A categorical approach to abstract convex spaces and interval spaces

Bing Wang; Qing-Hua Li; Zhen-Yu Xiu

doi:10.1515/math-2019-0029

Article Open Access

A categorical approach to abstract convex spaces and interval spaces

Bing Wang , Qing-Hua Li and Zhen-Yu Xiu

Published/Copyright: May 16, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

In this paper, we establish the axiomatic conditions of hull operators and introduce the category of interval spaces. We also investigate their relations with convex spaces from a categorical sense. It is shown that the category CS of convex spaces is isomorphic to the category HS of hull spaces, and they are all topological over Set. Also, it is proved that there is an adjunction between the category IS of interval spaces and the category CS of convex spaces. In particular, the category CS(2) of arity 2 convex spaces can be embedded in IS as a reflective subcategory.

Keywords: Convex structure; Hull operator; Interval operator; Adjunction

MSC 2010: 52A01; 54A05; 18D35

1 Introduction

Convexity is an important and basic property in many mathematical areas. However, in some concrete mathematical setting, such as vector spaces, it is not the most suitable setting for studying the basic properties of convex sets. In order to avoid this deficiency, abstract convex structures (convex structures, in short) are defined by three axioms [1], which is a similar way of defining topological structures. Up to now, the convexity theory has become a branch of mathematics dealing with set-theoretic structures satisfying axioms similar to that usual convex sets fulfill. Actually, convex structures appeared in many research areas, such as lattices [2], graphs [3], and topology [4]. Besides, convexity theory is also investigated from the lattice-valued aspect, including L-convex structures [5, 6, 7, 8, 9, 10, 11, 12, 13] and M-fuzzifying convex structures [14, 15, 16, 17, 18, 19].

Category theory plays an important role in demonstrating the relations between different types of spatial structures. It emerges frequently in general topology and fuzzy topology, especially in crisp and fuzzy convergence theory [20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. This motivates us to apply category theory to convex structures since convex structures can be considered as topology-like structures. Actually, like continuous mappings between topological spaces, there is also a special kind of mappings between convex spaces, which is called convexity-preserving mapping. Under a convexity-preserving mapping, convex sets in the range space are inverted to convex sets of the domain. Such mappings arise in various constructions of convexities. For spaces derived from an algebraic structure, convexity-preserving mappings usually agree with the corresponding notions of homomorphisms. In fact, convexity-preserving mappings are exactly the appropriate notions of morphisms in the category of convex spaces. This is just one of our motivations of this paper. That is, we would like to investigate the categorical properties of convex spaces.

A convex structure is completely determined by a special kind of closure operators, which is called algebraic closure operators. Actually, algebraic closure operators can be treated as the hull operators of convex spaces. Except for algebraic closure operators, interval operators, as a generalization of intervals, provide a natural and frequent method of describing or constructing convex structures. There are also close relations between convex structures and interval operators. Inspired by this, we will not only provide a new characterization of convex structures by closure operators and present an axiomatic hull operators, but also focus on the categorical properties of interval spaces and study its relations with convex spaces from a categorical sense.

2 Preliminaries

Throughout this paper, let X denote a nonempty set and 2^X the powerset of X. Let {A_j}_j∈J ⊆dir 2^X denote that {A_j}_j∈J is a directed subset of 2^X, which means for each B, C ∈ {A_j}_j∈J, there exists D ∈ {A_j}_j∈J such that B ⊆ D and C ⊆ D.

Let X, Y be two nonempty sets and f : X ⟶ Y be a mapping. Define f^→ : 2^X ⟶ 2^Y and f^← : 2^Y ⟶ 2^X as follows:

∀A∈2X, f→(A)={f(x)∣x∈A}; ∀B∈2Y, f←(B)={x∣f(x)∈B}.

Definition 2.1

([1]). A convex structure 𝓒 on X is a subset of 2^X which satisfies:

∅, X ∈ 𝓒;
{A_i}_i∈I ⊆ 𝓒 implies ⋂_i∈I A_i ∈ 𝓒;
{A_j}_j∈J ⊆dir 𝓒 implies ⋃dirj∈J∈C.

