A multi-valued framework for coalgebraic logics over generalised metric spaces Adriana Balan University Politehnica of Bucharest TACL2017, Prague
Motivation Coalgebras encompass a wide variety of dynamical systems. Their behaviour can be universally characterised using the theory of coalgebras. However, in real life, the complexity of dynamical systems often makes bisimilarity is a too strict concept. Consequently, the focus should be on quantitative behaviour (e.g. ordered, fuzzy, or probabilistic behavior): (bi)similarity pseudometric that measures how similar two systems are from the point of view of their behaviours These can be properly captured using coalgebras based on quantale-enriched categories . A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 1 / 9
� Coalgebras and their logics – the abstract recipe The coalgebraic data: ◮ Category C C ◮ Functor T : C → C T A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 2 / 9
� � � � � Coalgebras and their logics – the abstract recipe The coalgebraic data: Coalg( T ) ◮ Category C C ◮ Functor T : C → C T ◮ T -coalgebra c : X → TX ◮ T -coalgebra morphism f : ( X , c ) → ( X ′ , c ′ ) c X TX f Tf c ′ � TX ′ X ′ A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 2 / 9
� � � � � � � Coalgebras and their logics – the abstract recipe The coalgebraic data: Coalg( T ) op ◮ Category C C op D T op ◮ Functor T : C → C ⊥ ⊥ ◮ T -coalgebra c : X → TX The logical data: ◮ T -coalgebra morphism ◮ Contravariant adjunction S ⊣ P : D → C op f : ( X , c ) → ( X ′ , c ′ ) c X TX f Tf c ′ � TX ′ X ′ A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 2 / 9
� � � � � � � � � Coalgebras and their logics – the abstract recipe The coalgebraic data: Coalg( T ) op Alg( L ) ◮ Category C C op D T op ◮ Functor T : C → C ⊥ ⊥ L ◮ T -coalgebra c : X → TX The logical data: ◮ T -coalgebra morphism ◮ Contravariant adjunction S ⊣ P : D → C op f : ( X , c ) → ( X ′ , c ′ ) c ◮ Functor L : D → D X TX f Tf c ′ � TX ′ X ′ A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 2 / 9
� � � � � � � � � � Coalgebras and their logics – the abstract recipe The coalgebraic data: Coalg( T ) op Alg( L ) ◮ Category C C op D T op ◮ Functor T : C → C ⊥ ⊥ L ◮ T -coalgebra c : X → TX The logical data: ◮ T -coalgebra morphism ◮ Contravariant adjunction S ⊣ P : D → C op f : ( X , c ) → ( X ′ , c ′ ) c ◮ Functor L : D → D X TX f Tf ◮ Natural transformation c ′ � TX ′ X ′ δ : LP → PT op A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 2 / 9
✷ This talk Today’s purpose: to look for a contravariant adjunction (to be used in the future for logics) for coalgebras over quantale-enriched categories. Let V denote a commutative integral quantale. Let V -cat be the category of V -categories and V -functors. A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 3 / 9
� � � � � � � This talk Today’s purpose: to look for a contravariant adjunction (to be used in the future for logics) for coalgebras over quantale-enriched categories. Let V denote a commutative integral quantale. Let V -cat be the category of V -categories and V -functors. An ideal picture: base for � Spaces op base for Algebras ⊤ coalgebras logics [ − , V ] V -cat op V -cat ⊤ Coalgebraic side Logical side [ − , V ] ◮ For V = ✷ , this is relatively well understood. ◮ What about for other quantale V ? For V = ([0 , 1] , ⊗ , 1), for example? A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 3 / 9
✷ � A hint from positive coalgebraic logics ◮ The simplest case: the quantale ✷ Poset op � DLat ⊥ ◮ Posets: antisymmetric ✷ -enriched categories. ◮ Distributive lattices: antisymmetric finitely complete and cocomplete ✷ -categories such that finite limits distribute over finite colimits. A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 4 / 9
� A hint from positive coalgebraic logics ◮ The simplest case: the quantale ✷ Poset op � DLat ⊥ ◮ Posets: antisymmetric ✷ -enriched categories. ◮ Distributive lattices: antisymmetric finitely complete and cocomplete ✷ -categories such that finite limits distribute over finite colimits. ◮ Move from ✷ to an arbitrary quantale V – a naive approach: ◮ Replace posets by antisymmetric V -categories. ◮ Replace distributive lattices by finitely complete and cocomplete V -categories such that finite conical limits distribute over finite conical colimits. ◮ Does it work? A minimal requirement: the quantale V itself should have a distributive lattice reduct. A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 4 / 9
