Constructing noncommutative topology David Kruml Masaryk - PowerPoint PPT Presentation

Constructing noncommutative topology David Kruml Masaryk University, Brno Constructing noncommutative topology – p. 1/13

Noncommutative topology Ex.: X 1 : X 2 : X 3 : • • • • • • � � • • • • 0 1 0 1 0 1 (Connes 1994) Constructing noncommutative topology – p. 2/13

Noncommutative topology Ex.: X 1 : X 2 : X 3 : • • • • • • � � • • • • 0 1 0 1 0 1 (Connes 1994) A 1 = C ( X 2 ) ∼ = C ( X 3 ) A 2 = { continuous f : [0 , 1] → M 2 ( C ) | f (0) , f (1) diagonal } Constructing noncommutative topology – p. 2/13

Motivation Formalize the essence of noncommutative spaces in terms of category theory. Constructing noncommutative topology – p. 3/13

Motivation Formalize the essence of noncommutative spaces in terms of category theory. Construct a quantale with a given structure of right- and left-sided elements, or a C*-algebra from its spectrum of q-open sets (Akemann 1970, Giles and Kummer 1971). Constructing noncommutative topology – p. 3/13

Motivation Formalize the essence of noncommutative spaces in terms of category theory. Construct a quantale with a given structure of right- and left-sided elements, or a C*-algebra from its spectrum of q-open sets (Akemann 1970, Giles and Kummer 1971). Generalize the idea of quantale couples (Egger and Kruml 2008, CT 2007). Constructing noncommutative topology – p. 3/13

Quantales and modules S up . . . category of complete (join) semilattices. The main results hold in any closed monoidal category. Constructing noncommutative topology – p. 4/13

Quantales and modules S up . . . category of complete (join) semilattices. The main results hold in any closed monoidal category. Quantale . . . semigroup in S up , unital quantale . . . monoid (assoc., distrib.). Morphisms, modules, tensor product, quantaloids, . . . Constructing noncommutative topology – p. 4/13

� Quantales and modules S up . . . category of complete (join) semilattices. The main results hold in any closed monoidal category. Quantale . . . semigroup in S up , unital quantale . . . monoid (assoc., distrib.). Morphisms, modules, tensor product, quantaloids, . . . Saying that a graph Q M N � • � • • is enriched over S up we mean that Q is a quantale, M is a left, and N a right Q -module. Constructing noncommutative topology – p. 4/13

Ideals of a ring Let A be a ring. T . . . two-sided ideals L . . . left ideals R . . . right ideals Q . . . additive subgroups (or only those which are modules of the center) (Van den Bossche 1995) Constructing noncommutative topology – p. 5/13

Ideals of a ring Let A be a ring. T . . . two-sided ideals L . . . left ideals R . . . right ideals Q . . . additive subgroups (or only those which are modules of the center) (Van den Bossche 1995) They are all quantales, some of them also modules, bimorphisms L × R → T, R × L → Q . Constructing noncommutative topology – p. 5/13

� � � � Van den Bossche quantaloid L • • Q T R Constructing noncommutative topology – p. 6/13

� � � � Van den Bossche quantaloid L • • Q T R T ⊗ T → T Q ⊗ Q → Q R ⊗ T → R Q ⊗ R → R T ⊗ L → L L ⊗ Q → L L ⊗ R → T R ⊗ L → Q Constructing noncommutative topology – p. 6/13

� � � � Van den Bossche quantaloid L • • Q T R T ⊗ T → T Q ⊗ Q → Q R ⊗ T → R Q ⊗ R → R T ⊗ L → L L ⊗ Q → L L ⊗ R → T R ⊗ L → Q 16 pentagonal coherence axioms + some of the 6 triangular axioms for unital objects Constructing noncommutative topology – p. 6/13

Triads and solutions Given a triad ( L, T, R ) , is there some solution Q ? Constructing noncommutative topology – p. 7/13

Triads and solutions Given a triad ( L, T, R ) , is there some solution Q ? � Q 0 R ⊗ T ⊗ L �� R ⊗ L Constructing noncommutative topology – p. 7/13

�� � � � � � � Triads and solutions Given a triad ( L, T, R ) , is there some solution Q ? � Q 0 R ⊗ T ⊗ L �� R ⊗ L ( T ⊗ L ) ⊸ L Q 1 L ⊸ L ( L ⊗ R ) ⊸ T R ⊸ R ( R ⊗ T ) ⊸ R Constructing noncommutative topology – p. 7/13

Category of solutions A quantale Q is a solution of ( L, T, R ) iff both diagrams commute for Q , i.e. there is a unique factorization Q 0 → Q → Q 1 . The actions are given via Q → L ⊸ L Q → R ⊸ R R ⊗ L → Q, L ⊗ Q → L Q ⊗ R → R Constructing noncommutative topology – p. 8/13

