Factorizing the $\mathbf {Top}$–$\mathbf {Loc}$ adjunction through positive topologies

Abstract

We characterize the category of Sambin’s positive topologies as the result of the Grothendieck construction applied to a doctrine over the category Loc of locales. We then construct an adjunction between the category of positive topologies and that of topological spaces Top, and show that the well-known adjunction between Top and Loc factors through the constructed adjunction.

1 Introduction

Positive topologies are introduced by Sambin [22] (see also [7]) as a natural structure for developing constructive pointfree topology. The category PTop of positive topologies can be regarded as a natural extension of the category Loc of locales; actually Loc is a reflective subcategory of PTop (see e.g.  [7]). In a predicative setting, the role of a locale is played by a formal cover $(S,\triangleleft )$, sometimes called a formal topology, which can be read as a presentation of a frame by generators and relations, see e.g. [5]. A positive topology is then a formal cover endowed with a positivity relation, that is a relation $\ltimes$ between S and $\mathcal {P}(S)$ such that for every $a\in S$ and $U,V\subseteq S$

• 1. $a\ltimes U$ $\Longrightarrow$ $a\in U$;

• 2. $a\ltimes U\wedge (\forall b\in S)(b\ltimes U\rightarrow b\in V)$ $\Longrightarrow$ $a\ltimes V$;

• 3. $a\triangleleft U\wedge a\ltimes V$ $\Longrightarrow$ $(\exists b\in U)(b\ltimes V)$.

The motivating example of a positive topology is built from a topological space in such a way as to keep the information about its closed subsets (classically, all such information is already encoded by the opens); see Sect. 5.2.

In [8] the first author and Vickers characterize positive topologies as locales endowed with a suitable family of suplattice homomorphisms. Here we show that this characterization can be organized into a fibration arising from a doctrine¹ over $\mathbf {Loc}$ via the so-called Grothendieck construction (see, e.g. [11]).

We will then use this representation of $\mathbf {PTop}$ to give an adjunction between the category Top of topological spaces and PTop; in particular, the notion of sobriety provided by this adjunction coincides with the one introduced in [22], which is known [1] to be constructively weaker than the notion of sobriety provided by the usual Top–Loc adjunction [13]. Moreover, the Top–Loc adjunction can be factorized as the composition of the Top–PTop adjunction above and the reflection PTop–Loc.

As a by-product, we get the completeness and cocompleteness of the category PTop and of the wider category BTop of basic topologies, which can be similarly characterized as a Grothendieck construction over the category of suplattices. This completes the picture in [10], where the pointwise counterparts of BTop and PTop were shown to be complete and cocomplete.

Our foundational framework is intuitionistic and impredicative, like that provided by the internal language of a topos. We use the term “constructive” in this sense.

2 Basic topologies and positive topologies

A suplattice (or complete join semilattice) is a poset $(L,\le )$ with all joins, that is, $\bigvee X$ exists for all subsets $X\subseteq L$.² A map $f:L\rightarrow M$ between two suplattices preserves joins if

$$\begin{aligned} f\big (\bigvee _{i\in I}x_i\big ) = \bigvee _{i\in I}f(x_i) \end{aligned}$$

for every family $(x_i)_{i\in I}$ in L. Suplattices and join-preserving maps form a category $\mathbf {SL}$. We hence refer to join-preserving maps between suplattices as suplattice homomorphisms.

If X is a set and L is (the carrier of) a suplattice, then the collection of maps $\mathbf {Set}(X,L)$ has a natural suplattice structure where joins are computed pointwise, that is,

$$\begin{aligned} \big ({\bigvee _{i\in I}}\varphi _{i}\big )(x):={\bigvee _{i\in I}}\big (\varphi _{i}(x)\big ). \end{aligned}$$

If X has a suplattice structure, then $\mathbf {SL}(X,L)$ is a sub-suplattice of $\mathbf {Set}(X,L)$.

A base for a suplattice L is a subset $S\subseteq L$ such that p = $\bigvee \{a\in S\ |\ a\le p\}$ for all $p \in L$. For instance, the powerset ${\mathcal {P}}(S)$ of a set S is a suplattice (with respect to union) and a base for ${\mathcal {P}}(S)$ is given by all singletons.³ Given a base S, let $\triangleleft \subseteq S\times {\mathcal {P}}(S)$ be the relation defined as $a\triangleleft U$ iff $a\le \bigvee U$. It is easy to check that $\triangleleft$ satisfies the following properties:

• 1. $a\in U\ \Longrightarrow \ a\triangleleft U$;

