Interor and \(\mathfrak h\) operators of the category of locales

1. Introduction

Kuratowski operators (closure, interior, exterior, boundary and others) have been used intensively in General(set-theoretic) Topology ([1,2,3]). For a topological space it is well-known that, for example, the associated closure and interior operators provide equivalent descriptions of the topology; but this is not always true in other categories, consequently it makes sense to define and study separately these operators. In this context, we study an interior operator \(I\) on the the coframe $\mathcal{S}_{\mathbf{l}}(L)$ of sublocales of every object \(L\) in the category \(\mathbf{Loc}\).

On the other hand, a new topological operador \(\mathfrak h\) was introduced by M. Suarez [4] in order to complete a Boolean algebra with all topological operators in General Topology. Following his ideas, we study an operator \(\mathfrak h\) on the collection \(\mathcal{S}_{\mathbf{l}}^{c}(L)\) of all complemented sublocales of every object \(L\) in the category \(\mathbf{Loc}\).

The paper is organized as follows, we begin presenting, in S3, the basic concepts of Heyting algebras, Frames, locales, sublocales, images and preimages of sublocales for the morphisms of \(\mathbf{Loc}\) and the notions of closed and open sublocales; these notions can be found in Picado and Pultr [5] and A. L. Suarez [6]. In S3, we present the concept of interior operator \(I\) on the category \(\mathbf{Loc}\) and then we construct a topological category \(\big(\mathbf{I\text{-}Loc},\ U\big)\), where \(U:\mathbf{I\text{-}Loc}\rightarrow \mathbf{Loc}\) is a forgetful functor. Finally in S4 we present the notion of \(\mathfrak h\) operator on the category \(\mathbf{Loc}\) and discuss some of their properties for constructing the topological category \(\big(\mathbf{\mathfrak h\text{-}Loc},\ U\big)\) associated to the forgetful functor \(U:\mathbf{\mathfrak h\text{-}Loc}\rightarrow \mathbf{Loc}\).

2. Preliminaries

For a comprehensive account on the the categories of frames and locales we refer to Picado and Pultr [5] and A. L. Suarez [6], from whom we take the following notions:

