Quantale-valued Approach Spaces via Closure and Convergence Hongliang Lai (based on joint work with Walter Tholen) Sichuan University, Chengdu, China (Permanent) York University, Toronto, Canada (Visiting) Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 1 / 12
Quantale V = ( V , ⊗ , k ) A unital quantale (or a monoid in Sup ) Each frame ( H , ∧ , ⊤ ) is an idempotent quantale. The Lawvere quantale ([ 0 , ∞ ] op , + , 0 ) , which is isomorphic to the quantale ([ 0 , 1 ] , · , 1 ) . Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 2 / 12
Quantale V = ( V , ⊗ , k ) A unital quantale (or a monoid in Sup ) Each frame ( H , ∧ , ⊤ ) is an idempotent quantale. The Lawvere quantale ([ 0 , ∞ ] op , + , 0 ) , which is isomorphic to the quantale ([ 0 , 1 ] , · , 1 ) . Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 2 / 12
Quantale V = ( V , ⊗ , k ) A unital quantale (or a monoid in Sup ) Each frame ( H , ∧ , ⊤ ) is an idempotent quantale. The Lawvere quantale ([ 0 , ∞ ] op , + , 0 ) , which is isomorphic to the quantale ([ 0 , 1 ] , · , 1 ) . Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 2 / 12
The quantale ∆ The quantale ∆ consists of all distance distribution functions � [ 0 , 1 ] satisfying the left-continuity condition ϕ : [ 0 , ∞ ] ∀ β ∈ [ 0 , ∞ ] , ϕ ( β ) = sup ϕ ( α ) . α<β Its order is inherited from [ 0 , 1 ] , and its monoid structure is given by the commutative convolution product ( ϕ ⊙ ψ )( γ ) = sup ϕ ( α ) · ψ ( β ) . α + β ≤ γ ∆ is a coproduct of ([ 0 , ∞ ] op , + , 0 ) and ([ 0 , 1 ] , · , 1 ) in the category of quantales. Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 3 / 12
The quantale ∆ The quantale ∆ consists of all distance distribution functions � [ 0 , 1 ] satisfying the left-continuity condition ϕ : [ 0 , ∞ ] ∀ β ∈ [ 0 , ∞ ] , ϕ ( β ) = sup ϕ ( α ) . α<β Its order is inherited from [ 0 , 1 ] , and its monoid structure is given by the commutative convolution product ( ϕ ⊙ ψ )( γ ) = sup ϕ ( α ) · ψ ( β ) . α + β ≤ γ ∆ is a coproduct of ([ 0 , ∞ ] op , + , 0 ) and ([ 0 , 1 ] , · , 1 ) in the category of quantales. Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 3 / 12
V-power monad � Set is given by The V -powerset functor P V : Set � Y ) �→ ( f ! : V X � V Y ) , f ! ( σ )( y ) = � ( f : X σ ( x ) , f ( x )= y � V , y ∈ Y . for all σ : X The functor P V carries a monad structure, given by � k � if y = x � V X , y X : X ( y X x )( y ) = , ⊥ otherwise s X : V V X � V X , � ( s X Σ)( x ) = Σ( σ ) ⊗ σ ( x ) , σ ∈ V X � V. for all x , y ∈ X and Σ : V X Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 4 / 12
V-power monad � Set is given by The V -powerset functor P V : Set � Y ) �→ ( f ! : V X � V Y ) , f ! ( σ )( y ) = � ( f : X σ ( x ) , f ( x )= y � V , y ∈ Y . for all σ : X The functor P V carries a monad structure, given by � k � if y = x � V X , y X : X ( y X x )( y ) = , ⊥ otherwise s X : V V X � V X , � ( s X Σ)( x ) = Σ( σ ) ⊗ σ ( x ) , σ ∈ V X � V. for all x , y ∈ X and Σ : V X Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 4 / 12
Lax distributive law Let T = ( T , m , e ) be a monad on Set . A lax distributive law λ of T over � V TX ( X ∈ Set ) P V = ( P V , s , y ) is a family of maps λ X : T ( V X ) which, when one orders maps to a power of V pointwise by the order of V, must satisfy the following conditions: � Y : ( Tf ) ! · λ X ≤ λ Y · T ( f ! ) (lax naturality of λ ), ∀ f : X ∀ X : y TX ≤ λ X · T y X (lax P V -unit law), ∀ X : s TX · ( λ X ) ! · λ V X ≤ λ X · T s X (lax P V -multiplication law), ∀ X : ( e X ) ! ≤ λ X · e V X (lax T -unit law), ∀ X : ( m X ) ! · λ TX · T λ X ≤ λ X · m V X (lax T -multiplication law), � V X : g ≤ h = ∀ g , h : Z ⇒ λ X · Tg ≤ λ X · Th (monotonicity). [D. Hofmann, G.J. Seal, and W. Tholen, editors. Monoidal Topology: A Categorical Approach to Order, Metric, and Topology . Cambridge University Press, Cambridge, 2014.] Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 5 / 12