• 2. $a\triangleleft U\ \wedge \ (\forall u\in U)(u\triangleleft V)\ \Longrightarrow \ a\triangleleft V$;

for every $a\in S$ and $U,V\subseteq S$. A pair $(S,\triangleleft )$ satisfying 1 and 2 above is called a basic cover. A basic cover has to be understood as a presentation of a suplattice by generators and relations. Indeed, any basic cover induces an equivalence relation $=_\triangleleft$ on ${\mathcal {P}}(S)$ where $U=_\triangleleft V$ is

$$\begin{aligned} (\forall u\in U)(u\triangleleft V)\wedge (\forall v\in V)(v\triangleleft U). \end{aligned}$$

The quotient ${\mathcal {P}}(S){/{=_\triangleleft }}$ is a suplattice (with a base indexed by S) where joins $\bigvee _i[U_i]$ can be computed as $[\bigcup _i U_i]$. To complete the picture, one should note that the basic cover induced by a suplattice L (with any base S) presents a suplattice which is isomorphic to L itself.

Two basic covers ${\mathcal {S}}_1=(S_1,\triangleleft _1)$ and ${\mathcal {S}}_2=(S_2,\triangleleft _2)$ are isomorphic if they induce isomorphic suplattices. More generally we say that a morphism from ${\mathcal {S}}_1$ to ${\mathcal {S}}_2$ is a suplattice homomorphism from ${\mathcal {P}}(S_2)/=_{\triangleleft _2}$ to ${\mathcal {P}}(S_1)/=_{\triangleleft _1}$.⁴ This corresponds to having a relation $s\subseteq S_1\times S_2$ which respects the covers in the following sense:

$$\begin{aligned} \text {if }a\,s\,b\text { and } b\triangleleft _2 V\text {, then }a\triangleleft _1 s^-V \end{aligned}$$

where $s^- V:=\{x\in S_1\ |\ (\exists v\in V)(x\,s\,v)\}$. Actually, the same homomorphism corresponds to several relations which we want to consider equivalent; explicitly, two relations s and $s'$ are equivalent if $s^-V=_{\triangleleft _1}s'^-V$ for all $V\subseteq S_2$.

Basic covers and their morphisms form a category which is dual to the category $\mathbf{SL}$ of suplattices, that is, a category equivalent to $\mathbf {SL}^{op}$. We refer the reader to [2] for further details.

2.1 Basic topologies

A basic topology [22] is a triple $(S,\triangleleft ,\ltimes )$ where $(S,\triangleleft )$ is a basic cover and $\ltimes$ is a relation between S and $\mathcal {P}(S)$ such that

• 1. $a\ltimes U$ $\Longrightarrow$ $a\in U$;

• 2. $a\ltimes U\wedge (\forall b\in S)(b\ltimes U\rightarrow b\in V)$ $\Longrightarrow$ $a\ltimes V$;

• 3. $a\triangleleft U\wedge a\ltimes V$ $\Longrightarrow$ $(\exists b\in U)(b\ltimes V)$.

The relation $\ltimes$ is called a positivity relation on $(S, \triangleleft )$. Thus, a basic topology can be regarded as a suplattice together with the extra structure specified by a positivity relation.

The powerset $\varOmega :=\mathcal {P}(1)$ of a singleton can be identified with the algebra of propositions up to logical equivalence.⁵ Condition 3. in the definition above says that the map⁶

$$\begin{aligned} \begin{array}{ccccc} \varphi _Z &{} : &{} {\mathcal {P}}(S)/=_{\triangleleft } &{} \longrightarrow &{} \varOmega \\ &{} &{} {[U]} &{} \longmapsto &{} U\between Z\ \end{array} \end{aligned}$$

is well-defined if Z is of the form $\{a\in S\ |\ a\ltimes V\}$, in which case $\varphi _Z$ is a suplattice homomorphism. Given any positivity relation $\ltimes$ on $(S,\triangleleft )$, the collection of all such $\varphi _Z$ forms a sub-suplattice of $\mathbf {SL}({{\mathcal {P}}(S)/ =_{\triangleleft }}\,,\,\varOmega )$. The first author and Vickers [8, Theorem 2.3] have shown that there is a bijective correspondence between positivity relations on $(S,\triangleleft )$ and sub-suplattices of $\mathbf {SL}({{\mathcal {P}}(S)/ =_{\triangleleft }}\,,\,\varOmega )$. Thus, a basic topology can be identified with a pair $(L, \varPhi )$ where L is a suplattice and $\varPhi$ is a sub-suplattice of the collection $\mathbf {SL}(L,\varOmega )$ of suplattice homomorphisms from L to $\varOmega$.⁷

