Theory and Applications of Categories, Vol. 21, No. 5, 2008, pp. 76–90.

COMPLETIONS IN BIAFFINE SETS

ELISABETTA FELACO AND ERALDO GIULI

Abstract. The theory of completion of T_{0} objects in categories of affine objects over a given complete category developed by the second author in [12] is extended to the case of T_{0} objects in categories of 2affine objects. In the paper the case of the category Set and target object the two-point set is studied in detail and an internal characterization of 2affine sets is provided.

1. Introduction

Let X be a complete category endowed with a proper factorization structure for morphisms and A an X-object. The category X(A) of affine X-objects over A is defined as follows: the objects are the pairs (X, U) where X ∈ X and U is a subset of the set X(X, A) = A^{X} of all X-morphisms from X to A and the morphisms (affine X-morphisms) from (X, U) to (Y, V) are X-morphisms f : X ⟶ Y such that v ∘ f ∈ U whenever v ∈ V.

It is shown in [12] that the category X(A), which is topological over X, has a natural closure operator, called Zariski closure operator (first considered by Skula [19] in the category of topological spaces), and that the subcategory T_{0}X(A) of all affine X-objects whose diagonal is Zariski closed admits completions (admits a firm reflection in the terminology of [2] and [3]). Moreover Zariski compactness ([4]) implies completeness.

In [10, 11] (see also [5] and [14]) the second author studied in detail the case X = Set with A= two-point set, extended the above result to every hereditary coreflective subcategory of Set({0, 1}) (there denoted by SSet ) and provided an internal characterization of complete (algebraic in [6] ) objects, explaining that “completion” coincides with “sobrification” in the familiar examples.

As the theory of topological spaces has a parallel theory of bitopological spaces, the theory of affine objects admits a parallel theory of 2affine objects.

A 2affine object over A is a triple (X, P, Q), with X a X-object and P, Q affine structures over A. An affine X-morphism from (X, P, Q) to (X′, P′, Q′) is an affine Xmorphism both from (X, P) to (X′, P′) and from (X, Q) to (X′, Q′).

The aim of the paper is to study in detail (as in [10, 11]) the case X = Set, with factorization structure (onto, one to one) and target the two-point set {0, 1}.

In Section 2 the category 2SSET of 2affine sets over {0, 1} is introduced and some elementary properties are listed. In Section 3 and Section 4 the Zariski closure and the separated (T_{0}) 2affine sets are introduced and studied, with emphasis on epimorphisms and regular monomorphisms.

Received by the editors 2008-03-12 and, in revised form, 2008-07-07.

Published on 2008-07-22 in the Tholen Festschrift.

2000 Mathematics Subject Classification: 18A20, 18B30, 18B99, 18A32, 08B30, 54A05, 54B30.

Key words and phrases: 2affine set, Zariski closure, separated object, complete object, injective object, bitopological space.

\begin{matrix}c \\ ◯ \end{matrix} Elisabetta Felaco and Eraldo Giuli, 2008. Permission to copy for private use granted.

Section 5 contains the main result: (Theorem 5.5), the category 2SSET_{0} of separated 2affine sets admits completions. That is: 2SSET_{0} admits a (unique determined) subcategory C which is embedding reflective and such that every z-dense embedding into an element of C is a C-reflection map (i.e. C is a firm reflection in the terminology of [2, 3]). Moreover (Proposition 5.7) a separated 2affine set is complete if and only if it is injective with respect to Zariski-dense embeddings, if and only if it is Zariski-closed in every separated 2affine set in which it can be embedded.

In Section 6 the above results are extended to all hereditary coreflective subcategories of 2SSET , recovering, among others, the fact that the category of all T_{0} bitopological spaces admits completions (cf. [13]).

Section 7 contains a general method to provide hereditary coreflective subcategories of 2SSET and Section 8 contains an internal characterization of complete 2affine sets.

The categorical terminology is that of [1] and [16] and for the categorical theory of closure operators we refer to [8].

2. The category 2SSet of 2affine sets

A 2affine set over the two point set S = {0, 1} is a triple (X, P, Q), where X is a set and P and Q are subsets of the power set S^{X} .

A 2affine map from (X, P, Q) to (Y, L, R) is a function f : X ⟶ Y such that l∘f ∈ P whenever l ∈ L and r ∘ f ∈ Q whenever r ∈ R.

2SSet will denote the category of 2affine sets (over S) and 2affine maps.

Since the elements of P and Q are characteristic functions of X, we can also describe 2SSet using subsets: objects are triples (X, P, Q) where X is a set and P and Q are subsets of the power set P(X), and the morphisms from (X, P, Q) to (Y, L, R) are functions f : X ⟶ Y such that f^{−1}(L) ∈ P for every L ∈ L and f^{−1}(R) ∈ Q for every R ∈ R. When we use this description we speak about P-open and Q-open sets and 2-continuous maps.

