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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 ϕ : [0, ∞]

[0, 1] satisfying the left-continuity condition

∀β ∈ [0, ∞], ϕ(β) = sup

α<β

ϕ(α). Its order is inherited from [0, 1], and its monoid structure is given by the commutative convolution product (ϕ ⊙ ψ)(γ) = sup

α+β≤γ

ϕ(α) · ψ(β). ∆ 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 ϕ : [0, ∞]

[0, 1] satisfying the left-continuity condition

∀β ∈ [0, ∞], ϕ(β) = sup

α<β

ϕ(α). Its order is inherited from [0, 1], and its monoid structure is given by the commutative convolution product (ϕ ⊙ ψ)(γ) = sup

α+β≤γ

ϕ(α) · ψ(β). ∆ 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

The V-powerset functor PV : Set

Set is given by

(f : X

Y) → (f! : VX VY), f!(σ)(y) =

• f(x)=y

σ(x), for all σ : X

V, y ∈ Y.

The functor PV carries a monad structure, given by yX : X

VX,

(yXx)(y) = k if y = x ⊥

• therwise
• ,

sX : VVX

VX,

(sXΣ)(x) =

• σ∈VX

Σ(σ) ⊗ σ(x), for all x, y ∈ X and Σ : VX

V.

Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 4 / 12

V-power monad

The V-powerset functor PV : Set

Set is given by

(f : X

Y) → (f! : VX VY), f!(σ)(y) =

• f(x)=y

σ(x), for all σ : X

V, y ∈ Y.

The functor PV carries a monad structure, given by yX : X

VX,

(yXx)(y) = k if y = x ⊥

• therwise
• ,

sX : VVX

VX,

(sXΣ)(x) =

• σ∈VX

Σ(σ) ⊗ σ(x), for all x, y ∈ X and Σ : VX

V.

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 PV = (PV, s, y) is a family of maps λX : T(VX)

VTX (X ∈ Set)

which, when one orders maps to a power of V pointwise by the order of V, must satisfy the following conditions: ∀f : X

Y : (Tf)! · λX ≤ λY · T(f!) (lax naturality of λ),

∀X : yTX ≤ λX · TyX (lax PV-unit law), ∀X : sTX · (λX)! · λVX ≤ λX · TsX (lax PV-multiplication law), ∀X : (eX)! ≤ λX · eVX (lax T-unit law), ∀X : (mX)! · λTX · TλX ≤ λX · mVX (lax T-multiplication law), ∀g, h : Z

VX : g ≤ h =

⇒ λX · Tg ≤ λX · 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 PV = (PV, s, y) is a family of maps λX : T(VX)

VTX (X ∈ Set)

which, when one orders maps to a power of V pointwise by the order of V, must satisfy the following conditions: ∀f : X

Y : (Tf)! · λX ≤ λY · T(f!) (lax naturality of λ),

∀X : yTX ≤ λX · TyX (lax PV-unit law), ∀X : sTX · (λX)! · λVX ≤ λX · TsX (lax PV-multiplication law), ∀X : (eX)! ≤ λX · eVX (lax T-unit law), ∀X : (mX)! · λTX · TλX ≤ λX · mVX (lax T-multiplication law), ∀g, h : Z

VX : g ≤ h =

⇒ λX · Tg ≤ λX · 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 PV. A lax (λ, V)-algebra (X, c) is a set X with a map c : TX

VX satisfying

yX ≤ c · eX (lax unit law, reflexivity), sX · c! · λX · Tc ≤ c · mX (lax multiplication law, transitivity). A lax homomorphism f : (X, c)

(Y, d) of lax (λ, V)-algebras is a

map f : X

Y satisfying

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 PV. A lax (λ, V)-algebra (X, c) is a set X with a map c : TX

VX satisfying

yX ≤ c · eX (lax unit law, reflexivity), sX · c! · λX · Tc ≤ c · mX (lax multiplication law, transitivity). A lax homomorphism f : (X, c)

(Y, d) of lax (λ, V)-algebras is a

map f : X

Y satisfying

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 = P2 distributes laxly over the V-powerset monad PV, via αX : P(VX)

VPX,

(αXS)(A) =

• x∈A
• σ∈S

σ(x) (S ⊆ VX, A ⊆ X). A V-closure space (X, c) is a lax (α, V)-algebra. Precisely, c satisfies

1

∀x ∈ X : k ≤ c({x})(x),

2

∀A ⊆ PX, B ⊆ X, z ∈ X :

y∈B

• A∈A(cA)(y)
• ⊗ (cB)(z) ≤

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 = P2 distributes laxly over the V-powerset monad PV, via αX : P(VX)

VPX,

(αXS)(A) =

• x∈A
• σ∈S

σ(x) (S ⊆ VX, A ⊆ X). A V-closure space (X, c) is a lax (α, V)-algebra. Precisely, c satisfies

1

∀x ∈ X : k ≤ c({x})(x),

2

∀A ⊆ PX, B ⊆ X, z ∈ X :

y∈B

• A∈A(cA)(y)
• ⊗ (cB)(z) ≤

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 = P2 distributes laxly over the V-powerset monad PV, via αX : P(VX)