• The adjunction \(\Omega : \mathbf{Top} \rightarrow \mathbf{Frm}^{op} \), \(\mathbf{pt}:\mathbf{Frm}^{op}\rightarrow \mathbf{Top}\) with \(\Omega \dashv \mathbf{pt}\), connects the categories of frames with that of topological spaces. The functor \(\Omega\) assigns to each space its lattice of opens, and \(\mathbf{pt}\) assigns to a frame \(L\) the collection of the frame maps \(f : L \rightarrow 2\), topologized by setting the opens to be exactly the sets of the form \( \{f: L \rightarrow 2 \mid f(a) = 1\}\) for some \(a\in L\).
• A frame \(L\) is spatial if for \(a, b \in L\) whenever \(a\nleqslant b\) there is some frame map \(f: L \rightarrow 2\) such that \(f(a) = 1\ne f(b)\). Spatial frames are exactly those of the form \(\Omega(X)\) for some space \(X\).
• A space is sober if every irreducible closed set is the closure of a unique point. Sober spaces are exactly those of the form \(\mathbf{pt}(L)\) for some frame \(L\).
• The adjunction \(\Omega\dashv \mathbf{pt}\) restricts to a dual equivalence of categories between spatial frames and sober spaces.
• The category of sober spaces is a full reflective subcategory of \(\mathbf{Top}\). For each space \(X\) we have a sobrification map \(N : X\rightarrow \mathbf{pt}(\Omega(X))\) mapping each point \(x\in X\) to the map \(f_x :(X) \rightarrow 2\) defined as \(f(U) = 1\) if and only if \(x\in U\).
• The category of spatial frames is a full reflective subcategory of \(\mathbf{Frm}\). For each frame we have a spatialization map \(\phi : L \rightarrow \Omega(\mathbf{pt}(L))\) which sends each \(a\in L\) to \(\{f : L \rightarrow 2 \mid f(a) = 1\}\).
• We call \(\mathbf{Loc}\) the category \(\mathbf{Frm}^{op}\), and we call its objects locales. Maps in the category of locales have a concrete description: they can be characterized as the right adjoints of frame maps (since frame maps preserve all joins, they always have right adjoints).
• A sublocale of a locale \(L\) is a subset \(S\subseteq L\) such that it is closed under arbitrary meets, and such that \(s\in S\) implies \(x\rightarrow s \in S\) for every \(x \in L\). This is equivalent to \(S\subseteq L\) being a locale in the inherited order, and the subset inclusion being a map in \(\mathbf{Loc}\).
• Sublocales of \(L\) are closed under arbitrary intersections, and so the collection \(\mathcal{S}_{\mathbf{l}}(L)\) of all sublocales of \(L\), ordered under set inclusion, is a complete lattice. The join of sublocales is (of course) not the union, but we have a very simple formula \(\bigvee_{i} S_{i} = \{\bigvee M \mid M \subseteq\bigcup_{i} S_{i}\}\).
• In the coframe \(\mathcal{S}_{\mathbf{l}}(L)\) the bottom element is the sublocale \(\{1\}\) and the top element is \(L\).
• Let \(f: L\rightarrow M\) be a localic map. The set-theoretic preimage \(f^{-1}[T]\) of a sublocale \(T\subseteq M\) is not necessarily a sublocale of \(L\). To obtain a concept of a preimage suitable for our purposes we will, first, make the following observation: “Let \(A\subseteq L\) be a subset closed under meets. Then \(\{1\} \subseteq A\) and if \(S_i \subseteq A\) for \(i \in J\) then \(\bigwedge_{i\in J} S_i\subseteq A\)”. Consequently there exists the largest sublocale contained in \(A\). It will be denoted by \(A_{sloc}\).
• The set-theoretic preimage \(f^{-1}[T]\) of a sublocale \(T\) is closed under meets \big(indeed, \(f(1) = 1\), and if \(x_i \in f^{-1}[T])\) then \(f(x_i) \in T\), and hence \( f(\bigwedge_{i\in J} x_i)=\bigwedge_{i\in J} f(x_i)\) belongs to \(T\) and \(\bigwedge_{i\in J} x_i\in f^{-1}[T]\) \big) and we have the sublocale \(f_{-1}[T]:= f^{-1}[T]_{sloc}\). It will be referred to as the preimage of \(T\), and we shall sat that \(f_{-1}[-]\) is the preimage function of \(f\).
• For every localic map \(f: L \rightarrow M\), the preimage function \(f_{-1}[-] \) is a right Galois adjoint of the image function \(f [-] :\mathcal{S}_{\mathbf{l}}(L)\rightarrow \mathcal{S}_{\mathbf{l}}(M)\).
• Embedded in \(\mathcal{S}_{\mathbf{l}}(L)\) we have the coframe of closed sublocales which is isomorphic to \(L^{op}\). The closed sublocale \(\mathfrak c(a) \subseteq L\) is defined to be \(\uparrow a\) for \(a \in L\).
• Embedded in \(\mathcal{S}_{\mathbf{l}}(L)\) we also have the frame of open sublocales which is isomorphic to \(L\). The open sublocale is defined to be \(\{a \rightarrow x \mid x \in L\}\) for \(a \in L\).
• The sublocales \(\mathfrak o(a)\) and \(\mathfrak c(a)\) are complements of one another in the coframe \(\mathcal{S}_{\mathbf{l}}(L)\) for any element \(a\in L\). Furthermore, open and closed sublocales generate the coframe \(\mathcal{S}_{\mathbf{l}}(L)\) in the sense that for each \(S \in \mathcal{S}_{\mathbf{l}}(L)\) we have \(S = \bigcap\{\mathfrak o(x) \cup \mathfrak c(y) \mid S \subseteq \mathfrak o(x) \cup \mathfrak c(y)\}\).
• A pseudocomplement of an element \(a\) in a meet-semilattice \(L\) with \(0\) is the largest element \(b\) such that \(b\land a = 0\), if it exists. It is usually denoted by \(\neg a\). Recall that in a Heyting algebra \(H\) the pseudocomplement can be expressed as \(\neg x= x\rightarrow 0\).