Both descriptions of 2SSet will be used throughout the paper.

We will denote by U : 2SSet ⟶ Set the obvious forgetful functor.

2.1. Proposition. (2SSet , U) is a topological category.

Proof. To show that every U-structured source admits a unique initial lift, let X be a set, {(Y_{i}, L_{i}, R_{i}) | i ∈ I} a family of 2affine sets and F = {f_{i} : X ⟶ Y_{i} | i ∈ I} a family of functions. The (P, Q) structure on X, where P = {f_{i}^{−1}(L) | L ∈ L_{i}, i ∈ I} and Q = {f_{i}^{−1}(R) | R ∈ R_{i}, i ∈ I}, is the unique initial structure on X determined by F.

Thus a 2affine map f : (X, P, Q) ⟶ (Y, L, R) is initial (with respect to U) if and only if every P ∈ P is of the form f^{−1}(L), L ∈ L, and every Q ∈ Q is of the form f^{−1}(R), R ∈ R.

As usual an initial monomorphism (= injection) will be called embedding. It is clear that every subset M of the underlying set X of a 2affine set (X, P, Q) carries

as initial structure the families of the restrictions to M of the functions in P and in Q, or, equivalently, the families of subsets V = {P ∩ M | P ∈ P} and W = {Q ∩ M | Q ∈ Q}.

In this case we say that (M, V, W) is a 2affine subset of (X, P, Q). Some consequences (cf.[1]) of Proposition 2.1 are collected in the:

2.2. Corollary.

1. Every F -structured sink {g_{i} : F (X_{i}, P_{i}, Q_{i}) ⟶ Y | i ∈ I} admits a unique final lift. A surjective and final 2affine map is called quotient map.

2. In 2SSet the epimorphisms are the surjective 2affine maps and the monomorphisms are the injective 2affine maps.

3. In 2SSet the embeddings coincide with the regular monomorphisms (= equalizers of two 2affine maps).

4. Every 2affine map admits an essentially unique (surjective, embedding)-factorization.

5. 2SSet is complete. Every limit is obtained as the initial lift of the corresponding limit in Set.

In particular, the product of a family {(X_{i}, P_{i}, Q_{i}) | i ∈ I} of 2affine sets is the cartesian product X of the family {(X_{i}) | i ∈ I} endowed with the 2affine structure P = {p ∘ π_{i} | p ∈ P_{i}, i ∈ I} and Q = {q ∘ π_{i} | q ∈ Q_{i}, i ∈ I} (or, equivalently, P = {π_{i}^{−1}(P) | P ∈ P_{i}, i ∈ I} and Q = {π_{i}^{−1}(Q) | Q ∈ Q_{i}, i ∈ I}), where π_{i} : X ⟶ X_{i} is the i-th projection.

6. 2SSet is cocomplete. Every colimit is obtained as the final lift of the corresponding colimit in Set.

7. Every class C of 2affine sets admits an epireflective hull E(C) (given by the 2affine subsets of products of elements of C). If E(C) is stable under refinements then E(C) is quotient reflective.

8. A full and isomorphism closed subcategory A of 2SSet is hereditary coreflective if and only if it is stable under coproducts, quotients and embeddings.

2.3. Example. Every bitopological space (cf. [13]) is a 2affine set and a function between bitopological spaces is continuous if and only if it is 2affine. Thus the category 2Top of bitopological spaces is fully embedded in 2SSet .

2.4. Remark. The category 2SSet is not well-fibred even though every set admits a set of 2affine structures. In fact the empty set admits four 2affine structures. This defect can be removed by assuming that every 2affine structure contains the empty set and the whole set (using the functional description: every 2affine structure contains the constant functions c_{0} and c_{1}). We shall consider this full subcategory of 2SSet , denoted by 2SSET , in the next Section.

Note that 2SSET is universal in the sense of Marny [17].

3. Separated (= T_{0}) 2affine sets

3.1. Definition. A 2affine set (X, P, Q) is called separated, or T_{0}, if P ∪ Q separates the points of X i.e. for every x ≠ y ∈ X exists α ∈ P ∪ Q such that αx ≠ αy.

Equivalently for every x ≠ y ∈ X there exists A ∈ P ∪ Q such that x ∈ A and y ∉ A.

3.2. Remark. Obviously every refinement of a T_{0} 2affine structure is T_{0}. 2SSET_{0} will denote the full subcategory of separates 2affine sets.

3.3. Example. The 2affine set S = {S × S; {π_{1}, c_{0}, c_{1}}, {π_{2}, c_{0}, c_{1}}} is separated. S is called Sierpinski 2affine set.