VPX,

(αXS)(A) =

• x∈A
• σ∈S

σ(x) (S ⊆ VX, A ⊆ X). A V-closure space (X, c) is a lax (α, V)-algebra. Precisely, c satisfies

1

∀x ∈ X : k ≤ c({x})(x),

2

∀A ⊆ PX, B ⊆ X, z ∈ X :

y∈B

• A∈A(cA)(y)
• ⊗ (cB)(z) ≤

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-indexed closure towers

For a V-closure space (X, c), with cvA := {x ∈ X|(cA)(x) ≥ v}, v ∈ V, A ⊆ X, one obtains a family of maps (cv : PX

PX)v∈V

satisfying

(C0) if B ⊆ A, then cvB ⊆ cvA, (C1) if v ≤

i∈I ui, then i∈I cuiA ⊆ cvA,

(C2) A ⊆ ckA, (C3) cucvA ⊆ cv⊗uA,

for all A ⊆ X and u, v, ui ∈ V (i ∈ I). Conversely, for any family maps (cv : PX

PX)v∈V satisfying the

conditions (C0)–(C3), putting (cA)(x) :=

• {v ∈ V | x ∈ cvA}

(A ⊆ X, x ∈ X) makes (X, c) a V-closure space. The correspondences above are inverse to each other. Contractivity of a map f : (X, c)

(Y, d) is equivalently

described by ∀A ⊆ X, v ∈ V : f(cvA) ⊆ dv(f(A)).

Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 8 / 12

V-indexed closure towers

For a V-closure space (X, c), with cvA := {x ∈ X|(cA)(x) ≥ v}, v ∈ V, A ⊆ X, one obtains a family of maps (cv : PX

PX)v∈V

satisfying

(C0) if B ⊆ A, then cvB ⊆ cvA, (C1) if v ≤

i∈I ui, then i∈I cuiA ⊆ cvA,

(C2) A ⊆ ckA, (C3) cucvA ⊆ cv⊗uA,

for all A ⊆ X and u, v, ui ∈ V (i ∈ I). Conversely, for any family maps (cv : PX

PX)v∈V satisfying the

conditions (C0)–(C3), putting (cA)(x) :=

• {v ∈ V | x ∈ cvA}

(A ⊆ X, x ∈ X) makes (X, c) a V-closure space. The correspondences above are inverse to each other. Contractivity of a map f : (X, c)

(Y, d) is equivalently

described by ∀A ⊆ X, v ∈ V : f(cvA) ⊆ dv(f(A)).

Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 8 / 12

V-indexed closure towers

For a V-closure space (X, c), with cvA := {x ∈ X|(cA)(x) ≥ v}, v ∈ V, A ⊆ X, one obtains a family of maps (cv : PX

PX)v∈V

satisfying

(C0) if B ⊆ A, then cvB ⊆ cvA, (C1) if v ≤

i∈I ui, then i∈I cuiA ⊆ cvA,

(C2) A ⊆ ckA, (C3) cucvA ⊆ cv⊗uA,

for all A ⊆ X and u, v, ui ∈ V (i ∈ I). Conversely, for any family maps (cv : PX

PX)v∈V satisfying the

conditions (C0)–(C3), putting (cA)(x) :=

• {v ∈ V | x ∈ cvA}

(A ⊆ X, x ∈ X) makes (X, c) a V-closure space. The correspondences above are inverse to each other. Contractivity of a map f : (X, c)

(Y, d) is equivalently

described by ∀A ⊆ X, v ∈ V : f(cvA) ⊆ dv(f(A)).

Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 8 / 12

V-approach space

A V-approach space is a V-closure space (X, c) such that c : PX

VX preserves finite joins:

(c∅)(x) = ⊥ and c(A ∪ B)(x) = (cA)(x) ∨ (cB)(x) for all x ∈ X, A, B ⊆ X. We obtain a full subcategory V-App of V-Cls. If V is a completely distributive lattice, then a V-closure space (X, c) is a V-approach space iff its V-closure tower {cv : v ∈ V} satisfies that

1

cp∅ = ∅,

2

cp(A ∪ B) = cpA ∪ cpB, for all coprime elements p ∈ V and A, B ⊆ X.

Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 9 / 12

V-approach space

A V-approach space is a V-closure space (X, c) such that c : PX

VX preserves finite joins:

(c∅)(x) = ⊥ and c(A ∪ B)(x) = (cA)(x) ∨ (cB)(x) for all x ∈ X, A, B ⊆ X. We obtain a full subcategory V-App of V-Cls. If V is a completely distributive lattice, then a V-closure space (X, c) is a V-approach space iff its V-closure tower {cv : v ∈ V} satisfies that

1

cp∅ = ∅,

2

cp(A ∪ B) = cpA ∪ cpB, for all coprime elements p ∈ V and A, B ⊆ X.

Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 9 / 12

Lax (β, V)-algebra

We let U = (U, Σ, ˙ (-)) denote the ultrafilter monad on Set. If V is completely distributive, then the ultrafilter monad U distributes laxly over the V-powerset monad PV, via βX : U(VX)

VUX,

(βXs)(x) =

• S∈s

A∈x

• σ∈S

x∈A

σ(x) (s ∈ U(VX), x ∈ UX). A lax (β, V)-algebra (X, ℓ) is a set X with a map ℓ satisfying

1

∀x ∈ X : k ≤ ℓ( ˙ x)(x),

2

∀X ∈ UUX, y ∈ UX, z ∈ X:

• A∈X,B∈y
• x∈A,y∈B

(ℓx)(y)

• ⊗ (ℓy)(z) ≤ ℓ(ΣXX)(z).

Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 10 / 12

Lax (β, V)-algebra

We let U = (U, Σ, ˙ (-)) denote the ultrafilter monad on Set. If V is completely distributive, then the ultrafilter monad U distributes laxly over the V-powerset monad PV, via βX : U(VX)

VUX,

(βXs)(x) =

• S∈s

A∈x

• σ∈S

x∈A

σ(x) (s ∈ U(VX), x ∈ UX). A lax (β, V)-algebra (X, ℓ) is a set X with a map ℓ satisfying

1

∀x ∈ X : k ≤ ℓ( ˙ x)(x),

2

∀X ∈ UUX, y ∈ UX, z ∈ X:

• A∈X,B∈y
• x∈A,y∈B

(ℓx)(y)

• ⊗ (ℓy)(z) ≤ ℓ(ΣXX)(z).

Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 10 / 12

V-App ∼ = (β, V)-Alg

For a (β, V)-algebra (X, ℓ), (X, cℓ) given by cℓ : PX

PVX, (cℓA)(x) =

• x∈UX,x∋A

(ℓx)(x) is a V-approach space. For a V-approach space (X, c), (X, ℓc) given by ℓc : UX

PVX, (ℓcx)(x) =

• A∈x

(cA)(x) is a (β, V)-algebra. The processes above are inverse to each other. Thus, V-approach spaces are exactly lax (β, V)-algebras.

Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 11 / 12

V-App ∼ = (β, V)-Alg

For a (β, V)-algebra (X, ℓ), (X, cℓ) given by cℓ : PX

PVX, (cℓA)(x) =

• x∈UX,x∋A

(ℓx)(x) is a V-approach space. For a V-approach space (X, c), (X, ℓc) given by ℓc : UX

PVX, (ℓcx)(x) =

• A∈x

(cA)(x) is a (β, V)-algebra. The processes above are inverse to each other. Thus, V-approach spaces are exactly lax (β, V)-algebras.

Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 11 / 12

V-App ∼ = (β, V)-Alg

For a (β, V)-algebra (X, ℓ), (X, cℓ) given by cℓ : PX

PVX, (cℓA)(x) =

• x∈UX,x∋A

(ℓx)(x) is a V-approach space. For a V-approach space (X, c), (X, ℓc) given by ℓc : UX

PVX, (ℓcx)(x) =

• A∈x

(cA)(x) is a (β, V)-algebra. The processes above are inverse to each other. Thus, V-approach spaces are exactly lax (β, V)-algebras.

Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 11 / 12

Examples

If V = 2, then V-App ∼ = Top. If V is the Lawvere quantale ([0, ∞]op, +), then V-App reduces to the category App of Lowen’s approach spaces. If V is the quantale ∆ of distance distribution functions, then V-App reduces to the category ProbApp of probabilistic approach spaces. App is fully embedded into ProbApp as a reflective and coreflective subcategory. [G. J¨

• ager. Probabilistic approach spaces.

Preprint, University of Applied Science and Technology Stralsund (Germany), December 2015.]

Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 12 / 12

Examples

If V = 2, then V-App ∼ = Top. If V is the Lawvere quantale ([0, ∞]op, +), then V-App reduces to the category App of Lowen’s approach spaces. If V is the quantale ∆ of distance distribution functions, then V-App reduces to the category ProbApp of probabilistic approach spaces. App is fully embedded into ProbApp as a reflective and coreflective subcategory. [G. J¨

• ager. Probabilistic approach spaces.

Preprint, University of Applied Science and Technology Stralsund (Germany), December 2015.]

Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 12 / 12

Examples

If V = 2, then V-App ∼ = Top. If V is the Lawvere quantale ([0, ∞]op, +), then V-App reduces to the category App of Lowen’s approach spaces. If V is the quantale ∆ of distance distribution functions, then V-App reduces to the category ProbApp of probabilistic approach spaces. App is fully embedded into ProbApp as a reflective and coreflective subcategory. [G. J¨

• ager. Probabilistic approach spaces.

Preprint, University of Applied Science and Technology Stralsund (Germany), December 2015.]

Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 12 / 12

For more details, see [Hongliang Lai and Walter Tholen, Quantale-valued Approach Spaces via Closure and Convergence, arXiv:1604.08813].

Hongliang Lai (Sichuan Univ. & York Univ.) Quantale-valued Approach Spaces via Closure and Convergence 13 / 12