The contravariant adjunction – step I ◮ Consider the finitely complete and cocomplete V -categories such that finite conical limits distribute over finite conical colimits ( ⋆ ) with left and right exact V -functors between them. ◮ Recall that finite colimits/limits can be completely described in terms of tensors/cotensors and finite joins/meets with respect to the underlying order of a V -category. ◮ Hence each A as above is in particular a distributive lattice by ( ⋆ ), and each f : A → Y left and right exact is a morphism of distributive lattices. ◮ Tensors and cotensors are encoded by a family of adjoint pair of maps r ⊙ − ⊣ ⋔ ( r , − ) on the underlying distributive lattice of A , and lex/rex V -functors preserve them. A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 5 / 9
The contravariant adjunction – step I ◮ In view of the previous features, call the resulting structure a distributive lattice with V -operators (dlao( V )). In detail: ◮ ( A , ∧ , ∨ , 0 , 1) is a bounded distributive lattice. ◮ A is endowed with a family of adjoint maps r ⊙ − ⊣ ⋔ ( r , − ) : A → A , r ∈ V satisfying the following: ◮ 1 ⊙ a = a ◮ ( r ⊗ r ′ ) ⊙ a = r ⊙ ( r ′ ⊙ a ) ◮ For each family ( r i ) i ∈ I in V with � i ∈ I r i = r , r i ⊙ a ≤ r ⊙ a ∀ i ∈ I ∀ i ∈ I = r i ⊙ a ≤ b , ⇒ r ⊙ a ≤ b ◮ Morphisms of dlao( V ) are those preserving all operations. ◮ Hence we obtain a category DLatAO( V ) (more precisely a V -cat-category) A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 6 / 9
The contravariant adjunction – step II ◮ The dual of DLatAO( V ) can be obtained by restricted Priestley duality: A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 7 / 9
The contravariant adjunction – step II ◮ The dual of DLatAO( V ) can be obtained by restricted Priestley duality: ◮ Objects of DLatAO( V ) op are Priestley spaces ( X , τ, ≤ ), endowed with a family of binary relations ( R r ) r ∈ V satisfying ◮ x ′ ≤ x and R r ( x , y ) and y ≤ y ′ imply R r ( x ′ , y ′ ) ◮ R 1 = ≤ ◮ R r ◦ R r ′ = R r ⊗ r ′ ◮ R � i ∈ I r i = � i ∈ I R r i and several topological conditions. A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 7 / 9
The contravariant adjunction – step II ◮ The dual of DLatAO( V ) can be obtained by restricted Priestley duality: ◮ Objects of DLatAO( V ) op are Priestley spaces ( X , τ, ≤ ), endowed with a family of binary relations ( R r ) r ∈ V satisfying ◮ x ′ ≤ x and R r ( x , y ) and y ≤ y ′ imply R r ( x ′ , y ′ ) ◮ R 1 = ≤ ◮ R r ◦ R r ′ = R r ⊗ r ′ ◮ R � i ∈ I r i = � i ∈ I R r i and several topological conditions. ◮ Morphisms in DLatAO( V ) op are monotone continuous maps f : X → Y such that ◮ R r ( x , y ) = ⇒ R r ( fx , fy ) ⇒ ( ∃ x ′ ∈ X . u ≤ fx ′ and R r ( x ′ , x )) ◮ R r ( u , fx ) ⇐ ⇒ ( ∃ x ′ ∈ X . R r ( x , x ′ ) and fx ′ ≤ u ) ◮ R r ( fx , u ) ⇐ A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 7 / 9
❯ ❯ The contravariant adjunction – step II ◮ Denote by RelPriest( V ) the resulting category. Hence RelPriest( V ) op ∼ = DLatAO( V ) A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 8 / 9
❯ ❯ The contravariant adjunction – step II ◮ Denote by RelPriest( V ) the resulting category. Hence RelPriest( V ) op ∼ = DLatAO( V ) ◮ Each relational Priestley space X becomes a V -category by � X ( x , y ) = { r | R r ( x , y ) } A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 8 / 9
The contravariant adjunction – step II ◮ Denote by RelPriest( V ) the resulting category. Hence RelPriest( V ) op ∼ = DLatAO( V ) ◮ Each relational Priestley space X becomes a V -category by � X ( x , y ) = { r | R r ( x , y ) } ◮ Assume that V is completely distributive and recall that each relational Priestley space is in particular compact Hausdorff. ◮ For completely distributive V , the V -cat-ification ❯ V of the ultrafilter monad is a monad on V -cat, hence we may speak of compact V -categories as ❯ V -algebras. ◮ The V -category structure and the compact Hausdorff structure on X are compatible, in the sense that the convergence map assigning to each ultrafilter on X its limit point is a V -functor. A. Balan (UPB) A multi-valued framework for coalgebraic logics TACL 8 / 9
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