3.4. Lemma. Let (X, P, Q) be a 2affine set; then f : (X, P, Q) ⟶ S is 2affine if and only if π_{1} ∘ f ∈ P and π_{2} ∘ f ∈ Q.

Proof. Trivial.

3.5. Proposition. Let (X, P, Q) be a 2affine set; then:

1. the canonical map, defined by ϕ(x) = {(px, qx) | p ∈ P, q ∈ Q},

ϕ : (X, P, Q) ⟶ S^{(P×Q)}

(which is 2affine) is an initial map.

2. ϕ is an embedding if and only if (X, P, Q) is separated.

Proof.

1. The 2affine maps from (X, P, Q) to S correspond to pairs in (P × Q), by Lemma 3.4. Thus, for every p ∈ P (resp. q ∈ Q) there exists c_{0} ∈ P (resp. c_{0} ∈ Q) such that p×c_{0}, c_{0} ×q ∈ P ×Q; finally we have p = π_{1} ∘π_{(p×c_{0})} ∘ϕ and q = π_{2} ∘π_{(c_{0}×q)} ∘ϕ.

2. Let x_{1} ≠ x_{2} in X and assume, by separation of (X, P, Q), that p ∈ P satisfies px_{1} ≠ px_{2}. Then (px_{1}, 0) ≠ (px_{2}, 0) and these two elements are the (p)-components of ϕ(x_{1}) and ϕ(x_{2}) respectively.

3.6. Proposition. 2SSET_{0} is quotient reflective in 2SSET .

Proof. 2SSET_{0} is the epireflective hull (see Corollary 2.2.7) of the Sierpinski 2affine set S in virtue of Proposition 3.5. Then by Remark 3.2 it is quotient reflective.

4. Zariski closure and epimorphisms

Recall from [10] that an affine set over the two point set S = {0, 1} is a pair (X, U), where X is a set and U is a subset of P(X), containing X and ∅ (equivalently : U is a family of functions from X to S containing the constants).

An affine map is a function inversely preserving the elements of the structures.

If U separates the points of X, (X, U) is a T_{0} (separated) affine set.

The category of (T_{0}) affine sets is denoted by (SSET_{0}) SSET .

A closure operator z (called Zariski closure) is introduced and studied to characterize the epimorphisms and regular monomorphisms of SSET_{0} and the main result is that SSET_{0} admits a firm reflection (cf. [10], Theorem 5.2). Here we will establish a similar existence theorem.

Given an affine set (X, U) and M ⊂ X, the Zariski closure is defined, functionally, by:

z_{(X,U)}M = ⋂{Eq(u, v) | u, v ∈ U and M ⊂ Eq(u, v)}

and using “open” sets by:

z_{(X,U)}M = {x ∈ X | (∄ U, V ∈ U)(U ∩ M = V ∩ M, x ∈ (U \ V))}.

We want to introduce here an analogous Zariski closure in 2SSET to obtain epimorphisms and regular monomorphisms of 2SSET_{0}.

4.1. Definition. Let (X, P, Q) be a 2affine set, and M ⊂ X. Set

\begin{aligned}z_{(X,P,Q)}M & = (⋂\{Eq(p_{1}, p_{2}) | p_{1}, p_{2} ∈ P and M ⊂ Eq(p_{1}, p_{2})\}) \\ & ∩ (⋂\{Eq(q_{1}, q_{2}) | q_{1}, q_{2} ∈ Q and M ⊂ Eq(q_{1}, q_{2})\}) . \end{aligned}

Equivalently:

\begin{aligned}z_{(X,P,Q)}M & = (\{x ∈ X | (∄ P_{1}, P_{2} ∈ P)(P_{1} ∩ M = P_{2} ∩ M, x ∈ (P_{1} \ P_{2}))\}∩ \\ & ⋂\{x ∈ X | (∄ Q_{1}, Q_{2} ∈ Q)(Q_{1} ∩ M = Q_{2} ∩ M, x ∈ (Q_{1} \ Q_{2}))\}). \end{aligned}

Thus, by definition

z_{(X,P,Q)}M = z_{(X,P)}M ∩ z_{(X,Q)}M

so (Cf [10], Theorem 3.2) we obtain:

4.2. Theorem. The Zariski closure in 2SSET is a grounded (i.e. z(∅) = ∅), idempotent (i.e z_{(X,P,Q)}(z_{(X,P,Q)}M) = z_{(X,P,Q)}M) and hereditary (i.e. for every M ⊂ Y ⊂ X, z_{(Y,V,W)}M = (z_{(X,P,Q)}M) ∩ Y, where Y is endowed with the initial structure induced by the inclusion Y ⊂ X) closure operator in the sense of [7].