3. Interior Operators

We shall be conserned in this section with a version on locales of the interior operator studied in [7]. Before stating the next definition, we need to observe that since for localic maps \(f: L \rightarrow M\) and \(g:M\rightarrow N\):

• the preimage function \(f_{-1}[-] \) is a right Galois adjoint of the image function \(f [-] :\mathcal{S}_{\mathbf{l}}(L)\rightarrow \mathcal{S}_{\mathbf{l}}(M)\);
• \( g [-]\circ f [-]=(g\circ f) [-]\).

Therefore \(g_{-1}[-]\circ f_{-1}[-]= (g\circ f)_{-1}[-]\) because given two adjunctions the composite functors yield an adjunction.

Definition 1. An interior operator \(I\) of the category \(\mathbf{Loc}\) is given by a family \(I =(i_{L})_{L\in \mathbf{Loc}}\) of maps \(i_{L}:\mathcal{S}_{\mathbf{l}}(L)\rightarrow \mathcal{S}_{\mathbf{l}}(L)\) such that

• (\(I_1)\) \(\left(\text{Contraction}\right)\) \(i_{L}(S)\subseteq S\) for all \(S \in \mathcal{S}_{\mathbf{l}}(L)\);
• (\(I_2)\) \(\left(\text{Monotonicity}\right)\) If \(S\subseteq T\) in \(\mathcal{S}_{\mathbf{l}}(L)\), then \(i_{L}(S)\subseteq i_{L}(T)\)
• (\(I_3)\) \(\left(\text{Upper bound}\right)\) \(i_{L}(L)=L\).

Definition 2. An \(I\)-space is a pair \((L,i_{L})\) where \(L\) is an object of \(\mathbf{Loc}\) and \(i_{L}\) is an interior operator on \(L\).

Definition 3. A morphism \(f:L\rightarrow M\) of \(\mathbf{Loc}\) is said to be \(I\)-continuous if \begin{equation}\label{conti}\tag{1} f_{-1}\left[ i_{M}(T)\right]\subseteq i_{L}\left( f_{-1}[T]\right) \end{equation} for all \(T\in \mathcal{S}_{\mathbf{l}}(M)\). Where \(f_{-1}[-]\) is the preimage of \(f[-]\).

Proposition 1. Let \(f:L\rightarrow M\) and \(g:M\rightarrow N\) be two \(I\)-continuous morphisms of \(\mathbf{Loc}\) then \(g\centerdot f\) is an \(I\)-continuous morphism of \(\mathbf{Loc}\).

Proof. Since \(g:M\rightarrow N\) is \(I\)-continuous, we have $$g_{-1}\big[ i_{N}(S)\big]\subseteq i_{M}\big( g_{-1}[S]\big)$$ for all \(S\in \mathcal{S}_{\mathbf{l}}(N)\), it fallows that \[f_{-1}\Big[g_{-1}\big[( i_{N}(S)\big]\Big]\subseteq f_{-1}\Big[ i_{M}\big( g_{-1}[S]\big)\Big];\] now, by the \(I\)-continuity of \(f\),$$ f_{-1}\Big[ i_{M} \big( g_{-1}[S]\big)\Big]\subseteq i_{L}\Big( f_{-1}\big[g_{-1}[S]\big]\Big),$$ therefore \[f_{-1}\Big[g_{-1}\big[ i_{N}(S)\big]\Big]\subseteq i_{L}\Big( f_{-1}\big[g_{-1}[S]\big]\Big),\] that is to say \[(g\centerdot f)_{-1}\big[ i_{N}(S)\Big]\subseteq i_{L}\Big( (g\centerdot f)_{-1}[S]\Big)\]

As a consequence we obtain;

Definition 4. The category \(\mathbf{I\text{-}Loc}\) of \(I\)-spaces comprises the following data:

Objects: Pairs \((L,i_{L})\) where \(L\) is an object of \(\mathbf{Loc}\) and \(i_{L}\) is an interior operator on \(L\).