Let ${\mathcal {S}}_1 = (S_1, \triangleleft _1, \ltimes _1)$ and $\mathcal {S}_2 = (S_2, \triangleleft _2, \ltimes _2)$ be basic topologies, and $(L_1, \varPhi _1)$ and $(L_2,\varPhi _2)$ be the corresponding suplattices together with sub-suplattices of suplattice homomorphisms to $\varOmega$. According to [22], a morphism between basic topologies $\mathcal {S}_1$ and $\mathcal {S}_2$ is a morphism s between $(S_1,\triangleleft _1)$ and $(S_2,\triangleleft _2)$ satisfying the following additional condition

$$\begin{aligned} \text {if }a\,s\,b\text { and }a\ltimes _1 U\text {, then } b\ltimes _2 s\,U \end{aligned}$$

for all $a\in S_1$, $b\in S_2$ and $U\subseteq S_1$, where $s\, U:=\{y\in S_2 \mid (\exists \,u\in U)(u\,s\,y)\}$. This corresponds to having a suplattice homomorphism $f : L_2 \rightarrow L_1$ such that $\varPhi _1 \circ f \subseteq \varPhi _2$, where $\varPhi _1 \circ f := \left\{ \varphi \circ f \mid \varphi \in \varPhi _1 \right\}$; in other words

$$\begin{aligned} \text {if }L_1{\mathop {\longrightarrow }\limits ^{\varphi }}\varOmega \text { belongs to }\varPhi _1\text {, then }L_2{\mathop {\longrightarrow }\limits ^{f}}L_1{\mathop {\longrightarrow }\limits ^{\varphi }}\varOmega \text { belongs to }\varPhi _2 \end{aligned}$$

(see [8, Proposition 2.9]).

Let $\mathbf {BTop}$ be the category whose objects are pairs $(L,\varPhi )$ of a suplattice L and a sub-suplattice $\varPhi$ of $\mathbf {SL}(L,\varOmega )$, and whose arrows $f : (L_1,\varPhi _1) \rightarrow (L_2,\varPhi _2)$ are suplattice homomorphisms $f : L_2 \rightarrow L_1$ such that $\varPhi _1 \circ f \subseteq \varPhi _2$. Apart from the impredicativity involved, $\mathbf {BTop}$ is equivalent to the category of basic topologies in [22].

2.2 Positive topologies

A positive topology [22] is a basic topology $(S,\triangleleft ,\ltimes )$ such that the underlying basic cover $(S,\triangleleft )$ is a formal cover [5] (sometimes called a formal topology). This means that the suplattice presented by $(S,\triangleleft )$ is a frame, that is, binary meets distribute over arbitrary joins.

By an observation similar to the one we made for a basic topology in Sect. 2.1, a positive topology can be identified with a pair $(L, \varPhi )$ where L is a frame and $\varPhi$ is a sub-suplattice of $\mathbf {SL}(L,\varOmega )$. A morphism between such pairs $(L, \varPhi )$ and $(M, \varPsi )$ is a frame homomorphism $f : M \rightarrow L$ such that $\varPhi \circ f \subseteq \varPsi$, which corresponds to a formal map between positive topologies as described in [22].

Let $\mathbf {PTop}$ be the subcategory of $\mathbf {BTop}$ consisting of those objects whose underlying suplattice is a frame and arrows which are frame homomorphisms between the underlying frames. The category $\mathbf {PTop}$ is thus equivalent to that of positive topologies in [22].

3 A categorical characterization of BTop and PTop

In this section, we are going to give a categorical characterization of BTop and PTop in terms of Grothendieck constructions over two doctrines on the opposite of the category of suplattices and on the category of locales, respectively.

3.1 A doctrine on $\mathbf {SL}^{op}$

For L a suplattice, the (contravariant) hom-functor $\mathbf {SL}(\__ ,L)\,:\,\mathbf {SL}^{op}\rightarrow \mathbf {Set}\$ can be also regarded as a functor

$$\begin{aligned} \mathbf {SL}(\__ ,L)\,:\,\mathbf {SL}\rightarrow \mathbf {SL}^{op}\ \end{aligned}$$

where, for $f\in \mathbf {SL}(X,Y)$ and $\varphi \in \mathbf {SL}(Y,L)$, we have $\mathbf {SL}(f,L)(\varphi )$ = $\varphi \circ f$.

Another well-known contravariant functor is the subobject functor

$$\begin{aligned} \mathbf {Sub}\,:\,\mathbf {SL}^{op}\rightarrow \mathbf {PreOrd} \end{aligned}$$

which sends each suplattice L to the preorder (actually a poset) $\mathbf {Sub}(L)$ of subobjects of L in $\mathbf {SL}$. Recall that a suboject of L can be represented as a subset $I\subseteq L$ closed under joins in L, that is a sub-suplattice of L. Given $f:M\rightarrow L$ in $\mathbf {SL}$ and $I\in \mathbf {Sub}(L)$, $\mathbf {Sub}(f)$ sends I to the pullback $\{x\in M\ |\ f(x)\in I\}$ of I along f.