Proof. The intersection of closure operators is a closure operator and all the additional properties above are preserved.

Throughout the paper we will use a property, weak heredity, which is implied by idempotency and hereditariness of z: for every M ⊂ X

z_{z_{(X,P,Q)}M}M = z_{(X,P,Q)}M.

4.3. Remark. Note that z is additive in SSET while it is not in 2SSET . (cf. [13], 3.7)

The terms z-dense map, z-closed embedding (subset) have the usual meaning.

4.4. Corollary. (z-dense map, z-closed embedding) is a factorization structure of 2SSET . That is: every 2-affine map f : X ⟶ Y admits a factorization f = m∘e, where e is a z-dense map and m is a z-closed embedding (factorization property); moreover, for every g ∘ e = m ∘ f, with e z-dense map and m z-closed embedding, there exists a unique d such that m ∘ d = g and d ∘ e = f (diagonalization property).

Proof. Follows from a general result on idempotent and weakly hereditary closure operators (cf.[7], Proposition 3.1, 3.2.)

4.5. Corollary. The epimorphisms in 2SSET_{0} are the z-dense 2affine maps and the regular monomorphisms (= extremal monomorphism) are the z-closed embeddings.

Proof. It is clear from Proposition 3.5, Proposition 3.6 and from the definition of the Zariski closure that z is the regular closure operator induced by 2SSET_{0} in 2SSET . (cf. [7], section 5)

Thus the epimorphisms in 2SSET_{0} are the z-dense 2affine maps and, by weak heredity of z, the regular monomorphisms are the z-closed embeddings.

5. Complete 2affine sets

Recall, from [3], that a full, isomorphism closed subcategory C of 2SSET_{0} is (embedding-) firm reflective if:

(firm1) C is embedding reflective in 2SSET_{0}, i.e. it is reflective and the reflections maps r_{X} : X ⟶ RX, RX ∈ C are embeddings in 2SSET .

(firm2) If f : X ⟶ Y is a z-dense embedding in 2SSET_{0} with Y ∈ C then there is an isomorphism f′ : RX ⟶ Y such that f′ ∘ r_{X} = f.

5.1. Remark. Note that every category has at most one firm reflective subcategory (cf. [3]).

5.2. Remark. The category of metric spaces and non-expansive maps admits as firm reflective subcategory the one of all complete metric spaces. This is a motivating example for the abstract notion of firm epireflection and motivates even the name complete for the members of the firm epireflective subcategory (cf. [2], [3]).

We will show now that 2SSET_{0} admits a firm reflection (i.e. 2SSET_{0} admits completions).

5.3. Definition. X ∈ 2SSET_{0} is called z-injective if it is injective with respect to the class of z-dense embeddings. That is: for every z-dense embedding m : M ⟶ Y and 2affine map f : M ⟶ X there exists an 2affine map f′ : Y ⟶ X such that f′ ∘ m = f.

We denote with Inj(2SSET_{0}) the full subcategory of all z-injective 2affine sets.

The 2affine set S is trivially z-injective and the class of all z-injective 2affine sets is stable under products and z-closed embeddings by a standard argument, so that Inj(2SSET_{0}) is epireflective in 2SSET_{0} by Corollary 4.5. In addition we have:

5.4. Proposition. Inj(2SSET_{0}) is embedding reflective in 2SSET_{0}.

Proof. Let (X, P, Q) be a T_{0} 2affine set and r_{X} : X ⟶ RX the restriction of the canonical map ϕ of Proposition 3.5.2, to z_{S^{P×Q}}(ϕX) = RX. Since z is idempotent RX is z-closed in the z-injective 2affine set S^{P×Q}, so that RX is z-injective.

Let now f : X ⟶ Y be a 2affine map with Y ∈ Inj(2SSET_{0}). Then, by z-injectivity of Y, there is a 2affine map f′ : RX ⟶ Y such that f′∘r_{X} = f. Now r_{X} is an embedding, and it is z-dense since z is weakly hereditary.

5.5. Theorem. Inj(2SSET_{0}) is firm reflective in 2SSET_{0}.

Proof. Thanks to Proposition 5.4 we need to show that the uniqueness property (firm2) is fulfilled.

Now let f : X ⟶ Y be as in (firm 2). Then by the universal property of reflections there exists a (unique) f′ : RX ⟶ Y such that f′ ∘ r_{X} = f.

On the other hand RX is z-injective and f is a z-dense embedding so there exists r′ : Y ⟶ RX with r′ ∘ f = r_{X}.

Now

(r′ ∘ f′) ∘ r_{X} =
r′ ∘ (f′ ∘ r_{X}) =
r′ ∘ f = r_{X} =
1_{RX} ∘ r_{X}

so that r′ ∘ f′ = 1_{RX} by r_{X} (z-dense, hence) epimorphism.