Morphisms: Morphisms of \(\mathbf{Loc}\) which are \(I\)-continuous.

3.1. The lattice structure of all interior operators

For the category \(\mathbf{Loc}\) we consider the collection \[ Int(\mathbf{Loc}) \] of all interior operators on \(\mathbf{Loc}\). It is ordered by \[ I\leqslant J \Leftrightarrow i_{L}(S)\subseteq j_{L}(S), \;for\; all\; S\in \mathcal{S}_{\mathbf{l}}\; and\; all\; L \; object \;of \;\mathbf{Loc}. \] This way \(Int(\mathbf{Loc})\) inherits a lattice structure from \(\mathcal{S}_{\mathbf{l}}\):

Proposition 2. Every family \((I_{\lambda})_{\lambda\in \Lambda}\) in \(Int(\mathbf{Loc})\) has a join \(\bigvee\limits_{\lambda\in \Lambda }I_{\lambda }\) and a meet \(\bigwedge\limits_{\lambda\in \Lambda }I_{\lambda}\) in \(Int(\mathbf{Loc})\). The discrete interior operator \[ I_{D}=({i_{D}}_{L})_{\text{\(L\in \mathbf{Loc}\)}}\;with\;{i_{D}}_{L}(S)=S \;for \;all\;S\in \mathcal{S}_{\mathbf{l}} \] is the largest element in \(Int(\mathbf{Loc})\), and the trivial interior operator \[ I_{T}=({i_{T}}_L)_{L\in \mathbf{Loc}}\; with\; {i_{T}}_{L}(S)= \begin{cases} \{1\}& \text{for all}S\in \mathcal{S}_{\mathbf{l}},S\ne L\\ L&\text {if}S=L \end{cases} \] is the least one.

Proof. For \(\Lambda\ne\emptyset\), let \(\widehat{I}=\bigvee\limits_{\lambda\in\Lambda}I_{\lambda }\), then \[ \widehat{i_{L}}=\bigvee\limits_{\lambda\in \Lambda} {i_{\lambda }}_{L}, \] for all \(L\) object of \(\mathbf{Loc}\), satisfies

• \(\widehat{i_{L}}(S)\subseteq S\), because \({i_\lambda}_{L}(S)\subseteq S\) for all \(S\in \mathcal{S}_{\mathbf{l}}\) and for all \(\lambda \in \Lambda\).
• If \(S\leqslant T\) in \(\mathcal{S}_{\mathbf{l}}\) then \({i_\lambda}_{L}(S)\subseteq {i_\lambda}_{L}(T)\) for all \(S\in \mathcal{S}_{\mathbf{l}}\) and for all \(\lambda \in \Lambda\), therefore \(\widehat{i_{S}}(S)\subseteq \widehat{i_{L}}(T)\).
• Since \({i_\lambda}_{L}(L)=L \) for all \(\lambda \in \Lambda\), we have that \( \widehat{i_{L}}(L)=L\).

Similarly \(\bigwedge\limits_{\lambda\in \Lambda}I_\lambda\), \(I_{D}\) and \(I_{T}\) are interior operators.

Corollary 1. For every object L of \(\mathbf{Loc}\) \[ Int(L) = \{i_{L}\mid i_{L}\text{ is an interior operator on}L\} \] is a complete lattice.

3.2. Initial interior operators

Let \(\mathbf{I\text{-}Loc}\) be the ctegory of \(I\)-spaces. Let \((M,i_{M})\) be an object of \(\mathbf{I\text{-}Loc}\) and let L be an object of \(\mathbf{Loc}\). For each morphism \(f:L\rightarrow M\) in \(\mathbf{Loc}\) we define on L the operotor \begin{equation} \label{initial}\tag{2} i_{L_{f}}:=f_{-1}[-]\centerdot i_{M}\centerdot f[-]. \end{equation}

Proposition 3. The operator (\ref{initial}) is an interior operator on L for which the morphism \(f\) is \(I\)-continuous.

Proof.