The composition $\mathbf {Sub}\circ \mathbf {SL}(\__,\varOmega )$ is a functor

$$\begin{aligned} \mathbf {P}:\mathbf {SL}\rightarrow \mathbf {PreOrd} \end{aligned}$$

which, of course, can also be read as a contravariant functor on $\mathbf {SL}^{op}$

$$\begin{aligned} \mathbf {P}:(\mathbf {SL}^{op})^{op}\rightarrow \mathbf {PreOrd}, \end{aligned}$$

that is, a doctrine on $\mathbf {SL}^{op}$.

As the result of the so-called Grothendieck construction [11, Definition 1.10.1],⁸ we get a category $\int \mathbf {P}$ whose objects are pairs $(L,\varPhi )$ with L a suplattice and $\varPhi$ a subobject of $\mathbf {SL}(L,\varOmega )$ in $\mathbf {SL}$. An arrow $(L,\varPhi )\rightarrow (M,\varPsi )$ in $\int \mathbf {P}$ is a suplattice homomorphism $f:M\rightarrow L$ such that

$$\begin{aligned} \varPhi \subseteq \mathbf {P}(f)(\varPsi ). \end{aligned}$$

Since $\mathbf {P}(f)(\varPsi )$ = $\{\varphi \in \mathbf {SL}(L,\varOmega )\ |\ \varphi \circ f\in \varPsi \}$ by definition, such a condition is equivalent to the following

$$\begin{aligned} \varPhi \circ f\subseteq \varPsi \end{aligned}$$

Therefore, $\int \mathbf {P}$ is exactly the category $\mathbf {BTop}$ of basic topologies which we introduced in Sect. 2.1 above.

This construction yields a forgetful functor $\mathbf {U}:\int \mathbf {P}\rightarrow \mathbf {SL}^{op}$, which is in fact a fibration (see [11]). This functor has a right adjoint, the constant object functor

$$\begin{aligned} \varvec{\Delta }: \mathbf {SL}^{op}\rightarrow \int \mathbf {P}, \end{aligned}$$

which sends each suplattice L to the object $(L,\mathbf {SL}(L,\varOmega ))$ and each $f:L\rightarrow M$ in $\mathbf {SL}^{op}$ to itself as an arrow from $\varvec{\Delta }(L)$ to $\varvec{\Delta }(M)$ in $\int \mathbf {P}$. So $\varvec{\Delta }$ is full.

Moreover $\mathbf {U}\circ \varvec{\Delta }$ is just the identity functor on $\mathbf {SL}^{op}$. Thus, $\varvec{\Delta }$ is full, faithful and injective on objects, and so $\mathbf {SL}^{op}$ can be regarded as a reflective subcategory of $\int \mathbf {P}$. In this way, we recover the result in [6].

Note that the monad T induced by the adjunction $\mathbf {U}\dashv \varvec{\Delta }$ is an idempotent monad. By the results in Sect. 4.2 of [3], we have that $\mathbf{SL}^{op}$ is equivalent both to the category of free algebras (the Kleisli category) and to the category of algebras (the Eilenberg–Moore category) on T. Hence the adjunction $\mathbf {U}\dashv \varvec{\Delta }$ is monadic.

Remark

Since in a suplattice arbitrary meets always exist, if $(L,\le )$ is a suplattice, then $(L,\le )^{op} := (L,\ge )$ is a suplattice as well. Moreover, every suplattice homomorphism $f:X\rightarrow Y$ has a right adjoint (as a monotone function) $f^{op}:Y\rightarrow X$ which preserves all meets. This determines a contravariant functor $(\_)^{op}$, which is in fact an isomorphism between $\mathbf {SL}$ and $\mathbf {SL}^{op}$. In particular, $\mathbf {SL}(X,Y)\cong \mathbf {SL}(Y^{op},X^{op})$ for all X and Y.

Classically, $\mathbf {SL}(\__,\varOmega )$ is naturally isomorphic to the functor $(\_)^{op}$ because $\varOmega ^{op}\cong \varOmega$ so that $\mathbf {SL}(L,\varOmega )$ $\cong$ $\mathbf {SL}(\varOmega ,L^{op})$ $\cong$ $L^{op}$.⁹ Therefore, for every L, $\mathbf {P}(L)=\mathbf {Sub}(\mathbf {SL}(L,\varOmega ))\cong \mathbf {Sub}(L^{op})$ which is isomorphic to the lattice of suplattice quotients of L. In other words, an object $(L,\varPhi )$ corresponds to an epimorphism $e:L\rightarrow \varPhi ^{op}$, and an arrow $(L,\varPhi )\rightarrow (M,\varPsi )$ is a suplattice homomorphism $f:M\rightarrow L$ such that $e\circ f: M\rightarrow \varPhi ^{op}$ preserves the congruence relation on M corresponding to $\varPsi$.