Furthermore

(f′ ∘ r′) ∘ f =
f′ ∘ (r′ ∘ f) =
f′ ∘ r_{X} = f =
1_{Y} ∘ f

so that f′ ∘ r′ = 1_{Y} by f (z-dense, hence) epimorphism.

Thus f′ is an isomorphism and the proof is complete.

5.6. Definition. X ∈ 2SSET_{0} is called absolutely z-closed (resp. z-saturated) if for every embedding (resp. z-dense embedding) k : X ⟶ Y, Y ∈ 2SSET_{0}, kX is z-closed in Y (resp. k is an isomorphism).

5.7. Proposition. For X ∈ 2SSET_{0} equivalent are:

1. X is z-injective

2. X is absolutely z-closed

3. X is z-saturated.

Proof. 1 ⇒ 2 Let X be z-injective, f : X ⟶ Y be an embedding with Y ∈ 2SSET_{0} and f = m ∘ e a factorization of f, with e z-dense map and m z-closed embedding.

We need to show that e is an isomorphism. Since X is z-injective the identity 1_{X} admits an extension α : M ⟶ X, i.e. there exists α such that α ∘ e = 1_{X}. Thus e is a section and an embedding so it is an isomorphism.

2 ⇒ 3 Trivial.

3 ⇒ 1 Let X be z-saturated. Then r_{X} : X ⟶ RX is a z-dense embedding (cf. Proposition 5.4) and so it is an isomorphism and RX ∈ Inj(2SSET_{0}).

6. Completions in coreflective subcategories

Let X be a (bi-)coreflective subcategory of 2SSET ; since 2SSET is topological over Set, X bicoreflective means that X is stable under quotients (= final surjective 2affine maps) and coproducts (= disjoint unions with final structures). Furthermore it is not restrictive to assume that the coreflection c_{X} : X ⟶ CX of X in X is the identity 1_{X} in the underlying sets.

Then X is still a topological category over Set (cf. [1]), thus in particular (surjective map, embedding (in X)) is a factorization structure of X.

Let us assume, in addition, that X is hereditary in 2SSET ; that is: if f : X ⟶ Y is an embedding with Y ∈ X then X ∈ X.

Then the above factorization structure coincides with the restriction to X of the factorization structure of 2SSET .

Again by heredity of X the Zariski closure in 2SSET can be restricted to X.

Let Ŝ be the coreflection of S in X.

6.1. Lemma. Let (X, P, Q) ∈ X and f : X ⟶ S × S be a function. Then f : (X, P, Q) ⟶ Ŝ is 2affine if and only if f : (X, P, Q) ⟶ S is 2affine.

Proof. The proof directly follows by the universal property of bicoreflections.

84  ELISABETTA FELACO AND ERALDO GIULI

Set X_{0} = X ∩ 2SSET_{0}. 6.2. Proposition.

(a) The (z-dense map, z-closed embedding)-factorization of 2SSET_{0} can be restricted to X_{0}.

(b) The inclusion functor K : X_{0} ⟶ 2SSET_{0} preserves epimorphisms and regular (= extremal) monomorphisms.

Proof.

(a) Directly follows from the heredity of X.

(b) Lemma 3.4 together with Lemma 6.1 say that the Zariski closure of 2SSET , restricted to X, is the regular closure of X induced by X_{0}. So the proof follows from heredity of z.

Now we are ready for the proof of : 6.3. Theorem. Let X be a bicoreflective and hereditary subcategory of 2SSET and let X_{0} consist of T_{0} 2affine sets in X. Then X_{0} admits a firm epireflection given by Inj(X_{0}), i.e. the z-injective 2affine sets of X_{0}. Proof. Follows the lines of the proof of Proposition 5.4 and Theorem 5.5, with Inj(X_{0}), X_{0} and Ŝ instead of Inj(2SSET_{0}), 2SSET_{0} and S, taking in account Proposition 6.2 and Lemma 6.1. 6.4. Corollary. The complete 2affine sets of X_{0} are precisely the z-closed 2affine subsets of products in X of Ŝ. 6.5. Remark. Products in X are the coreflections of products in 2SSET . 6.6. Corollary. For X ∈ X_{0} equivalent are:

1. X is z-injective in X_{0}

2. X is absolutely z-closed in X_{0}

3. X is z-saturated in X_{0}. Proof. As in Proposition 5.7.

7. Hereditary coreflections defined by algebra structures

In this section we give a general method of constructing hereditary coreflections. Recall that an algebra structure in the set S is a family of functions

Ω = {ω_{T} : S^{T} ⟶ S}

where T runs in a given class of sets.