[(\(I_1)\)] \(\left(\text{Contraction}\right)\) \(i_{L_{f}}(S)= f_{-1}\big[i_{M}\big( f[S]\big)\big]\subseteq f_{-1}\big[f[S]\big]\subseteq S\) for all \(S\in \mathcal{S}_{\mathbf{l}}\);

[(\(I_2)\)] \(\left(\text{Monotonicity}\right)\) \(S\subseteq T\) in \(\mathcal{S}_{\mathbf{l}}\), implies \(f[S]\subseteq f[T]\), then \(i_{M}\big( f[S]\subseteq i_{M}\big( f[T]\big)\), consequently \( f_{-1}\Big[ i_{M}\big( f[S]\big)\big]\subseteq f_{-1}\big[ i_{M}\big( f[T]\big)\big]\);

[(\(I_3)\)] \(\left(\text{Upper bound}\right)\) \(i_{L_{f}}(L)=f_{-1}\big[ i_{M}\big( f[L]\big)\big]=L\).

Finally, \begin{align*} f_{-1}\big[i_{M}(T)\big]&\subseteq f_{-1}\big[i_{M} \big( f\big[[ f_{-1}[T]\big)\big]\subseteq i_{L_{f}}\big(f_{-1}[T]\big), \end{align*} for all \(T\in \mathcal{S}_{\mathbf{l}}\).

It is clear that \( i_{L_{f}}\) is the coarsest interior operator on L for which the morphism \(f\) is \(I\)-continuous; more precisaly

Proposition 4. Let \((L,i_{L})\) and \((M,i_{M})\) be objects of \(\mathbf{I\text{-}Loc}\), and let \(N\) be an object of \(\mathbf{Loc}\). For each morphism \(g:N\rightarrow L\) in \(\mathbf{Loc}\) and for \(f:(L,i_{L_{f}})\rightarrow (M,i_{N})\) an \(I\)-continuous morphism, \(g\) is \(I\)-continuous if and only if \(f\centerdot g\) is \(I\)-continuous.

Proof. Suppose that \(g\centerdot f\) is \(I\)-continuous, i. e. \[(f\centerdot g)_{-1}\big[i_{M}(T)\big]\subseteq i_{N}\big( (f\centerdot g)_{-1}[T] \big)\] for all \(T\in \mathbf{S(N)}\). Then, for all \(S\in \mathcal{S}_{\mathbf{l}}\), we have \begin{align*} g_{-1}\big[i_{L_{f}}(S)\big]&=g_{-1}\big[f_{-1}\centerdot i_{M}\centerdot f[S]\big]=(f\centerdot g)_{-1}\big[ i_{M}\big( f[S]\big) \big]\\ &\subseteq i_{N}\big( (f\centerdot g)_{-1}\big[f[S]\big) \big)=i_{N}\big( g_{-1}\centerdot f_{-1}\centerdot f[S]\big)\\ &\subseteq i_{N}\big( g_{-1}[S]\big),\\ \end{align*} i.e. \(g\) is \(I\)-continuous.

As a consequence of Corollary 1, Proposition 3 and Proposition 4 (cf. [8] or [9]), we obtain

Theorem 1. The forgetful functor \(U:\mathbf{I\text{-}Loc}\rightarrow \mathbf{Loc}\) is topological, i.e. the concrete category \(\big(\mathbf{I\text{-}Loc},\ U\big)\) is topological.

3.3. Open subobjects

Definition 5. An sublocale S of a locale L is called \(I\)-open \big(in L\big) if it is isomorphic to its \(I\)-interior, that is: if \(i_{L}(S)= S\).

The \(I\)-continuity condition (\ref{conti}) implies that \(I\)-openness is preserve by inverse images:

Proposition 5. Let \(f:L\rightarrow M\) be a morphism in \(\mathbf{Loc}\). If \(T\) is \(I\)-open in \(M\), then \(f_{-1}(T)\) is \(I\)-open in L.

Proof. If \(T= i_{M}(T)\) then \(f_{-1}[T]=f_{-1}\big[i_{M}(T)\big]\subseteq i_{L}\big(f_{-1}[T]\big)\), therefore \(i_{L}\big(f_{-1}[T]\big)=f_{-1}[T]\).