3.2 The case of frames (and locales)

The category $\mathbf {Frm}$ of frames is the subcategory of $\mathbf {SL}$ whose objects are frames and whose arrows preserve finite meets (in addition to arbitrary joins). The category $\mathbf {Loc}$ of locales is defined as $\mathbf {Frm}^{op}$. By restricting the functor $\mathbf {P}$ to $\mathbf {Frm}$, we get a doctrine

$$\begin{aligned} \widetilde{\mathbf {P}}:\mathbf {Loc}^{op}=\mathbf {Frm}\longrightarrow \mathbf {PreOrd} \end{aligned}$$

on $\mathbf {Loc}$, which gives rise to a fibration $\mathbf {U}:\int \widetilde{\mathbf {P}}\rightarrow \mathbf {Loc}$ fitting in a pullback square of categories as follows.

Here $\int \widetilde{\mathbf {P}}$ is exactly the category $\mathbf {PTop}$ as introduced in Sect. 2.2.

As we have shown before in the case of $\mathbf {SL}^{op}$ and $\int \mathbf {P}$, there is an adjunction $\mathbf {U}\dashv \varvec{\Delta }$ between $\int \widetilde{\mathbf {P}}$ and $\mathbf {Loc}$ with $\varvec{\Delta }$ full, faithful and injective on objects. Thus, the category $\mathbf {Loc}$ can be regarded as a reflective subcategory of $\mathbf {PTop}$, as already shown in [7].

4 Weakly sober spaces

4.1 Irreducible closed subsets

The open sets of a topological space $(X,\tau )$ form a frame with respect to set-theoretic unions and intersections. A subset $C\subseteq X$ is closed if

$$\begin{aligned} (\forall I\in \tau )(x\in I\ \Rightarrow \ C\between I)\ \Longrightarrow x\in C \end{aligned}$$

for all $x\in X$. The collection $\mathsf {Closed}(X,\tau )$ of closed subsets of $(X,\tau )$ is a complete lattice (where meets are given by intersections, and joins are given by closure of unions), but it need not be a co-frame constructively.¹⁰

As usual, it makes sense to define the closure $\mathsf {cl}D$ of a subset $D\subseteq X$ as the intersection of all closed subsets containing D.

Every closed subset C of X determines a map

$$\begin{aligned} \begin{array}{rcrcl} \varphi _C &{} : &{} \tau &{} \longrightarrow &{} \varOmega \\ &{} &{} I &{}\longmapsto &{} C\between I \end{array} \end{aligned}$$

which preserves joins, that is, $\varphi _C\in \mathbf {SL}(\tau ,\varOmega )$. Note that $\varphi _D$ makes sense also when D is an arbitrary subset; however $\varphi _{D}=\varphi _{\mathsf {cl}D}$ because $I\between D$ if and only if $I\between \mathsf {cl}D$ for every $I\in \tau$. So the mapping

$$\begin{aligned} \begin{array}{rcl} \mathsf {Closed}(X,\tau ) &{} \longrightarrow &{} \mathbf {SL}(\tau ,\varOmega ) \\ C &{} \longmapsto &{} \varphi _C \end{array} \end{aligned}$$

is injective and preserves joins. Thus $\mathsf {Closed}(X,\tau )$ is a sub-suplattice of $\mathbf{SL}(\tau ,\varOmega )$.¹¹

A closed subset $C\subseteq X$ is irreducible if any of the following equivalent conditions holds:

• 1. $\varphi _C$ preserves finite meets;

• 2. C is inhabited and for every $I,J\in \tau$, if $I\between C$ and $J\between C$, then $(I\cap J)\between C$;

• 3. $\{I\in \tau \ |\ I\between C\}$ is a completely-prime filter of opens.

In other words, a closed subset C is irreducible if and only if $\varphi _C$ is a frame homomorphism, that is, a point in the sense of locale theory. However we cannot show constructively that all frame homomorphisms $\tau \rightarrow \varOmega$ arise in this way; see Sect. 4.2.

Classically, C is irreducible if and only if it is non-empty and cannot be written as a disjoint union of two non-empty closed subsets [13]; moreover $\{C\subseteq X\ |\ C\text { is irreducible closed}\}$ can be identified with $\mathbf {Frm}(\tau ,\varOmega )$.