Lax distributive law Let T = ( T , m , e ) be a monad on Set . A lax distributive law λ of T over � V TX ( X ∈ Set ) P V = ( P V , s , y ) is a family of maps λ X : T ( V X ) which, when one orders maps to a power of V pointwise by the order of V, must satisfy the following conditions: � Y : ( Tf ) ! · λ X ≤ λ Y · T ( f ! ) (lax naturality of λ ), ∀ f : X ∀ X : y TX ≤ λ X · T y X (lax P V -unit law), ∀ X : s TX · ( λ X ) ! · λ V X ≤ λ X · T s X (lax P V -multiplication law), ∀ X : ( e X ) ! ≤ λ X · e V X (lax T -unit law), ∀ X : ( m X ) ! · λ TX · T λ X ≤ λ X · m V X (lax T -multiplication law), � V X : g ≤ h = ∀ g , h : Z ⇒ λ X · Tg ≤ λ X · Th (monotonicity). [D. Hofmann, G.J. Seal, and W. Tholen, editors. Monoidal Topology: A Categorical Approach to Order, Metric, and Topology . Cambridge University Press, Cambridge, 2014.] Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 5 / 12
Lax ( λ, V ) -algebra Let λ be a lax distributive law of T over P V . A lax ( λ, V ) -algebra ( X , c ) � V X satisfying is a set X with a map c : TX y X ≤ c · e X (lax unit law, reflexivity), s X · c ! · λ X · Tc ≤ c · m X (lax multiplication law, transitivity). � ( Y , d ) of lax ( λ, V ) -algebras is a A lax homomorphism f : ( X , c ) � Y satisfying map f : X f ! · c ≤ d · Tf (lax homomorphism law, monotonicity). Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 6 / 12
Lax ( λ, V ) -algebra Let λ be a lax distributive law of T over P V . A lax ( λ, V ) -algebra ( X , c ) � V X satisfying is a set X with a map c : TX y X ≤ c · e X (lax unit law, reflexivity), s X · c ! · λ X · Tc ≤ c · m X (lax multiplication law, transitivity). � ( Y , d ) of lax ( λ, V ) -algebras is a A lax homomorphism f : ( X , c ) � Y satisfying map f : X f ! · c ≤ d · Tf (lax homomorphism law, monotonicity). Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 6 / 12
V-closure space The ordinary powerset monad P = P 2 distributes laxly over the V-powerset monad P V , via � V P X , � � α X : P ( V X ) ( S ⊆ V X , A ⊆ X ) . ( α X S )( A ) = σ ( x ) x ∈ A σ ∈ S A V- closure space ( X , c ) is a lax ( α, V ) -algebra. Precisely, c satisfies ∀ x ∈ X : k ≤ c ( { x } )( x ) , 1 � � � ∀A ⊆ P X , B ⊆ X , z ∈ X : � ⊗ ( cB )( z ) ≤ A ∈A ( cA )( y ) 2 y ∈ B c ( � A )( z ) . A lax α -homomorphism of V-closure spaces is also called a contractive map. We obtain a category V- Cls . Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 7 / 12
V-closure space The ordinary powerset monad P = P 2 distributes laxly over the V-powerset monad P V , via � V P X , � � α X : P ( V X ) ( S ⊆ V X , A ⊆ X ) . ( α X S )( A ) = σ ( x ) x ∈ A σ ∈ S A V- closure space ( X , c ) is a lax ( α, V ) -algebra. Precisely, c satisfies ∀ x ∈ X : k ≤ c ( { x } )( x ) , 1 � � � ∀A ⊆ P X , B ⊆ X , z ∈ X : � ⊗ ( cB )( z ) ≤ A ∈A ( cA )( y ) 2 y ∈ B c ( � A )( z ) . A lax α -homomorphism of V-closure spaces is also called a contractive map. We obtain a category V- Cls . Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 7 / 12
V-closure space The ordinary powerset monad P = P 2 distributes laxly over the V-powerset monad P V , via � V P X , � � α X : P ( V X ) ( S ⊆ V X , A ⊆ X ) . ( α X S )( A ) = σ ( x ) x ∈ A σ ∈ S A V- closure space ( X , c ) is a lax ( α, V ) -algebra. Precisely, c satisfies ∀ x ∈ X : k ≤ c ( { x } )( x ) , 1 � � � ∀A ⊆ P X , B ⊆ X , z ∈ X : � ⊗ ( cB )( z ) ≤ A ∈A ( cA )( y ) 2 y ∈ B c ( � A )( z ) . A lax α -homomorphism of V-closure spaces is also called a contractive map. We obtain a category V- Cls . Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 7 / 12
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