---
title: 'FaR-Loc: Factorizing the Top–Loc Adjunction'
url: https://www.emergentmind.com/topics/far-loc
type: topic
---

# FaR-Loc: Factorizing the Top–Loc Adjunction

Searching arXiv for the cited paper and closely related work on positive topologies, locales, and formal topology.
arXiv search query: 1812.09190 positive topologies locales Sambin formal topology
FaR-Loc denotes the factorization of the classical adjunction between topological spaces and locales through an intermediate category of Sambin’s positive topologies. In categorical terms, the standard adjunction
\[
\Omega \dashv \mathbf{Pt} : \mathbf{Top} \rightleftarrows \mathbf{Loc}
\]
is decomposed into an adjunction between \(\mathbf{Top}\) and \(\mathbf{PTop}\) together with a reflection of \(\mathbf{PTop}\) into \(\mathbf{Loc}\). The construction characterizes positive topologies as the Grothendieck construction of a doctrine over \(\mathbf{Loc}\), and then identifies functors
\[
\mathbf{\Lambda} \dashv \mathbf{Pt}^+ : \mathbf{Top} \rightleftarrows \mathbf{PTop},
\qquad
\mathbf{U} \dashv \mathbf{\Delta} : \mathbf{PTop} \rightleftarrows \mathbf{Loc}
\]
such that \(\Omega = \mathbf{U}\circ \mathbf{\Lambda}\) and \(\mathbf{Pt} = \mathbf{Pt}^+\circ \mathbf{\Delta}\) [1812.09190].

## 1. Classical background: the adjunction between spaces and locales

The ambient setting is the adjunction between the category \(\mathbf{Top}\) of topological spaces and continuous maps, and the category \(\mathbf{Loc}\) of locales, defined as \(\mathbf{Frm}^{op}\), where \(\mathbf{Frm}\) is the category of frames and frame homomorphisms [1812.09190]. A frame is a suplattice \(L\) in which finite meets distribute over arbitrary joins:
\[
y \wedge \bigvee_{i\in I} x_i \;=\; \bigvee_{i\in I} (y \wedge x_i).
\]
A frame homomorphism preserves arbitrary joins and finite meets.

The functor
\[
\Omega : \mathbf{Top} \to \mathbf{Loc}
\]
sends a space \((X,\tau)\) to its frame of opens \(\tau\), regarded contravariantly as a locale. A continuous map \(f : (X,\tau_X)\to (Y,\tau_Y)\) is sent to the inverse-image frame homomorphism \(f^{-1} : \tau_Y \to \tau_X\). In the opposite direction, the points functor
\[
\mathbf{Pt} : \mathbf{Loc} \to \mathbf{Top}
\]
sends a locale \(L\) to the space of its points, where a point is a frame homomorphism \(p : L \to \Omega\).

The classical adjunction is expressed by the hom-set isomorphisms
\[
\mathbf{Top}(X,\mathbf{Pt}(L)) \cong \mathbf{Loc}(\Omega(X),L) \cong \mathbf{Frm}(L,\tau_X).
\]
Its unit characterizes sobriety: a space is sober exactly when the canonical map \(X \to \mathbf{Pt}(\Omega(X))\) is an isomorphism. FaR-Loc retains this background but refines it by inserting an intermediate category that records a chosen positivity structure on a locale rather than all of its points at once.

## 2. Positive topologies from formal and basic covers

The constructive origin of the theory lies in formal topology. Instead of starting from arbitrary frames, one begins with a basic cover \((S,\triangleleft)\), where \(S\) is a set of generators and \(\triangleleft\) is a cover relation \(a \triangleleft U\) between \(a\in S\) and \(U\subseteq S\). The quotient \(P(S)/=_\triangleleft\) yields the associated suplattice, and a formal cover is a basic cover whose associated suplattice is a frame [1812.09190].

A basic topology is a triple \((S,\triangleleft,\ltimes)\) where \((S,\triangleleft)\) is a basic cover and \(\ltimes\) is a positivity relation satisfying three conditions: soundness of positivity, monotonicity in the target, and compatibility with cover. In particular, if \(a\ltimes U\), then \(a\in U\); and if \(a\triangleleft U\) and \(a\ltimes V\), then some \(b\in U\) satisfies \(b\ltimes V\).

