f ( x ) x ∈ A Par ( f : A → B )( x ) = x = ⋆ ⋆ A monad with ◮ η A : A → A � { ⋆ } : a �→ a ◮ µ A : ( A � { ⋆ } ) � { ⋆ } → A � { ⋆ } Its Kleisli arrows are total representations of partial functions; a partial function f : A → B can be thought of as � f : A → B { ⋆ } Restriction Monads

� � � � � � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Example: Par : Set → Set in Cat Giving Par a restriction monad structure: Set E = D = Par . (R.1): “ f = f f ” A � { ⋆ } ( A � { ⋆ } ) � { ⋆ } ∆ A ∆ Par 2 Par A Par ρ 1 Par �− → 1 Par ( A ) Par ( A ) ρ A A � { ⋆ } ( A � { ⋆ } ) � { ⋆ } Par 2 Par µ. Par ι µ. Par ( A ) ι A Implies that ρ = 1 . Restriction Monads

� � � � � � � � � � � � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Example: R ( X ) : X 0 → X 0 in Span ( Set ) Let X be a small restriction category. T = X 1 s t X 0 X 0 X 0 1 1 η : 1 T ⇒ T : X 0 → X 1 : A �→ 1 A X 0 η X 0 s t X 1 C s π 1 t π 2 µ : T 2 ⇒ T : C → X 1 : ( f , g ) �→ gf X 0 X 0 µ s t X 1 Restriction Monads

� � � � � � � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Example: R ( X ) : X 0 → X 0 in Span ( Set ) D = X 1 s s X 0 X 0 X 1 s t ∆ : T ⇒ TD : X 1 → D : f �→ ( f , f ) X 0 X 0 ∆ s π 1 t π 2 D D = { ( f , g ) ∈ X 1 × X 1 : sf = sg } Restriction Monads

� � � � � � � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Example: R ( X ) : X 0 → X 0 in Span ( Set ) E = X 1 s t X 0 X 0 where X 1 = { f : f ∈ X } And define X 1 s s ρ : D ⇒ E : X 1 → X 1 : f �→ f X 0 X 0 ρ s t X 1 Restriction Monads

� � � � � � � � � � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Example: R ( X ) : X 0 → X 0 in Span ( Set ) X 1 s t ι : E ⇒ T : X 1 → X 1 : f �→ f X 0 ι X 0 s t X 1 D s π 1 s π 2 τ : D 2 ⇒ D 2 : D → D : ( f , g ) �→ ( g , f ) X 0 X 0 τ s π 1 s π 2 D Restriction Monads

� � � � � � � � � � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Example: R ( X ) : X 0 → X 0 in Span ( Set ) E ℓ s π 1 s π 2 [ µ | ∗ DE ] : DE ⇒ D : E ℓ → X 1 : ( f , g ) �→ gf X 0 X 0 [ µ |∗ DE ] s s X 1 C s π 1 t π 2 ψ : DT ⇒ TD : C → D : ( f , g ) �→ ( gf , f ) X 0 X 0 ψ s π 1 s π 2 D Restriction Monads

� � � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Example: R ( X ) : X 0 → X 0 in Span ( Set ) (R.1): “ f = f f ” ∆ T TD 1 T T ρ T TE µ. T ι f �→ ( f , f ) �→ ( f , f ) �→ f f Restriction Monads

� � � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Example: R ( X ) : X 0 → X 0 in Span ( Set ) (R.2): “ f g = g f ” ρ 2 D 2 E 2 µ.ι 2 � T τ µ.ι 2 � E 2 D 2 ρ 2 ( f , g ) �→ ( f , g ) �→ g f Restriction Monads

� � � � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Example: R ( X ) : X 0 → X 0 in Span ( Set ) (R.3): “ gf = g f ” D ρ � DE D 2 [ µ | ∗ DE ] ρ 2 � E 2 D µ.ι 2 ρ T E ι ( f , g ) �→ ( f , g ) �→ gf �→ gf Restriction Monads

� � � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Example: R ( X ) : X 0 → X 0 in Span ( Set ) (R.4): “ gf = f gf ” ρ T µ.ι T � T DT ET ψ µ. T ι � TE TD T ρ ( f , g ) �→ ( gf , f ) �→ ( gf , f ) �→ f gf Restriction Monads

� � � � � � � � � � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Restriction Category Objects A restriction category in C (a category with pullbacks over s and t ) contains the following data: s � X 1 C X 0 u c t r C and D are defined by the pullback squares C D X 1 X 1 and X 1 X 1 s � s � s t X 0 X 0 Satisfying the usual category axioms and sr = s = tr Restriction Monads

� � � � � � � � � � � � � � � � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Restriction Category Objects: (R.1) – (R.4) ∆ τ X 1 D D D r 2 � r 2 r × 1 1 C C X 1 C c c c X 1 r 2 ψ D C C D c 1 × r � r × 1 r × 1 � X 1 C C C c c c r X 1 X 1 Restriction Monads

Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Restriction Category Objects Definition A double restriction category is a restriction category internal to rCat . Restriction Monads

� � � � � � � � � � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Restricted Pullbacks � C Given any cospan A , a restricted pullback is cone B consisting of an object P and total arrows p A , B , C : P → A / B / C satisfying the following universal property: For each lax cone ( P ′ , p ′ A , p ′ B , p ′ � B C ) over A C , there is a unique ϕ : P ′ → P such that ϕ ◦ p ≤ p ′ and ϕ = p ′ A p ′ B p ′ C P ′ P ′ ϕ p ′ p ′ p ′ A B A p ′ P ≥ ≤ C ≤ p A p B � C A � C A B Restriction Monads

� � � � � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Let X be a restriction category. A collection M of monics in X is stable under restricted pullbacks whenever: ◮ M contains all isomorphisms of M , ◮ M is closed under composition, ◮ for each m : B → C in M and f : A → C in X , the restricted pullback p 2 A ⊗ C B B p 1 m � C A f of m along f exists and p 1 ∈ M . Restriction Monads

� � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Define a restriction category Par ( X , M ) (Cockett and Lack, 2002) with the following data: ◮ Objects: Same objects as X ◮ Arrows: Isomorphism classes of spans i f � Y , X D with i ∈ M . ◮ Composition: restricted pullback ◮ Restriction: ( i , f ) = ( i , i ) Restriction Monads

� � � � � � � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Example: Double Category P ar ( X , M ) ◮ Objects: Same as X ◮ Vertical arrows: The total arrows of X ◮ total maps form a subcategory so composition is inherited from X . ◮ Horizontal arrows: the arrows of Par ( X , M ) ◮ composition by restricted pullbacks ◮ Double cells: i f � Y X D • ≥ ≤ • v u α X ′ D ′ � Y ′ i ′ f ′ Restriction Monads

� � � � � � � � � � � � � � � Motivation Restriction Monads Restriction Category Objects Double Restriction Categories Double Cell Composition Vertical Composition : compose all arrows vertically – straightforward Horizontal Composition: given by universal property of restricted pullback i f d x X S Y T Z u • ≥ α ≤ • v ≥ β ≤ • w X ′ S ′ � Y ′ T ′ � Z ′ g c y j Restriction Monads