4.2 Weak sobriety

Recall that a space is T0 or Kolmogorov if $x=y$ follows from the assumption that $\mathsf {cl}\{x\}=\mathsf {cl}\{y\}$. Since $\mathsf {cl}\{x\}$ is always irreducible, we have the following embeddings for a T0 space $(X,\tau )$:

$$\begin{aligned} X\hookrightarrow \{C\subseteq X\ |\ C\text { is irreducible closed}\}\hookrightarrow \mathbf {Frm}(\tau ,\varOmega ). \end{aligned}$$

A T0 space is weakly sober if every irreducible closed subset is the closure of a singleton, that is, if the embedding $X\hookrightarrow \{C\subseteq X\ |\ C\text { is irreducible closed}\}$ is a bijection. It is sober if the embedding $X\hookrightarrow \mathbf {Frm}(\tau ,\varOmega )$ is a bijection. Note that every weakly sober space is sober classically.

Constructively, every T2 space is weakly sober [1, Proposition 11.27], provided that the T2 separation property for $(X,\tau )$ is understood as the following statement: $(\forall I\in \tau )(\forall J\in \tau )(x\in I\wedge y\in J\longrightarrow I\between J)$ $\longrightarrow$ $x=y$, for all $x,y\in X$.

However, if every weakly sober space were sober, then the non-constructive principle LPO (the Limited Principle of Omniscience) would be derivable [1, Proposition 11.25]. Thus, we cannot prove that all $\varphi \in \mathbf {SL}(\tau ,\varOmega )$ are of the form $\varphi _C$ for some closed subset C; otherwise $\mathbf {Frm}(\tau ,\varOmega )$ could be identified with the irreducible closed subsets, which would make sobriety and weak sobriety coincide.

5 Factorizing the $\mathbf {Top}$–$\mathbf {Loc}$ adjunction

The usual $\varvec{\Omega }\dashv \mathbf {Pt}$ adjunction between the category $\mathbf {Top}$ of topological spaces and the category $\mathbf {Loc}$ of locales does not compose with the adjunction $\mathbf {U}\dashv \varvec{\Delta }$ between $\mathbf {Loc}$ and $\mathbf {PTop}$ ($=$ $\int \widetilde{\mathbf {P}}$) to give an adjunction between $\mathbf {Top}$ and $\mathbf {PTop}$.

Nevertheless, a meaningful adjunction between $\mathbf {Top}$ and $\mathbf {PTop}$ can be given, as explained in the following, through which the usual $\mathbf {Top}$–$\mathbf {Loc}$ adjunction factors.

5.1 Points of a positive topology

The suplattice $\varOmega$ is an initial frame, that is, a terminal locale. Hence $\varvec{\Delta }(\varOmega )$ is a terminal object in $\mathbf {PTop}$. We define a point of a positive topology $(L,\varPhi )$ as a global point $\varvec{\Delta }(\varOmega )\rightarrow (L,\varPhi )$ in $\mathbf {PTop}$, and we write $\mathbf {Pt}^{+}(L,\varPhi )$ instead of $\mathbf {PTop}(\varvec{\Delta }(\varOmega ),(L,\varPhi ))$. Thus, a point of $(L,\varPhi )$ is a frame homomorphism $f:L\rightarrow \varOmega$ such that $\mathbf {SL}(\varOmega ,\varOmega )\circ f\subseteq \varPhi$. Since $\mathbf {SL}(\varOmega ,\varOmega )$ contains the identity map, we have $f\in \varPhi$. Conversely, if $f\in \varPhi$ and $\varphi \in \mathbf {SL}(\varOmega ,\varOmega )$, then we have $\varphi \circ f=\bigvee \{g\in \{f\}|\,\varphi =\mathsf {id}_{\varOmega }\}\in \varPhi$. In other words, the points of $(L,\varPhi )$ are exactly those points of the locale L that are in $\varPhi$. Hence, $\mathbf {Pt}^+(L,\varPhi )$ can be regarded as a subspace of the topological space $\mathbf {Pt}(L)$.

The construction $\mathbf {Pt}^{+}$ can be extended to a functor from $\mathbf {PTop}$ to $\mathbf {Top}$ as follows. Given an arrow $(L,\varPhi )\rightarrow (M,\varPsi )$ with underlying frame homomorphism $f:M\rightarrow L$, the continuous map $\mathbf {Pt}(f):\mathbf {Pt}(L)\rightarrow \mathbf {Pt}(M)$, which sends a point $p:L\rightarrow \varOmega$ to the point $p\circ f:M\rightarrow \varOmega$, can be restricted to a continuous map $\mathbf {Pt}^+(L,\varPhi )\rightarrow \mathbf {Pt}^+(M,\varPsi )$ because $\varPhi \circ f\subseteq \varPsi$.