For a convex structure 𝓒 on X, the pair (X, 𝓒) is called a convex space.

A mapping f : (X, 𝓒_X) ⟶ (Y, 𝓒_Y) is called convexity-preserving (CP, in short) provided that B ∈ 𝓒_Y implies f^←(B) ∈ 𝓒_X. The category whose objects are convex spaces and whose morphisms are CP mappings will be denoted by CS.

Definition 2.2

([1]). A closure operator on X is a mapping C : 2^X ⟶ 2^X which satisfies:

C(∅) = ∅;
A ⊆ C(A);
A ⊆ B ⇒ C(A) ⊆ C(B);
CC(A) = C(A).

For a closure operator C on X, the pair (X, C) is called a closure space. It will be called algebraic if it also satisfies that,

(CLA) C(A) = ⋃{C(B) | B is a finite subset of A}.

The pair (X, C) is called an algebraic closure space.

A mapping f : (X, C_X) ⟶ (Y, C_Y) between closure spaces is called convexity-preserving (CP, in short) provided that f^→(C_X(A)) ⊆ C_Y(f^→(A)) for all A ∈ 2^X. The category whose objects are closure spaces and whose morphisms are CP mappings will be denoted by CLS, and the full subcategory of algebraic closure spaces by ACLS.

For a convex space (X, 𝓒), define C^𝓒 : 2^X ⟶ 2^X by C^𝓒(A) = ⋂_A⊆B∈𝓒 B. Then C^𝓒 is an algebraic closure operator on X. Conversely, for a closure operator (X, C), define 𝓒^C ⊆ 2^X by 𝓒^C = {A ∈ 2^X | A = C(A)}. Then 𝓒^C is a convex structure on X. Furthermore, they are one-to-one corresponding. In a categorical sense, we have

Theorem 2.3

The category CS is isomorphic to the category ACLS.

Definition 2.4

([29, 30]). A category C is called a topological category over Set with respect to the usual forgetful functor from C to Set if it satisfies (TC1), (TC2) and (TC3) or (TC1)^′, (TC2) and (TC3).

(TC1) Existence of initial structures: For any set X, any class J, and family ((X_j, ξ_j))_j∈J of C-object and any family (f_j : X ⟶ X_j)_j∈J of mappings, there exists a unique C-structure ξ on X which is initial with respect to the source (f_j : X ⟶ (X_j, ξ_j))_j∈J, this means that for a C-object (Y, η), a mapping g : (Y, η) ⟶ (X, ξ) is a C-morphism iff for all j ∈ J, f_j ∘ g :(Y, η) ⟶ (X_j, ξ_j) is a C-morphism.

(TC1)^′ Existence of final structures: For any set X, any class J, and family ((X_j, ξ_j))_j∈J of C-object and any family (f_j : X_j ⟶ X)_j∈J of mappings, there exists a unique C-structure ξ on X which is final with respect to the sink (f_j : (X_j, ξ_j) ⟶ X))_j∈J, this means that for a C-object (Y, η), a mapping g : (X, ξ) ⟶ (Y, η) is a C-morphism iff for all j ∈ J, g ∘ f_j : (X_j, ξ_j) ⟶ (Y, η) is a C-morphism.

(TC2) Fibre-smallness: For any set X, the C-fibre of X, i.e., the class of all C-structures on X, which we denote by C(X), is a set.

(TC3) Terminal separator property: For any set X with cardinality at most one, there exists exactly one C-object with underlying set X (i.e. there exists exactly one C-structure on X).