Category of solutions A quantale Q is a solution of ( L, T, R ) iff both diagrams commute for Q , i.e. there is a unique factorization Q 0 → Q → Q 1 . The actions are given via Q → L ⊸ L Q → R ⊸ R R ⊗ L → Q, L ⊗ Q → L Q ⊗ R → R In particular, Q 0 → Q 1 is a unital couple of quantales. Constructing noncommutative topology – p. 8/13

Special instances For any sup-lattice S , triad ( S ∗ , 2 , S ) provides a Girard couple ( S ⊗ S ∗ ) → ( S ⊸ S ) . More generally, Q 0 → Q 1 is a Girard couple whenever T is a Girard quantale and L ∗ ∼ = R as T -modules. Constructing noncommutative topology – p. 9/13

Special instances For any sup-lattice S , triad ( S ∗ , 2 , S ) provides a Girard couple ( S ⊗ S ∗ ) → ( S ⊸ S ) . More generally, Q 0 → Q 1 is a Girard couple whenever T is a Girard quantale and L ∗ ∼ = R as T -modules. For suplattices S, T , every Galois connection between them determines a unique map S ⊗ T → 2 . Then ( S, 2 , T ) form a triad and Q 1 is the Galois quantale (Resende 2004). Constructing noncommutative topology – p. 9/13

Special instances For any sup-lattice S , triad ( S ∗ , 2 , S ) provides a Girard couple ( S ⊗ S ∗ ) → ( S ⊸ S ) . More generally, Q 0 → Q 1 is a Girard couple whenever T is a Girard quantale and L ∗ ∼ = R as T -modules. For suplattices S, T , every Galois connection between them determines a unique map S ⊗ T → 2 . Then ( S, 2 , T ) form a triad and Q 1 is the Galois quantale (Resende 2004). In many situations, T is commutative and L ∼ = R as T -modules. The solutions are involutive. Constructing noncommutative topology – p. 9/13

Special instances For any sup-lattice S , triad ( S ∗ , 2 , S ) provides a Girard couple ( S ⊗ S ∗ ) → ( S ⊸ S ) . More generally, Q 0 → Q 1 is a Girard couple whenever T is a Girard quantale and L ∗ ∼ = R as T -modules. For suplattices S, T , every Galois connection between them determines a unique map S ⊗ T → 2 . Then ( S, 2 , T ) form a triad and Q 1 is the Galois quantale (Resende 2004). In many situations, T is commutative and L ∼ = R as T -modules. The solutions are involutive. When L ∼ = R ∼ = T is unital then also Q 0 ∼ = Q 1 ∼ = T . Constructing noncommutative topology – p. 9/13

Triads from semiquantales Assume that L is a right semiquantale, i.e. a sup-lattice with a (non-assoc.) right distributive multiplication, T a commutative quantale, T → L an open embedding, the images of elements of T are central in L , and the left adjoint | | : L → T satisfies | xy | = | yx | for all x, y ∈ L . Then the inner product x ⊗ y �→ | xy | and the action of T on L define a triad ( L, T, L ) . The coherence axioms hold by Frobenius reciprocity. Constructing noncommutative topology – p. 10/13

Triads from semiquantales Assume that L is a right semiquantale, i.e. a sup-lattice with a (non-assoc.) right distributive multiplication, T a commutative quantale, T → L an open embedding, the images of elements of T are central in L , and the left adjoint | | : L → T satisfies | xy | = | yx | for all x, y ∈ L . Then the inner product x ⊗ y �→ | xy | and the action of T on L define a triad ( L, T, L ) . The coherence axioms hold by Frobenius reciprocity. When L, T are selfdual w.r.t. x ′ = x → 0 then T → L preserves the duality iff it is open. Then Q 0 → Q 1 is a Girard couple. Constructing noncommutative topology – p. 10/13

Triads from semiquantales Assume that L is a right semiquantale, i.e. a sup-lattice with a (non-assoc.) right distributive multiplication, T a commutative quantale, T → L an open embedding, the images of elements of T are central in L , and the left adjoint | | : L → T satisfies | xy | = | yx | for all x, y ∈ L . Then the inner product x ⊗ y �→ | xy | and the action of T on L define a triad ( L, T, L ) . The coherence axioms hold by Frobenius reciprocity. When L, T are selfdual w.r.t. x ′ = x → 0 then T → L preserves the duality iff it is open. Then Q 0 → Q 1 is a Girard couple. ∧ y = ( x ∨ y ′ ) ∧ y and the central Ex.: Sasaki projection x ˙ cover | | in a complete orthomodular lattice. Constructing noncommutative topology – p. 10/13

Applications T . . . centre (classical data, invariants), L . . . statics (states, propositions), Q . . . dynamics (actions, transitions). Constructing noncommutative topology – p. 11/13

Applications T . . . centre (classical data, invariants), L . . . statics (states, propositions), Q . . . dynamics (actions, transitions). L a frame, T an open subframe . . . supported quantales (e.g. Penrose tilings of Mulvey, Resende 2005). L a complete orthomodular lattice, T its centre . . . dynamics of quantum logics. Quantum frames (Rosický 1989). MV-algebras. Constructing noncommutative topology – p. 11/13