5.2 The canonical positive topology associated to a space

As shown in Sect. 4.1, the closed subsets $\mathsf {Closed}(X,\tau )$ of a topological space $(X,\tau )$ can be seen as a sub-suplattice of $\mathbf {SL}(\tau ,\varOmega )$ via the mapping $C\mapsto \varphi _C$. Thus, we can define a functor $\varvec{\Lambda }:\mathbf {Top}\rightarrow \mathbf {PTop}$ whose object part is

$$\begin{aligned} \varvec{\Lambda }(X,\tau )\ =\ \big (\tau ,\{\varphi _C \mid C \text { is closed}\}\big ). \end{aligned}$$

For a continuous map $f:(X,\tau _X)\rightarrow (Y,\tau _Y)$, the $\mathbf {PTop}$-morphism $\varvec{\Lambda }(f)$ is just the locale morphism corresponding to the frame homomorphism $f^{-1}:\tau _Y\rightarrow \tau _X$. This makes sense because for any closed subset $C\subseteq X$, the suplattice homomorphism $\varphi _C\circ f^{-1}:\tau _Y\rightarrow \varOmega$ is precisely $\varphi _D$, where $D=\mathsf {cl}f(C)$.

5.3 The adjunction between $\mathbf {Pt}^+$ and $\varvec{\Lambda }$

Theorem

The following hold:

• 1. $\mathbf {Pt}=\mathbf {Pt}^+\circ \varvec{\Delta }$;

• 2. $\varvec{\Omega }= \mathbf {U}\circ \varvec{\Lambda }$;

• 3. $\varvec{\Lambda }\dashv \mathbf {Pt}^+$.

As a consequence, the usual adjunction between $\mathbf {Top}$ and $\mathbf {Loc}$ factors through an adjunction between $\mathbf {PTop}$ and $\mathbf {Loc}$.

Proof

For every locale L, $\mathbf {Pt}(L)=\mathbf {Pt}(L)\cap \mathbf {SL}(L,\varOmega )=\mathbf {Pt}^+(\varvec{\Delta }(L))$, and for every topological space $(X,\tau )$, $\mathbf {U}(\varvec{\Lambda }(X,\tau ))=\tau =\varvec{\Omega }(X,\tau )$. Hence 1 and 2 hold.

For 3, if $f:\varvec{\Lambda }(X,\tau )\rightarrow (L,\varPhi )$ in $\mathbf {PTop}$, then one can define a continuous map ${\widetilde{f}}$ from $(X,\tau )$ to $\mathbf {Pt}^+(L,\varPhi )$ as follows:

$$\begin{aligned} {\widetilde{f}}(x):=\varphi _{\mathsf {cl}\{x\}}\circ f, \end{aligned}$$

that is, for every $y\in L$, ${\widetilde{f}}(x)(y):=\mathsf {cl}\{x\}\between f(y)\in \varOmega$.

Conversely, if g is a continuous map from $(X,\tau )$ to $\mathbf {Pt}^+(L,\varPhi )$, then an arrow ${\widehat{g}}$ from $\varvec{\Lambda }(X,\tau )$ to $(L,\varPhi )$ in $\mathbf {PTop}$ is defined as follows:

$$\begin{aligned} {\widehat{g}}(y):=g^{-1}(\{\varphi \in \mathbf {Pt}(L)\cap \varPhi \,|\,\varphi (y)=1\})\in \tau \end{aligned}$$

for every $y\in L$. This is an arrow in $\mathbf {PTop}$ because it preserves arbitrary joins and finite meets, and for every closed subset $C\subseteq X$ we have $\varphi _C\circ {\widehat{g}}$ = $\bigvee \{\varphi \in \mathbf {Pt}^+(L,\varPhi )\ |\ \varphi \in g(C)\}\in \varPhi$.

One can show that the maps $\widetilde{(\_)}$ and $\widehat{(\_)}$ define a natural isomorphism between the functors $\mathbf {PTop}(\varvec{\Lambda }(\_),\_)$ and $\mathbf {Top}(\_\,,\mathbf {Pt}^+(\_))$. $\square$

Since $\mathbf {Pt}^+(\varvec{\Lambda }(X, \tau ))$ is the space of irreducible closed subsets of X and $\mathbf {Pt}(\varvec{\Omega }(X,\tau ))$ is the space of frame homomorphisms from $\tau$ to $\varOmega$, a topological space $(X,\tau )$ is weakly sober when the unit of the adjunction $\varvec{\Lambda }\dashv \mathbf {Pt}^+$ gives a homeomorphism between $(X,\tau )$ and $\mathbf {Pt}^+(\varvec{\Lambda }(X,\tau ))$, while it is sober when the unit of the adjunction $\varvec{\Omega }\dashv \mathbf {Pt}$ gives a homeomorphism between $(X,\tau )$ and $\mathbf {Pt}(\varvec{\Omega }(X,\tau ))$.