Lemma 2.5

([29, 30]). Suppose that 𝔽 : A ⟶ B and 𝔾 : B ⟶ A are concrete functors. Then the following conclusions are equivalent:

{id_Y : 𝔽 ∘ 𝔾(Y) ⟶ Y | Y ∈ B} is a natural transformation from the functor 𝔽 ∘ 𝔾 to the identity functor id_B on B, and {id_X : X ⟶ 𝔾 ∘ 𝔽(X) | X ∈ A} is a natural transformation from the identity functor id_A on A to the functor 𝔾 ∘ 𝔽.
For each Y ∈ B, id_Y : 𝔽 ∘ 𝔾(Y) ⟶ Y is a B-morphism, and for each X ∈ A, id_X : X ⟶ 𝔾 ∘ 𝔽(X) is a A-morphism.

In this case, (𝔽, 𝔾) is called an adjunction between A and B.

If (𝔽, 𝔾) is an adjunction, then it is easy to verify that 𝔽 is a left adjoint of 𝔾 or equivalently, 𝔾 is a right adjoint of 𝔽.

The class of objects of a category A is denoted by |A|. For more notions related to category theory we refer to [29] and [30].

3 The categories of convex spaces and hull spaces

In this section, we first study some categorical properties of convex spaces. Then we introduce the axiomatic hull operators and investigate its relations with convex structures.

Definition 3.1

For a nonempty, let F_𝓒(X) denote the fibre

{(X,C)∣C is a convex structure on X.}

of X. For convex spaces (X, 𝓒₁) and (X, 𝓒₂), we say (X, 𝓒₁) is finer than (X, 𝓒₂), or (X, 𝓒₂) is coarser than (X, 𝓒₁), denoted by (X, 𝓒₁) ⩽_𝓒 (X, 𝓒₂), if the identity mapping id_X : (X, 𝓒₁) ⟶ (X, 𝓒₂) is CP. We also write 𝓒₁ ⩽_𝓒 𝓒₂.

Example 3.2

Let X be a nonempty set.

Define 𝓒_* by 𝓒_* = 2^X. Then 𝓒_* is the finest convex structure on X, which is called the discrete convex structure on X.
Define 𝓒^* by 𝓒^* = {∅, X}. Then 𝓒^* is the coarsest convex structure on X, which is called the indiscrete convex structure on X.

Theorem 3.3

The category CS is topological over Set.

Proof

We first prove the existence of final structures. Let ((X_λ, 𝓒_λ))_λ∈Λ be a family of convex spaces and let X be a nonempty set. Let further ((f_λ : (X_λ, 𝓒_λ) ⟶ X))_λ∈Λ be a sink. Define 𝓒 ⊆ 2^X by

C={A∈2X∣∀λ∈Λ, fλ←(A)∈Cλ}.

Since fλ← preserves arbitrary meets and directed joins, we can verify that 𝓒 is a convex structure on X.

Let further (Y, 𝓒_Y) be a convex space and g : X ⟶ Y be a mapping. Assume that g ∘ f_λ is CP for all λ ∈ Λ. We have for all B ∈ 𝓒_Y,

∀λ∈Λ,fλ←(g←(B))=(g∘fλ)←(B)∈Cλ.

By definition of 𝓒, we obtain g^←(B) ∈ 𝓒. This implies that g : (X, 𝓒) ⟶ (Y, 𝓒_Y) is CP, as desired.

Secondly, the class of all convex structures on a fixed set X is a subset of 2^{(2^X)}, which means that the CS fibre of X is a set.

Finally, for a one point set X = {x}, there exists only one convex structure 𝓒 = {∅, {x}} on X. Hence, CS satisfies the terminal separator property. Therefore, CS is a topological category in the sense of [29]. That is, a well-fibred topological category in the terminology of [30].□

Corollary 3.4

(F_𝓒(X), ⩽_𝓒) is a complete lattice.

Next we will introduce the axiomatic hull operators and study its relations with convex structures.

Definition 3.5

A hull operator on X is a mapping co : 2^X ⟶ 2^X which satisfies:

co(∅) = ∅, co(X) = X;
A ⊆ co(A);
A ⊆ B ⇒ co(A)⊆ co(B);
co(co(A)) = co(A).
co(⋃dirj∈JAj) = ⋃_j∈J co(A_j).