Then, for every set X, the power set S^{X} carries a Ω-algebra structure defined, for every operation ω_{T} on S and family of functions (f_{t} : X ⟶ S)_{t∈T}, by

ω_{T}^{X}(f_{t}) = ω_{T} ∘ (< f_{t} >).

If Ω_{1} and Ω_{2} are algebra structures in S, we denote by 2SSET (Ω_{1}, Ω_{2}) the subcategory of 2SSET consisting of those 2affine sets (X, P, Q) for which P and Q are respectively Ω_{1}- and Ω_{2}-subalgebras of the function algebra S^{X}.

7.1. Proposition. 2SSET (Ω_{1}, Ω_{2}) is a hereditary coreflective subcategory of 2SSET .

Proof. For every (X, P, Q) ∈ 2SSET the object (X, Ω_{1}(P), Ω_{2}(Q)), where Ω_{1}(P) is the Ω_{1}-subalgebra generated by P (i.e. the intersection of all Ω_{1}-subalgebras containing P) and Ω_{2}(Q), is the Ω_{2}-subalgebra generated by Q is in 2SSET (Ω_{1}, Ω_{2}).

1_{X} : (X, Ω_{1}(P), Ω_{2}(Q)) ⟶ (X, P, Q)

is the 2SSET (Ω_{1}, Ω_{2})-coreflection of (X, P, Q).

Moreover the structure in a 2affine subset is the restriction to the subset of the two structures in the set, so that 2SSET (Ω_{1}, Ω_{2}) is also hereditary.

7.2. Example.

1. Let us consider the algebra structures

Ω_{1} = Ω_{2} = {(⋁_{T} : S^{T} ⟶ S | T any set) ⋃(c_{1} : S ⟶ S)} = Ω.

Then 2SSET (Ω_{1}, Ω_{2}) is the category of 2closure spaces, in fact the objects are 2affine sets (X, P, Q), with (X, P) and (X, Q) closure spaces and the morphisms are functions continuous with respect to both the structures. We will denote the category of 2closure spaces by 2CS. (Recall that a closure space (X, C) is a pair, where X is a set and C is a subset of the power set P(X) satisfying the condition that X belongs to C and that C is stable under arbitrary unions, and that a morphism between closure spaces is an ordinary continuous map (cf. [5]).)

2. Take

Ω_{1} = Ω_{2} = {(⋁_{T} : S^{T} ⟶ S | T any set), (⋀_{F} : S^{F} ⟶ S | F any finite set)} = Ω′.

2SSET (Ω_{1}, Ω_{2}) is the category of bitopological spaces, usually denoted by 2Top (cf. [13], [15]). The fact that 2Top_{0} has a firm epireflection was firstly shown in [13].

3. Let us consider the algebra structures

Ω_{1} = Ω_{2} = {(⋁_{F} : S^{F} ⟶ S | F any finite set) ∪ (⋀_{F} : S^{F} ⟶ S | F any finite set)∪

∪(τ : S ⟶ S | τ(0) = 1 e τ(1) = 0)} = Ω′′.

This is a Boolean algebra structure; 2SSET (Ω_{1}, Ω_{2}) is the category of 2fields of sets.

4. Every combination of two of the previous Ω-structures gives a new coreflective subcategory. For example 2SSET (Ω′, Ω′′) is the category of 2affine sets (X, P, Q), where P is a topology and Q is a field of subsets of X. Moreover there is a functor

F : Top ⟶ 2SSET (Ω_{1}, Ω_{2})

such that F (X, P) = (X, P, Q), where Q is the family of clopen sets in (X, P). We can also define the functor

F′ : 2SSET (Ω_{1}, Ω_{2}) ⟶ Top

such that F (X, P, Q) = (X, U) where U is the topology generated by P ∪ Q. As F ∘ F′ = 1_{Top} F is a retraction.

Question: We don’t know any non trivial (i.e. P = P (X) or Q = P (X)) example of hereditary coreflective subcategory X_{0} of 2SSET_{0} which is not of the form 2SSET (Ω_{1}, Ω_{2}).

8. The structure of complete T_{0} 2affine sets

8.1. Definition. Let (X, P, Q) ∈ 2SSET and a ⊂ P, b ⊂ Q both containing the whole subset X and not the empty set.

The one-point extension of (X, P, Q) defined by the pair (a, b) is the 2affine set denoted (X_{∞}, P_{a}, Q_{b}) with X_{∞} = X ∪ ∞, ∞ ∉ X, and P_{a}, Q_{b} respectively defined by

P_{a} = {A ∪ ∞ | A ∈ a} ∪ (P \ a)
Q_{b} = {B ∪ ∞ | B ∈ b} ∪ (Q \ b).