4. \(\mathfrak h\) Operators

In this section we shall be conserned with a weak categorical version of a topological function studied by M, Suarez M. in [4]. For that purpose we will use the colection \(\mathcal{S}_{\mathbf{l}}^{c}(L)\) of all complemented sublocales of a locale L (See P, T. Johnston [10], for example).

Definition 6. An \(\mathfrak h\) operator of the category \(\mathbf{Loc}\) is given by a family \(\mathfrak h =(h_{L})_{L\in \mathbf{Loc}}\) of maps \(h_{L}:\mathcal{S}_{\mathbf{l}}^{c}(L)\rightarrow \mathcal{S}_{\mathbf{l}}^{c}(L)\) such that

• [(\(h_1\))] \(S\cap h_{L}(S)\subseteq S\), for all \(S \in\mathcal{S}_{\mathbf{l}}^{c}(L)\);
• [(\(h_2\))] If \(S\subseteq T\) then \(S\cap h_{L}(S)\subseteq T\cap h_{L}(T)\), for all \(S,T \in\mathcal{S}_{\mathbf{l}}^{c}(L)\);
• [(\(h_3\))] \( h_{L}(L)=L\).

Definition 7. An \(\mathfrak h\)-space is a pair \((L,h_{L})\) where L is an object of \(\mathbf{Loc}\) and \(h_{L}\) is an \(\mathfrak h\) operator on L.

Definition 8. A morphism \(f:L\rightarrow M\) of \(\mathbf{Loc}\) is said to be \(\mathfrak h\)-continuous if \begin{equation}\label{h-conti}\tag{3} f_{-1}\left[T\cap h_{M}(T)\right]\subseteq f_{-1}[T]\cap h_{L}\left( f_{-1}[T]\right) \end{equation} for all \(T\in \mathcal{S}_{\mathbf{l}}^{c}(M)\). Where \(f_{-1}[-]\) is the inverse image of \(f[-]\).

Proposition 6. Let \(f:L\rightarrow M\) and \(g:M\rightarrow N\) be two \(\mathfrak h\)-continuous morphisms of \(\mathbf{Loc}\) then \(g\centerdot f\) is an \(\mathfrak h\)-continuous morphism of \(\mathbf{Loc}\).

Proof. Since \(g:M\rightarrow N\) is \(I\)-continuous, we have \[ g_{-1}\left[V\cap h_{N}(V)\right]\subseteq g_{-1}[V]\cap h_{M}\left( g_{-1}[V]\right) \] for all \(V\in \mathcal{S}_{\mathbf{l}}^{c}(N)\), it fallows that \[ f_{-1}\big[ g_{-1}\left[V\cap h_{N}(V)\right]\big]\subseteq f_{-1}\big[g_{-1}[V]\cap h_{M}\left( g_{-1}[V]\right)\big] \] now, by the \(\mathfrak h\)-continuity of \(f\), \[ f_{-1}\big[g_{-1}[V]\cap h_{M}(g_{-1}[V])\big]\subseteq f_{-1}\big[g_{-1}[V]\big]\cap h_{L}\left( f_{-1}\big[g_{-1}[V]\big]\right)\] therefore \[(g\centerdot f)_{-1}\big[V\cap h_{N}(V)\big]\subseteq (g\centerdot f)_{-1}\cap h_{L}\big((g\centerdot f)_{-1}[V] \big).\] This complete the proof.

As a consequence we obtain

Definition 9. The category \(\mathbf{\mathfrak h\text{-}Loc}\) of \(\mathfrak h\)-spaces comprises the following data:

1. Objects: Pairs \((L,h_{L})\) where L is an object of \(\mathbf{Loc}\) and \(h_{L}\) is an \(\mathfrak h\)-operator on L.
2. Morphisms: Morphisms of \(\mathbf{Loc}\) which are \(\mathfrak h\)-continuous.