For a hull operator co on X, the pair (X, co) is called a hull space. Actually, a hull operator on X is a closure operator on X which satisfies (H5).

Definition 3.6

A mapping f : (X, co_X) ⟶ (Y, co_Y) is called convexity-preserving (CP, in short) provided that f^→(co_X(A)) ⊆ co_Y(f^→(A)) for all A ∈ 2^X.

The category whose objects are hull spaces and whose morphisms are CP mappings will be denoted by HS.

Proposition 3.7

Let (X, 𝓒) be a convex space and define co^𝓒 : 2^X ⟶ 2^X by

∀A∈2X,coC(A)=⋂A⊆B∈CB.

Then co^𝓒 is a hull operator on X.

Proof

By Theorem 2.3, co^𝓒 is an algebraic closure operator on X. Thus co^𝓒 satisfies (H1)–(H4). It suffices to verify (H5). For {A_j}_j∈J ⊆dir 2^X, take any x ∉ ⋃_j∈J co^𝓒(A_j) = ⋃_j∈J ⋂_{A_j⊆B∈𝓒} B. Then there exists B_j ∈ 2^X such that A_j ⊆ B ∈ 𝓒 and x ∉ B_j for each j ∈ J. Let C_j = co^𝓒(A_j). By (C2) and (H3), we know A_j ⊆ C_j ∈ 𝓒 and {C_j}_j∈J is directed. Put B = ⋃_j∈J C_j. By (C3), we obtain ⋃_j∈J A_j ⊆ B ∈ 𝓒. Further, since C_j ⊆ B_j, it follows that x ∉ C_j for each j ∈ J. This implies that x ∉ B. As a consequence, we obtain B ∈ 2^X such that ⋃_j∈J A_j ⊆ B ∈ 𝓒 and x ∉ B. This means that x ∉ co^𝓒( ⋃dirj∈J A_j). By the arbitrariness of x, we have co^𝓒( ⋃dirj∈J A_j) ⊆ ⋃_j∈J co^𝓒(A_j). The inverse inequality holds obviously. Therefore, co^𝓒 is a hull operator.□

Proposition 3.8

If f : (X, 𝓒_X) ⟶ (Y, 𝓒_Y) is a CP mapping, then so is f : (X, co^𝓒_X) ⟶ (Y, co^𝓒_Y).

Proof

Since f : (X, 𝓒_X) ⟶ (Y, 𝓒_Y) is a CP mapping, it follows that

∀B∈2Y, B∈CY implies f←(B)∈CX.

Then for each A ∈ 2^X, we have

f←(coCY(f→(A)))=⋂f→(A)⊆B∈CYf←(B)⊇⋂A⊆f→(B)∈CXf←(B)⊇⋂A⊆C∈CXC=coCX(A).

This shows f^→(co^𝓒_X(A)) ⊆ co^𝓒_Y(f^→(A)), as desired.□

By Propositions 3.7 and 3.8, we obtain a functor 𝔽 : CS ⟶ HS as follows:

F:CS⟶HS(X,C)⟼(X,coC)f⟼f.

Proposition 3.9

Let (X, co) be a hull space and define 𝓒^co = {A ∈ 2^X | A = co(A)}. Then 𝓒^co is a convex structure on X.

Proof

(C1) is obvious. We need only verify (C2) and (C3).

(C2) Take any {A_i}_i∈I ⊆ 𝓒^co. Then for each i ∈ I, co(A_i) = A_i. By (H3), co(⋂_i∈I A_i) ⊆ ⋂_i∈I co(A_i). In order to show the inverse inequality, take any x ∉ co(⋂_i∈I A_i) ⊇ ⋂_i∈I A_i. Then there exists i₀ ∈ I such that x ∉ A_i₀ = co(A_i₀). This implies that x ∉ ⋂_i∈I co(A_i). By the arbitrariness of x, we obtain ⋂_i∈I co(A_i) ⊆ co(⋂_i∈I A_i). Hence, it follows that ⋂_i∈I co(A_i) = co(⋂_i∈I A_i). This means that ⋂_i∈I A_i ∈ 𝓒^co.