Classically, $\mathbf {SL}(\tau ,\varOmega )$ = $\{\varphi _C\ |\ C \text { is closed}\}$ holds (see footnote 11). Hence $\varvec{\Lambda }=\varvec{\Delta }\circ \varvec{\Omega }$, and thus $\mathbf {Pt}^+\circ \varvec{\Lambda }=\mathbf {Pt}^+\circ \varvec{\Delta }\circ \varvec{\Omega }=\mathbf {Pt}\circ \varvec{\Omega }$. Therefore, as already noted, sobriety and weak sobriety coincide classically.

Remark

A positivity relation on a formal cover is also called a binary positivity [21, 22], which is often explained as generalization of a (unary) positivity predicate. Impredicatively, formal covers with a unary positivity predicate (often called just formal topologies [20]) correspond to open locales [12, 14, 15], which are also called overt locales [23].

Overt locales form a coreflective subcategory oLoc of Loc [19]. On the other hand, our result above presents Loc as a reflective subcategory of PTop. Thus, the relation between oLoc and Loc and that between PTop and Loc seem to be of different kinds. In particular, the two adjunctions oLoc–Loc and PTop–Loc do not compose to give any adjunction between PTop and oLoc, apart from the fact that oLoc embeds into PTop (via the embedding $\varvec{\Delta }$). Moreover, classically, every locale is overt so that oLoc and Loc coincide, but this is clearly not the case for PTop .

The relation between PTop and oLoc has much to be clarified. However, the above observation suggests that the result of this section seems to be independent from what is already known about oLoc.

6 Limits and colimits in BTop and PTop

Let $\mathbf {Q}:\mathbb {C}^{op}\rightarrow \mathbf {PreOrd}$ be a doctrine which factors through the embedding of the category of inflattices (that is, the category whose objects are posets having all meets and whose arrows are functions preserving them) in $\mathbf {PreOrd}$.

Under this assumption, if $\mathbb {C}$ is complete, then the Grothendieck construction $\int \mathbf {Q}$ gives a complete category. Indeed, it is easy to check that if $(A_{i},\varphi _{i})_{i\in I}$ is a set-indexed family of objects in $\int \mathbf {Q}$, its product is given by the object

$$\begin{aligned} \prod _{i\in I}(A_{i},\varphi _{i})\ :=\ \left( \prod _{i\in I}A_{i},\bigwedge _{i\in I}\mathbf {Q}(\pi _{i})(\varphi _{i})\right) \end{aligned}$$

together with the projections $\pi _{i}$ inherited from $\mathbb {C}$; and the equalizer of two parallel arrows $f,g:(A,\varphi )\rightarrow (B,\psi )$ in $\int \mathbf {Q}$ is $e:\big (E,\mathbf {Q}(e)(\varphi )\big )\rightarrow (A,\varphi )$, where $e:E\rightarrow A$ is the equalizer of f and g in $\mathbb {C}$.

On the other hand, if $\mathbb {C}$ is cocomplete, then $\int \mathbf {Q}$ is cocomplete as well. Indeed, if we denote with $\exists _f$ the left adjoint to $\mathbf {Q}(f)$ for every arrow f of $\mathbb {C}$, the coproduct of a family of objects $(A_{i},\varphi _{i})_{i\in I}$ in $\int \mathbf {Q}$ is given by the object

$$\begin{aligned} \sum _{i\in I}(A_i,\varphi _i)\ := \left( \sum _{i\in I}A_{i},\bigvee _{i\in I}\exists _{j_{i}}(\varphi _{i}) \right) \end{aligned}$$

together with the injections $j_{i}$ inherited from $\mathbb {C}$; and the coequalizer of two arrows $f,g:(A,\varphi )\rightarrow (B,\psi )$ is $q:(B,\psi )\rightarrow (Q,\exists _{q}(\psi ))$, where $q:B\rightarrow Q$ is the coequalizer of f and g in $\mathbb {C}$.

The doctrines $\mathbf {P}$ and $\widetilde{\mathbf {P}}$ introduced in Sects. 3.1 and 3.2, respectively, satisfy the above requirements. Indeed, every $\mathbf {P}(L)$ and every $\widetilde{\mathbf {P}}(L)$ is an inflattice because an arbitrary intersection of sub-suplattices is a sub-suplattice. Moreover, every $\mathbf {P}(f)$ has a left adjoint, namely $\exists _f(\varPhi ):=\varPhi \circ f$, essentially by the very definition of $\mathbf {P}$; hence every $\mathbf {P}(f)$ preserves meets. Finally, it is well known that both $\mathbf {SL}^{op}$ and $\mathbf {Loc}$ are complete and cocomplete [13]. Thus, the categories $\mathbf {PTop}$ and $\mathbf {BTop}$ are complete and cocomplete.