Copower functors H. Peter Gumm Philipps-Universit at Marburg - PowerPoint PPT Presentation

Copower Functors Coalgebraic Logic Functors as data containers Copower functors H. Peter Gumm Philipps-Universit¨ at Marburg Oxford, August 10-11, 2007

Copower Functors Coalgebraic Logic Functors as data containers Functor properties Relevant properties Preservation properties standard weak pullbacks separating preimages connected weak kernel pairs bounded wide pullbacks finitary intersections finite ones can be assumed General program Functor properties ← → Coalgebraic structure theory

Copower Functors Coalgebraic Logic Functors as data containers Functors preserve ... ( − ) 3 2 and P k , k > 2 P weak (wide) pullbacks preimages intersections not kernel pairs not bounded Z [ − ] (bags with credit) F kernel pairs weak pullbacks not preimages finite intersections X 2 − X + 1 intersections not preimages not kernel pauirs

Copower Functors Coalgebraic Logic Functors as data containers Fuzzy sets and bags Purpose Provide parametrized class of functors tune parameters for desired properties start with standard examples P ( − ) = 2 − , subfunctor: P ω ( − ) = 2 − ω 2 is . . . . . . a complete semilattice L . . . a commutative monoid M Generalizing yields two types of functors L X := { σ : X → L} M X ω := { σ : X → M | σ ( x ) = 0 a . e . }

Copower Functors Coalgebraic Logic Functors as data containers L - fuzzy sets L a complete � -semilattice, define L X := { σ : X → L} For f : X → Y L f ( σ ) = λ y . � { σ ( x ) | f ( x ) = y } L ( − ) is a Set -functor L -coalgebras are L - valued relations L preserves preimages intersections L weakly preserves kernel pairs ⇐ ⇒ x ∧ � i ∈ I y i = � i ∈ I ( x ∧ y i )

Copower Functors Coalgebraic Logic Functors as data containers M -bags M commutative monoid, M X ω := { σ : X → M | σ ( x ) = 0 a . e . } For f : X → Y ω ( σ ) = λ y . � { σ ( y ) | f ( y ) = x } M f x y a 1 finite bags, multiplicities from M z u a 2 N : standard bags b 1 b 2 = Z bags“with credit” M -coalgebras: M -valued relations Theorem (HPG, T.Schr¨ oder) M f ω preserves preimages ⇐ ⇒ M is positive. M f ω weakly pres. kernel pairs ⇐ ⇒ M is refinable.

Copower Functors Coalgebraic Logic Functors as data containers Common generalization M commutative monoid image finiteness essential, commutativity, unit element ω ( σ )( y ) := � M f f ( x )= y σ ( x ) L complete semilattice idempotency essential zero element L f ( σ )( y ) := � f ( x )= y σ ( x ) Observe � ω ∼ M X = M x ∈ X S � � L X ∼ L ∼ = = L x ∈ X x ∈ X

� Copower Functors Coalgebraic Logic Functors as data containers The copower functor Given category C and A ∈ C copowers of A exist in C U : C → S et any (forgetful) functor C � A C [ X ] := U ( A ) x ∈ X For any map f : X → Y let A C [ f ] � A C [ Y ] A C [ X ] � ����� � � � e x � e f ( x ) � A Theorem A C [ − ] is a S et-endofunctor.

� � � � � � � � � � � Copower Functors Coalgebraic Logic Functors as data containers Free product functor V variety of algebras, A ∈ V A V [ X ] ≈ free product M Mc [ X ] = M X ω � � A V [ X ] L S [ X ] = L X F V ( A × X ) π θ � � � � � Which properties of A and V � � A × X � � ǫ x � � guarantee . . . � e x � � � � . . . weak pullback preservation Q A g x . . . image preservation . . . weak kernel preservation

Copower Functors Coalgebraic Logic Functors as data containers Weak pullback preservation for A V [ − ] ⇒ x ∧ � i ∈ I y i = � Sl : Complete semilattices ⇐ i ∈ I ( x ∧ y i ) Mc : Commutative monoids ⇐ ⇒ positive and refinable M : Monoids ⇐ ⇒ positive and equidivisible Sg : Semigroups ⇐ ⇒ equidivisible. Equidivisibility : Given a · b = c · d , there exists k such that c a ���� ���� a · k · d or c · k · b ���� ���� b d

Copower Functors Coalgebraic Logic Functors as data containers Product refinement Refinable U 0 × U 1 A × × × A × B ∼ = C × D ⇐ ⇒ V 0 × V 1 B ∼ C × D =

Copower Functors Coalgebraic Logic Functors as data containers Product refinement Equidivisible A A C × K A × × × × A × B ∼ = C × D ⇐ ⇒ or K × D B B B ∼ ∼ C × D C × D = = Theorem 1 Equidivisible semigroups are refinable. 2 Any two product decompositions have a common refinement

� � � � � � Copower Functors Coalgebraic Logic Functors as data containers A category with one object a a · b Semigroup S : one-object-category • b Elements of S are morphisms Composition is multiplication a • • Equidivisibility is categorically: � � diagonal property c b � k � • • d

� � � � � Copower Functors Coalgebraic Logic Functors as data containers Refinement Given a 1 · a 2 · . . . · a m = p = b 1 · b 2 · . . . · b n , a 2 � • a 3 � a 4 � • • • · · · • � a 1 a m � � ���� � � � � � • • h 1 h 2 � ���� � � � � b 1 b n � • b 3 � • � • • b 2 · · · • b 4 Theorem Any two product decompositions have a common refinement. a 3 � �� � p = a 1 · a 2 · h 1 · b 2 · b 3 · h 2 · a 4 · . . . · . . . � �� � � �� � b 1 b 4

� � � Copower Functors Coalgebraic Logic Functors as data containers Copower functors are almost universal What is special about copower functors ? � FY FX F faithful: Y X F 0 F faithful � e 0 � � � � ⇐ ⇒ Fe 0 � = Fe 1 0 � � � � � 1 e 1 Theorem Every faithful S et-functor F has a representation F ( − ) ∼ = A C [ − ] with A ∈ C for some (non-full) subcategory C of Set.