(C3) Take any {A_j}_j∈J ⊆dir 𝓒^co. Then A_j = co(A_j) for each j ∈ J. By (H5), it follows that

co(⋃dirj∈JAj)=⋃dirj∈Jco(Aj)=⋃dirj∈JAj.

This means that ⋃dirj∈J A_j ∈ 𝓒^co.□

Proposition 3.10

If f : (X, co_X) ⟶ (Y, co_Y) is a CP mapping, then so is f : (X, 𝓒^co_X) ⟶ (Y, 𝓒^co_Y).

Proof

Since f : (X, co_X) ⟶ (Y, co_Y) is a CP mapping, we have

∀A∈2X, coX(A)⊆f←(coY(f→(A))).

Then for each B ∈ 𝓒^co_Y, it follows that

coX(f←(B))⊆f←(coY(f→(f←(B))))⊆f←(coY(B))=f←(B).

This implies that co_X(f^←(B)) = f^←(B). Hence, f^←(B) ∈ 𝓒^co_X.□

By Propositions 3.9 and 3.10, we obtain a functor 𝔾 : HS ⟶ CS as follows:

G:HS⟶CS(X,co)⟼(X,Cco)f⟼f.

Theorem 3.11

The category CS is isomorphic to the category HS.

Proof

It suffices to show that 𝔾 ∘ 𝔽 = 𝕀_CS and 𝔽 ∘ 𝔾 = 𝕀_HS. That is, for each (X, 𝓒) ∈ |CS| and each (X, co) ∈ |HS|, co^{𝓒^co} = co and 𝓒^{co^𝓒} = 𝓒.

For each A ∈ 2^X, we have

coCco(A)=⋂A⊆B∈CcoB=⋂A⊆B=co(B)B=co(A)

and

A∈CcoC⟺A=coC(A)=⋂A⊆B∈CB⟺A∈C.

This completes the proof.□

4 The category of interval spaces

In this section, we will introduce the category of interval spaces with interval spaces as objects and with interval-preserving mappings as morphisms. Then we will study its categorical properties.

Definition 4.1

([1]). An interval operator on X is a mapping 𝓘 : X × X ⟶ 2^X which satisfies:

x, y ∈ 𝓘(x, y);
𝓘(x, y) = 𝓘(y, x).

For an interval operator 𝓘 on X, the pair (X, 𝓘) is called an interval space.

Definition 4.2

([1]). A mapping f : (X, 𝓘_X) ⟶ (Y, 𝓘_Y) is called interval-preserving (IP in short) provided that

∀x,y∈X, f→(JX(x,y))⊆JY(f(x),f(y)).

It is easy to check that all interval spaces and IP mappings form a category, denoted by IS.

Definition 4.3

For a nonempty set X, let F_𝓘(X) denote the fibre

{(X,J)∣J is an interval operator on X}

of X. For interval spaces (X, 𝓘₁) and (X, 𝓘₂), we say (X, 𝓘₁) is finer than (X, 𝓘₂), or (X, 𝓘₂) is coarser than (X, 𝓘₁), denoted by (X, 𝓘₁) ⩽_𝓘 (X, 𝓘₂), if the identity mapping id_X : (X, 𝓘₁) ⟶ (X, 𝓘₂) is IP. We also write 𝓘₁ ⩽_𝓘 𝓘₂.

Example 4.4

Let X be a nonempty set.

Define 𝓘_* : X × X ⟶ 2^X by 𝓘_*(x, y) = {x, y} for each x, y ∈ X. Then 𝓘_* is the finest interval operator on X, which is called the discrete interval operator on X.
Define 𝓘^* : X × X ⟶ 2^X by 𝓘^*(x, y) = X. Then 𝓘^* is the coarsest interval operator on X, which is called the indiscrete interval operator on X.
Suppose that ℝ is the set of real numbers. Define 𝓘_ℝ : ℝ × ℝ ⟶ 2^ℝ by

∀a,b∈R, JR(a,b)=[min{a,b},max{a,b}].