4.1. The lattice structure of all \(\mathfrak h\) operators

For the category \(\mathbf{Loc}\) we consider the collection \[ \mathfrak h(\mathbf{Loc}) \] of all \(\mathfrak h\) operators on \(\mathbf{Loc}\). It is ordered by \[ \mathfrak h\leqslant \mathfrak{h^{‘}} \Leftrightarrow h_{L}(S)\subseteq h^{‘}_{L}(S),\;for\; all\; S\in \mathcal{S}_{\mathbf{l}}^{c}(L)\; and\; all\; L \; object\; of \mathbf{Loc}. \] This way \(\mathfrak h(\mathbf{Loc})\) inherits a lattice structure from \( \mathcal{S}_{\mathbf{l}}^{c}\).

Proposition 7. Every family \(( \mathfrak h_\lambda)_{\lambda\in \Lambda}\) in \(\mathfrak h(\mathbf{Loc})\) has a join \(\bigvee\limits_{\lambda\in \Lambda} \mathfrak h_\lambda\) and a meet \(\bigwedge\limits_{\lambda\in \Lambda} \mathfrak h_\lambda\) in \(Int(\mathbf{Loc})\). The discrete \(\mathfrak h\) operator \[ \mathfrak h_{D}=({h_{D}}_{L})_{L\in \mathbf{Loc}}\;\; with\;\; {h_{D}}_{L}(S)=S\;\; for\;\; all\;\; S\in \mathcal{S}_{\mathbf{l}}^{c}(L) \] is the largest element in \(\mathfrak h(\mathbf{Loc})\), and the trivial \(\mathfrak{h}\) operator \[ \mathfrak h_{T}=({h_{T}}_L)_{L\in \mathbf{Loc}}\;\; with\;\; {h_{T}}_{L}(S)= \begin{cases} \{1\}& \text{for all}S\in \mathcal{S}_{\mathbf{l}}^{c}(L),S\ne L\\ L&\text {if}S=L \end{cases} \] is the least one.

Proof. For \(\Lambda\ne\emptyset\), let \(\widehat{\mathfrak h}=\bigvee\limits_{\lambda\in\Lambda }\mathfrak h_\lambda\), then \[ \widehat{h_{L}}=\bigvee\limits_{\lambda\in \Lambda} {h_{\lambda}}_{L}, \] for all L object of \(\mathbf{Loc}\), satisfies

• \(S\cap \widehat{h_{L}}(S)\subseteq S\), because \(S\cap {h_\lambda}_{S}(L)\subseteq S\), for all \(S \in\mathcal{S}_{\mathbf{l}}^{c}(L)\) and for all \(\lambda \in \Lambda\).
• If \(S\subseteq T\) then \(S\cap \widehat{h_{L}}(S)\subseteq T\cap \widehat{h_{L}}(T)\), since \(S \cup {h_\lambda}_{L}(S)\subseteq T \cup {h_\lambda}_{L}(T)\), for all \(S,T \in\mathcal{S}_{\mathbf{l}}^{c}(L)\) and for all \(\lambda \in \Lambda\).
• \(L\cap \widehat{h_{L}}(L)= L\), because \(L\cap {h_\lambda}_{L}(L)= L\) for all \(\lambda \in \Lambda\).

Similarly \(\bigwedge\limits_{\lambda\in \Lambda}\mathfrak h_\lambda\), \( \mathfrak h_{D}\) and \(\mathfrak h _{T}\) are \(\mathfrak h\) operators.

Corollary 3. For every object L of \(\mathbf{Loc}\) \[ \mathfrak h(L) = \{h_{L}\mid h_{L}\text{ is an \(\mathfrak h\) operator on}L\} \] is a complete lattice.

4.2. Initial \(\mathfrak h\) operators

Let \(\mathbf{\mathfrak h\text{-}Loc}\) be the category of \(\mathfrak h\)-spaces. Let \((M,h_{M})\) be an object of \(\mathbf{\mathfrak h\text{-}Loc}\) and let L be an object of \(\mathbf{Loc}\). For each morphism \(f:L\rightarrow M\) in \(\mathbf{Loc}\) we define on L the operotor \begin{equation} \label{h-initial}\tag{4} h_{L_{f}}:=f_{-1}[-]\centerdot h_{M}\centerdot f[-]. \end{equation}

Proposition 8. The operator (\ref{h-initial}) is an \(\mathfrak h\) operator on L for which the morphism \(f\) is \(\mathfrak h\)-continuous.