� Copower Functors Coalgebraic Logic Functors as data containers Coalgebraic logic Formulae � φ :: true | ¬ φ | φ i i ∈ I | ... modalities ... f semantics: [ [ φ ] ] : A → 2 A � 2 x | = φ : ⇐ ⇒ [ [ φ ] ]( x ) = 1 [ [ φ ] ] x ≈ y : ⇐ ⇒ ∀ φ. x | = φ ⇐ ⇒ y | = φ f definable : ⇐ ⇒ ∃ φ. f = [ [ φ ] ] ... U -definable ⇐ ⇒ ∃ φ. f | U = [ [ φ ] ] | U

� Copower Functors Coalgebraic Logic Functors as data containers Coalgebraic logic Formulae φ :: true | ¬ φ | φ 1 ∧ φ 2 | ... modalities ... semantics: [ [ φ ] ] : A → 2 f � A U � � � 2 x | = φ : ⇐ ⇒ [ [ φ ] ]( x ) = 1 [ [ φ ] ] x ≈ y : ⇐ ⇒ ∀ φ. x | = φ ⇐ ⇒ y | = φ f definable : ⇐ ⇒ ∃ φ. f = [ [ φ ] ] ... U -definable ⇐ ⇒ ∃ φ. f | U = [ [ φ ] ] | U Fact ( � vs. ∧ ) 1 � :: f is definable ⇐ ⇒ f respects ≈ . 2 ∧ :: f is U-definable for each U ⊆ fin A ⇐ ⇒ f respects ≈ .

� Copower Functors Coalgebraic Logic Functors as data containers Pattinson-Schr¨ oder Logic Formulae � φ :: true | ¬ φ | φ i x � A [ [ φ ] ] � 2 i ∈ I | [ w ] φ for each w ∈ F (2) α � F [ [ φ ] ] � F 2 FA Semantics w x | = [ w ] φ : ⇐ ⇒ F [ φ ]( α ( x )) = w

� � � � � � � Copower Functors Coalgebraic Logic Functors as data containers Stability: ∇ ⊆ ≈ | = is homomorphism stable ϕ : A → B = ⇒ ( ∀ x ∈ A . x | = φ ⇐ ⇒ ϕ ( x ) | = φ ) Proof by formula induction ϕ ( x ) x ϕ � B A α β F ϕ FA FB � � � � � � � � � B � F [ [ φ ] ] � � � A F [ [ φ ] ] � � F 2 w

� � � � � � � � � � Copower Functors Coalgebraic Logic Functors as data containers Completeness: ≈ ⊆ ∇ Assume: F separating define coalgebra on A / ≈ so that [ [ φ f ] ] π ≈ is a homomorphism x x | = [ w x ] φ f but y | = [ w y ] φ f y � A π ≈ � � f � 2 A / ≈ α F π ≈ F ( A / ≈ ) Ff � F (2) F ( A ) w y w x F [ [ φ f ] ]

� � � � � � � � � Copower Functors Coalgebraic Logic Functors as data containers Finitary conjunctions/disjunctions If F is finitary , then finite conjunctions suffice: ∃ U ⊆ fin X with α ( x ) , β ( x ) ∈ F ( U ) f ◦ π ≈ definable relative to U x | = [ w x ] φ f and y | = [ w y ] φ f [ [ φ f ] ] y x � ι U π ≈ f U � � � A � 2 � � A / ≈ α � � F ι U F π ≈ � � F ( A ) Ff � F (2) F ( U ) F ( A / ≈ ) w y w x F [ [ φ f ] ]

� Copower Functors Coalgebraic Logic Functors as data containers Modal logic for Copower functors M a commutative monoid M [ X ] = X -bags, multiplicities from M M [ X ] separates points √ x • M [ X ] finitary √ � � � q � �������� � � � � � � A � � • p � � φ :: true | ¬ φ | φ 1 ∧ φ 2 � � � � � � � � � � • • �� � � � �� | [ p , q ] φ, where p , q ∈ M ϕ x | = [ p , q ] φ ⇐ ⇒ p = � { m | x m → y | = φ } q = � { m | x m → y | = ¬ φ }

� �� � Copower Functors Coalgebraic Logic Functors as data containers Separating Functors F arbitrary functor, κ ∈ Card . . . before we started with X κ F κ × X κ FX . . . approximated F by F κ ’s η X � � � � � � � now start with κ X . . . � � � � � � � � F κ represent F ( X ) by all κ -patterns F separating ⇐ ⇒ injective � F κ κ X FX � Fact ⇒ F is a subfunctor of some A κ X F is κ -separating ⇐