Then 𝓘_ℝ is an interval operator on ℝ.
Suppose that d is a metric on X. Define 𝓘_d : X × X ⟶ 2^X by

∀x,y∈X, Jd(x,y)={z∈X∣d(x,y)=d(x,z)+d(z,y)}.

Then 𝓘_d is an interval operator on X.

Proposition 4.5

Let (X, 𝓘_X), (Y, 𝓘_Y) and (Z, 𝓘_Z) be interval spaces. If f : X ⟶ Y and g : Y ⟶ Z are IP, then g ∘ f : (X, 𝓘_X) ⟶ (Z, 𝓘_Z) is IP.

Proof

The proof is easy and omitted.□

In the category IS, important constructions like the formulations of products and subspaces, are always possible.

Theorem 4.6

The category IS is topological over Set.

Proof

We first prove the existence of initial structures. Let ((X_λ, 𝓘_λ))_λ∈Λ be a family of interval spaces and let X be a nonempty set. Let further ((f_λ : X ⟶ (X_λ, 𝓘_λ)))_λ∈Λ be a source. Define 𝓘 : X × X ⟶ 2^X by

∀x,y∈X, J(x,y)=⋂λ∈Λfλ←(Jλ(fλ(x),fλ(y))).

Then it is easy to verify that 𝓘 is an interval operator on X.

Let further (Y, 𝓘_Y) be an interval space and g : Y ⟶ X be a mapping. Assume that f_λ ∘ g is IP for all λ ∈ Λ. We then have for each y₁, y₂ ∈ Y and for each λ ∈ Λ,

(fλ∘g)→(JY(y1,y2))⊆Jλ(fλ(g(y1)),fλ(g(y2))).

From this we obtain

g→(JY(y1,y2))⊆⋂λ∈Λfλ←(Jλ(fλ(g(y1)),fλ(g(y2))))=J(g(y1),g(y2)).

This means that g : (Y, 𝓘_Y) ⟶ (X, 𝓘) is IP, as desired.

Secondly, the class of all interval operators on a fixed set X is a subset of 2^{((2^X)^X×X)}, which means that the IS fibre of X is a set.

Finally, for a one point set X = {x}, there exists only one interval operator 𝓘 on X, which is defined by 𝓘(x, x) = X. Hence, IS satisfies the terminal separator property. Therefore, IS is a topological category over Set.□

Corollary 4.7

(F_𝓘(X), ⩽_𝓘) is a complete lattice.

Example 4.8

(Product Spaces). Let {(X_λ, 𝓘_λ)}_λ∈Λ be a family of interval spaces. The interval operator Π – 𝓘 on ∏_λ∈Λ X_λ which is initial with respect to the projections (p_λ)_λ∈Λ is called the product interval operator and the pair (∏_λ∈Λ X_λ, Π – 𝓘) is called the product space. By definition, we have for x, y ∈ ∏_λ∈Λ X_λ,

Π−J(x,y)=⋂λ∈Λpλ←(Jλ(pλ(x),pλ(y)))=∏λ∈ΛJλ(pλ(x),pλ(y)).

Example 4.9

(Subspaces) Let (X, 𝓘_X) be an interval space and let Y ⊆ X. The interval operator 𝓘_Y on Y which is initial with respect to the inclusion mapping id_Y : Y ⟶ X is called the sub-interval operator and the pair (Y, 𝓘_Y) is called the subspace of (X, 𝓘_X). By definition, we have for x, y ∈ Y,

JY(x,y)=JX(x,y)∩Y.

5 Relations between IS and CS

In this section, we will focus on the relations between IS and CS. In particular, we will propose a full subcategory of CS, consisted of arity 2 convex spaces and study its relations with interval spaces.

Definition 5.1

A convex space (X, 𝓒) is called arity 2 if it satisfies

(AR2) ∀A ∈ 2^X, ∀x, y ∈ A, co^𝓒({x, y}) ⊆ A implies A ∈ 𝓒.