Since a and b contain X and not ∅, P_{a} and Q_{b} contain X ∪ {∞} and the empty set so (X_{∞}, P_{a}, Q_{b}) ∈ 2SSET and (X, P, Q) is clearly a 2affine subset of (X_{∞}, P_{a}, Q_{b}) ∈ 2SSET .

Moreover ∞ ∈ zX since, for every P ∈ P and Q ∈ Q never both P and P ∪ {∞} or both Q and Q ∪ {∞} belong respectively to P_{a} or to Q_{b}.

Let (X, P, Q) be a 2affine set and x ∈ X. We will denote by P_{x} and Q_{x} respectively the set of P-open sets containing x and of Q-open sets containing x.

Assume now that (X, P, Q) is separated. Then the one point extension (X_{∞}, P_{a}, Q_{b}), defined by (a, b) is separated if and only if for all x ∈ X (a, b) ≠ (P_{x}, Q_{x}).

8.2. Theorem. (X, P, Q) ∈ 2SSET_{0} is complete if and only if for every pair (a, b) as above there exists (unique) x ∈ X such that (a, b) = (P_{x}, Q_{x}).

Proof. (⇒)

Assume there exists (a, b) such that (a, b) ≠ (P_{x}, Q_{x}) for every x ∈ X and consider the inclusion map:

k : (X, P, Q) ⟶ (X_{∞}, P_{a}, Q_{b}).

¿From the above observations we know that k is a z-dense embedding in a separated 2affine set, so, by Theorem 5.5 and Proposition 5.7, (X, P, Q) is not complete.

(⇐)

Assume that (X, P, Q) is not complete. Since the Zariski closure is weakly hereditary, we may assume that there exists a separated 2affine set (X ∪ {∞}, V, W) such that the inclusion map:

k : (X, P, Q) ⟶ (X ∪ {∞}, V, W)

is a z-dense embedding. Set

a = {P ∈ P | P ∉ V}

and

b = {Q ∈ Q | Q ∉ W}.

Since (X, P, Q) is a 2affine subset of (X ∪ {∞}, V, W) we have P ∪ {∞} ∈ V whenever P ∈ a and Q ∪ {∞} ∈ W whenever Q ∈ b, i.e. a ⊂ {P ∈ P | P ∪ {∞} ∈ V} and b ⊂ {Q ∈ Q | Q ∪ {∞} ∈ W}.

On the other hand, by X z-dense in (X ∪ {∞}, V, W) we have P ∈ a whenever P ∪ {∞} ∈ V and Q ∈ b whenever Q ∪ {∞} ∈ W, i.e. {P ∈ P | P ∪ {∞} ∈ V} ⊂ a and {Q ∈ Q | Q ∪ {∞} ∈ W} ⊂ b.

This means that (X ∪ {∞}, V, W) is the one-point extension of (X, P, Q) defined by the pair (a, b). As (X ∪ {∞}, V, W) is separated we have that (a, b) ≠ (P_{x}, Q_{x}) for every x ∈ X.

8.3. Definition. Let X be a hereditary coreflective subcategory of 2SSET , and set X_{0} = X ∩ 2SSET_{0} as done before.

Let (X, P, Q) ∈ X and a ⊂ P, b ⊂ Q, both containing the whole set X and not the empty set. We say that (a, b) is a X-compatible pair if (X_{∞}, P_{a}, Q_{b}) ∈ X.

Using the same argument as in Theorem 8.2 we obtain:

8.4. Theorem. Let X be a hereditary coreflective subcategory of 2SSET_{0}, and X_{0} = X ∩ 2SSET_{0}.

(X, P, Q) ∈ X_{0} is (X_{0}-)complete if and only if for every X-compatible pair (a, b) there exists a unique x ∈ X such that (a, b) = (P_{x}, Q_{x}).

9. Comparison with completions in the category of affine sets.

Recall from [11] that in the one-dimensional case, (X, P) ∈ T_{0} is complete if and only if for every a ⊂ P there is a (unique) x ∈ X such that a = P_{x}.

Moreover we can define complete objects in SSET in the following way:

9.1. Definition. (X, P) ∈ SSET is complete if for every a ⊂ P there is an x ∈ X such that a = P_{x}.

(X, P, Q) ∈ 2SSET is complete if for every a ⊂ P and b ⊂ Q there is an x ∈ X such that a = P_{x} and b = Q_{x}.

Let T_{0} : SSET ⟶ SSET_{0} be the reflector to SSET_{0} and 2T_{0} : 2SSET ⟶ 2SSET_{0} be the reflector to 2SSET_{0}.

9.2. Lemma. (X, P) ∈ SSET is complete if and only if T_{0}(X, P) is complete and (X, P, Q) ∈ 2SSET is complete if and only if 2T_{0}(X, P, Q) is complete.