Proof.

(\(h_1)\) \(S\cap h_{L_{f}}(S)= f_{-1} \Big[f[S]\cap h_{M}\big[ f[S]\big]\Big]\subseteq f_{-1}\big[f[S]\big]\subseteq S\), for all \(S\in \mathcal{S}_{\mathbf{l}}^c(L)\).

(\(h_2)\) \(S\subseteq T\) in \(\mathcal{S}_{\mathbf{l}}^c(L)\), implies \(f[S]\subseteq f[T]\), then \(f[S]\cap h_{M}\big( f[S]\big)\subseteq f[T]\cap h_{M}\big( f[T]\big)\), therefore \(f_{-1}\Big[f[S]\cap h_{M}\big( f[S]\big)\Big]\subseteq f_{-1}\Big[f[T]\cap h_{M}\big( f[T]\big)\Big]\), consequently \(S\cap h_{L_{f}}(S)\subseteq T\cap h_{L_{f}}(T)\), for all \(S,T \in\mathcal{S}_{\mathbf{l}}^{c}(L)\);

(\(h_3)\) \(L\cap h_{L_{f}}(L)= f_{-1} \Big[f[L]\cap h_{M}\big[ f[L]\big]\Big]=L\).

It is clear that \(h_{L_{f}}(L)\) is the coarsest \(\mathfrak h\) operator on L for which the morphism \(f\) is \(\mathfrak h\)-continuous; more precisaly

Proposition 9. Let \((L,h_{L})\) and \((M,h_{M})\) be objects of \(\mathbf{\mathfrak h\text{-}Loc}\),and let \(N\) be an object of \(\mathbf{Loc}\). For each morphism \(g:N\rightarrow L\) in \(\mathbf{Loc}\) and for \(f:(L,h_{L_{f}})\rightarrow (M,h_{N})\) an \(\mathfrak h\)-continuous morphism, \(g\) is \(\mathfrak h\)-continuous if and only if \(f\centerdot g\) is \(\mathfrak h\)-continuous.

Proof. Suppose that \(g\centerdot f\) is \(I\)-continuous, i. e. \[(f\centerdot g)_{-1}\big[T\cap h_{M}(T)\big]\subseteq T\cap h_{N}\big( (f\centerdot g)_{-1}[T] \big)\] for all \(T\in T \in\mathcal{S}_{\mathbf{l}}^{c}(N)\). Then, for all \(S\in T \in\mathcal{S}_{\mathbf{l}}^{c}(L)\), we have \begin{align*} g_{-1}\Big[S\cap \big(h_{L_{f}}(S)\big)\Big]&=g_{-1}\Big[f_{-1}\big[ f[S]\cap h_{M}(f[S])\big]=(f\centerdot g)_{-1}\big[ f[S]\cap h_{M}( f[S]) \big]\\ &\subseteq f\centerdot g)_{-1}[f[S]] \cap \Big(h_{N}\big( (f\centerdot g)_{-1}\big[f[S] \big] \big)\\ &=(f\centerdot g)_{-1}\big[f[S]\big] \cap h_{N}\big( g_{-1}\centerdot f_{-1}\centerdot f[S]\big)\\ &\subseteq g_{-1}[S]\cap h_{N}\big( g_{-1}[S]\big),\\ \end{align*} i.e. \(g\) is \(I\)-continuous.

As a consequence of Corollary 3, Proposition 8 and Proposition 9 (cf. [8] or [9]), we obtain

Theorem 2. The forgetful functor \(U:\mathbf{\mathfrak h\text{-}Loc}\rightarrow \mathbf{Loc}\) is topological, i.e. the concrete category \(\big(\mathbf{\mathfrak h\text{-}Loc},\ U\big)\) is topological.

Author Contributions:

All authors contributed equally in this paper. All authors read and approved the final version of this paper.

Conflicts of Interest:

The authors declare no conflict of interest.