Let CS(2) denote the full subcategory of CS, consisted of arity 2 convex spaces.

Next we will study the relations between CS (CS(2)) and IS.

Proposition 5.2

Let (X, 𝓒) be a convex space and define 𝓘^𝓒 : X × X ⟶ 2^X by

∀x,y∈X, JC(x,y)=coC({x,y})=⋂x,y∈A∈CA.

Then 𝓘^𝓒 is an interval operator on X.

Proof

The verifications of (I1) and (I2) are straightforward and omitted.□

Proposition 5.3

If f : (X, 𝓒_X) ⟶ (Y, 𝓒_Y) is a CP mapping, then f : (X, 𝓘^𝓒_X) ⟶ (Y, 𝓘^𝓒_Y) is a IP mapping.

Proof

Since f : (X, 𝓒_X) ⟶ (Y, 𝓒_Y) is a CP mapping, it follows that f^←(B) ∈ 𝓒_X for each B ∈ 𝓒_Y. Then for each x, y ∈ X, we have

f←(JCY(f(x),f(y)))=⋂f(x),f(y)∈B∈CYf←(B)⊇⋂x,y∈f←(B)∈CXf←(B)⊇⋂x,y∈A∈CXA=JCX(x,y).

This implies that f^→(𝓘^𝓒_X(x, y)) ⊆ 𝓘^𝓒_Y(f(x), f(y)), as desired.□

By Propositions 5.2 and 5.3, we obtain a functor ℍ as follows:

H:CS⟶IS(X,C)⟼(X,JC)f⟼f.

Proposition 5.4

Let (X, 𝓘) be an interval space and define 𝓒^𝓘 as follows:

CJ={A∈2X∣∀x,y∈A, J(x,y)⊆A}.

Then (X, 𝓒^𝓘) is an arity 2 convex space.

Proof

(C1) is obvious. We need only verify (C2), (C3) and (AR2).

(C2) Take any {A_i}_i∈I ⊆ 𝓒^𝓘. Then for each i ∈ I and for each x, y ∈ A_i, 𝓘(x, y) ⊆ A_i. This implies that

x,y∈⋂i∈IAi⟺∀i∈I,x,y∈Ai⟺∀i∈I,J(x,y)⊆Ai⟺J(x,y)⊆⋂i∈IAi.

Hence, ⋂_i∈I A_i ∈ 𝓒^𝓘.

(C3) Take any {A_j}_j∈J ⊆dir 𝓒^𝓘. Then for each x, y ∈ ⋃dirj∈J A_j, there exist j₁, j₂ ∈ J such that x ∈ A_j₁ and y ∈ A_j₂. Since {A_j}_j∈J is directed, there exists j₃ ∈ J such that A_j₁ ⊆ A_j₃ and A_j₂ ⊆ A_j₃. This implies that x, y ∈ A_j₃ ∈ 𝓒^𝓘. Then it follows that 𝓘(x, y) ⊆ A_j₃ ⊆ ⋃dirj∈J A_j. This means ⋃dirj∈J A_j ∈ 𝓒^𝓘.

(AR2) Take any A ∈ 2^X such that

∀x,y∈A, coCJ({x,y})⊆A.

In order to show A ∈ 𝓒^𝓘, take any x, y ∈ A. It follows that

coCJ({x,y})=⋂x,y∈B∈CJB⊇J(x,y).

Then we have 𝓘(x, y) ⊆ co^{𝓒^𝓘}({x, y}) ⊆ A for each x, y ∈ A. This implies that A ∈ 𝓒^𝓘, as desired.□

Proposition 5.5

If f : (X, 𝓘_X) ⟶ (Y, 𝓘_Y) is a IP mapping, then f : (X, 𝓒^𝓘_X) ⟶ (Y, 𝓒^𝓘_Y) is a CP mapping.

Proof

Since f : (X, 𝓘_X) ⟶ (Y, 𝓘_Y) is a IP mapping, it follows that

∀x,y∈X, f→(JX(x,y))⊆JY(f(x),f(y)).