For (X, P, Q) ∈ 2SSET , let us consider the three functors F_{1}, F_{2}, F_{3} : 2SSET ⟶ SSET defined by:

F_{1} : ((X, P, Q)) = (X, P)
F_{2} : ((X, P, Q)) = (X, Q)
F_{3} : ((X, P, Q)) = (X, P ∪ Q).

9.3. Theorem. F_{1}, F_{2}, and F_{3} preserve completeness.

Proof. Assume that (X, P, Q) is complete. We want to show that also (X, P) is complete, i.e., by Definition 9.1 that for every a ⊂ P there is an x ∈ X such that a = P_{x}. Let a ⊂ P be fixed and b = {X}. As (X, P, Q) is complete, there is x ∈ X such that b = Q_{x} and a = P_{x}. So F_{1} preserves completeness.

The proof for F_{2} is similar.

In the case of F_{3}, take d ⊂ P ∪ Q and assume d ∩ P = a and d ∩ Q = b; as (X, P, Q) is complete, there is x ∈ X such that a = P_{x} and b = Q_{x}, so d = D_{x} = P_{x} ∪ Q_{x} and (X, P ∪ Q) is complete.

9.4. Lemma. If (X, P, Q) ∈ 2SSET is complete then for every x ≠ x′ ∈ X there is z ∈ X such that P_{x} = P_{z} and Q_{x′} = Q_{z}.

Proof. Take x, x′ ∈ X, x ≠ x′ and set a = P_{x} and b = Q_{x′}. Note that X ∈ a, b and ∅ ∉ a, b; so, as (X, P, Q) is complete, there is a unique z ∈ X such that a = P_{x} = P_{z} and b = Q_{x′} = Q_{z}.

9.5. Theorem. A separate 2affine set (X, P, Q) is complete if and only if:

1. (X, P) is complete;

2. (X, Q) is complete;

3. for every x ≠ x′ ∈ X there is z ∈ X such that P_{x} = P_{z} and Q_{x′} = Q_{z}.

Proof. (⇒)

If (X, P, Q) is complete we have 1 and 2 by Theorem 9.3 and 3 by Lemma 9.4.

(⇐)

Assume 1,2,3, hold and take a ⊂ P and b ⊂ Q; by 1 and 2 there are x, x′ ∈ X such that a = P_{x} and b = Q_{x′}. Finally, by 3, there is x ∈ X such that P_{x} = P_{z} and Q_{x′} = Q_{z}, so (X, P, Q) is complete.

9.6. Remark. Bitopological spaces fulfilling property 3 were called intertwining bispaces by Sergio Salbany [18] and that property was introduced to give an internal characterization of the injective T_{0} bispaces introduced and studied in [13].

References

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[3] G.C.L. Brummer, E. Giuli, H.Herrlich, Epireflections which are completions, Cahiers Topologie Geom. Diff. Categ. 33 (1992), 71-93.

[4] M.M.Clementino, E.Giuli, W.Tholen, Topology in a category: compactness, Portugal. Math., 53 (1996), 397-433.

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[8] D.Dikranjan, W.Tholen, Categorical structure of closure operators, Kluwer Academic Publishers, Dordrecht (1995).

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[11] E.Giuli, The structure of affine algebraic sets, Categorical Structures and their Applications, Berlin, 2003, World Scientific Publications, Singapore 2004, 113-121.

[12] E.Giuli, Zariski closure, completeness and compactness, Topology Appl., 153 (2006), 3158-3168.

[13] E.Giuli, S.Salbany, 2T_{0}-spaces and closure operators, Seminarberichte Fach. Math. und Inf. (FernUniversitat Hagen) 28 (1988) 11-40.

[14] E.Giuli, W. Tholen, A topologist’s view of Chu spaces, Appl. Categ. Structures, 15 (2007), 573-598.

[15] J.C. Kelly, Bitopological spaces, Proc. London Math. Soc. (3) 13 (1983) 71-89.

[16] S.Mac Lane Categories for the working mathematician, Graduate Texts in Mathematics, Springer (1997).

[17] Th.Marny, On epireflctive subcategories of topological categories, Gen. Topology Appl. 10 (1979), 392-405.

[18] S.Salbany, On injective topological spaces and bispaces, Topology Proceedings, 142 (1998), 309-322.

[19] L.Skula, On a reflective subcategory of the category of all topological spaces, Amer. Math. Soc. 142 (1969), 137-141.

Dipartimento di Matematica Pura ed Applicata, Universita degli Studi dell’Aquila, 67100 L’Aquila, Italia

Email: elisabetta.felaco@univaq.it
      giuli@univaq.it

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