Then for each B ∈ 𝓒^𝓘_Y, take any x, y ∈ f^←(B). It follows that f(x), f(y) ∈ B ∈ 𝓒^𝓘_Y. This means that 𝓘_Y(f(x), f(y)) ⊆ B. Further we have

JX(x,y)⊆f←(JY(f(x),f(y)))⊆f←(B).

This implies that f^←(B) ∈ 𝓒^𝓘_X, as desired.□

By Propositions 5.4 and 5.5, we obtain a functor 𝕂 : IS ⟶ CS as follows:

K:IS⟶CS(X,J)⟼(X,CJ)f⟼f.

Theorem 5.6

(𝕂, ℍ) is an adjunction between IS and CS.

Proof

Since 𝕂 and ℍ are both concrete functors, we need only verify that 𝕂 ∘ ℍ ⩽_𝓒 𝕀_CS and ℍ ∘ 𝕂 ⩾_𝓘 𝕀_IS. That is to say, for each (X, 𝓒) ∈ |CS| and (X, 𝓘) ∈ |IS|, 𝓒^{𝓘^𝓒} ⩽_𝓒 𝓒 and 𝓘 ⩽_𝓘 𝓘^{𝓒^𝓘}.

On one hand, take any x, y ∈ X. Then

J(x,y)⊆⋂x,y∈A∈CJA=coCJ({x,y})=JCJ(x,y).

On the other hand, take any A ∈ 2^X. Then

A∈C⟹∀x,y∈A,coC({x,y})⊆A⟺∀x,y∈A,JC(x,y)⊆A⟺A∈CJC.

This means that 𝓒 ⊆ 𝓒^{𝓘^𝓒}, that is, 𝓒^{𝓘^𝓒} ⩽_𝓒 𝓒, as desired.□

By Propositions 5.2 and 5.4, we know 𝕂^* ≜ 𝕂 : IS ⟶ CS(2) and ℍ^* ≜ ℍ|_CS(2) : CS(2) ⟶ IS are still functors. Moreover, we have the following result.

Theorem 5.7

(𝕂^*, ℍ^*) is an adjunction between IS and CS(2). Moreover, 𝕂^* is a left inverse of ℍ^*.

Proof

By Theorem 5.6, it suffices to show that 𝓒^{𝓘^𝓒} = 𝓒 for each arity 2 convex space (X, 𝓒). Take any A ∈ 2^X. Then

A∈C⟺∀x,y∈A,coC({x,y})⊆A((X,C)is arity 2)⟺∀x,y∈A,JC(x,y)⊆A⟺A∈CJC.

This means 𝓒^{𝓘^𝓒} = 𝓒.□

Corollary 5.8

The category CS(2) can be embedded in the category IS as a reflective subcategory.

6 Conclusions

In this paper we provided a categorical approach to abstract convex theory. On one hand, we introduced the axiomatic conditions of hull operators and showed that the resulting category is isomorphic to the category of convex spaces. On the other hand, we investigated the relations between convex spaces and interval spaces. We showed that there is an adjunction between the category of interval spaces and the category of convex spaces. Furthermore, the category of arity 2 convex spaces can be embedded in the category of interval spaces as a reflective subcategory. As is shown in this paper, category theory is an effective tool to deal with convex structures and interval operators. This also implies that category theory will be significant in the research on the theory of convex structures. In the future, we will consider applying category theory to fuzzy convex structures and establishing the relations between fuzzy convex structures and some other related structures.

Acknowledgement

This work is supported by the the Filling Project of Education Department of Heilongjiang Province (NO. 1352MSYYB008), the Scientific Research Project of Mudanjiang Normal University (NO. GP2017001), the Project of Shandong Province Higher Educational Science and Technology Program (NO. J18KA245).

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Received: 2018-08-30

Accepted: 2019-02-20

Published Online: 2019-05-16

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0029

Keywords for this article

Convex structure; Hull operator; Interval operator; Adjunction

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