Using Sambin’s overlap notation \(U\between V\), each positivity relation induces suplattice homomorphisms
\[
\varphi_Z : P(S)/=_\triangleleft \to \Omega, \quad [U] \mapsto U \between Z.
\]
Ciraulo–Vickers show that positivity relations on \((S,\triangleleft)\) are in bijection with sub-suplattices
\[
\Phi \subseteq \mathbf{SL}(L,\Omega),
\]
where \(L = P(S)/=_\triangleleft\) and \(\mathbf{SL}(L,\Omega)\) denotes the hom-suplattice of suplattice homomorphisms \(L\to\Omega\) [1812.09190].

This yields an equivalent description of a basic topology as a pair \((L,\Phi)\), with \(L\) a suplattice and \(\Phi\) a sub-suplattice of \(\mathbf{SL}(L,\Omega)\). A positive topology is the special case in which \(L\) is a frame. Thus an object of \(\mathbf{PTop}\) is a pair
\[
(L,\Phi)
\]
with \(L\in\mathbf{Frm}\) and \(\Phi \subseteq \mathbf{SL}(L,\Omega)\) a sub-suplattice. A morphism
\[
(L,\Phi)\to(M,\Psi)
\]
is a frame homomorphism \(f:M\to L\) such that
\[
\Phi\circ f \subseteq \Psi.
\]

Conceptually, a locale supplies all frame homomorphisms \(L\to\Omega\), whereas a positive topology specifies a chosen sub-suplattice of such homomorphisms. This suggests a constructive enrichment of locale theory in which positivity, overlap, and closed-set behavior are part of the structure rather than derived only after spatialization.

## 3. The doctrine over \(\mathbf{Loc}\) and the Grothendieck construction

The fibrational core of FaR-Loc is a doctrine built from truth-valued morphisms. For a suplattice \(L\), the hom-set \(\mathbf{SL}(L,\Omega)\) has a pointwise join structure, and the subobject functor assigns to a suplattice the preorder of its sub-suplattices. Composing these constructions gives
\[
\mathbf{P} = \mathbf{Sub} \circ \mathbf{SL}(-,\Omega) : \mathbf{SL} \to \mathbf{PreOrd}.
\]
For a homomorphism \(f:M\to L\),
\[
\mathbf{P}(f)(\Psi)
=
\{\varphi\in \mathbf{SL}(L,\Omega)\mid \varphi\circ f\in\Psi\},
\]
again a sub-suplattice [1812.09190].

Applying the Grothendieck construction to \(\mathbf{P}\) produces a total category \(\int\mathbf{P}\) whose objects are pairs \((L,\Phi)\) with \(\Phi\subseteq \mathbf{SL}(L,\Omega)\), and whose morphisms are precisely those homomorphisms satisfying the positivity-preservation condition \(\Phi\circ f\subseteq \Psi\). This is exactly the category \(\mathbf{BTop}\) of basic topologies.

Restricting from suplattices to frames yields a doctrine over \(\mathbf{Frm}\), equivalently over \(\mathbf{Loc}^{op}\), whose Grothendieck construction is the category \(\mathbf{PTop}\) of positive topologies. The forgetful functor
\[
\mathbf{U} : \mathbf{PTop} \to \mathbf{Loc}
\]
sends \((L,\Phi)\) to the underlying locale \(L\). It has a right adjoint
\[
\mathbf{\Delta} : \mathbf{Loc} \to \mathbf{PTop},
\qquad
\mathbf{\Delta}(L)=(L,\mathbf{SL}(L,\Omega)),
\]
called the constant object functor. Since \(\mathbf{\Delta}\) is full and faithful, \(\mathbf{Loc}\) is a reflective subcategory of \(\mathbf{PTop}\) [1812.09190].

This fibrational presentation is structurally significant. It organizes positive topologies as locales equipped with a predicate-like datum, namely a sub-suplattice of truth-valued maps. The paper further states that standard results on Grothendieck constructions imply that \(\mathbf{PTop}\) and \(\mathbf{BTop}\) are complete and cocomplete, and that the adjunction \(\mathbf{U}\dashv\mathbf{\Delta}\) gives an idempotent monad with \(\mathbf{Loc}\) equivalent to both the Kleisli and Eilenberg–Moore categories.

## 4. Positive points and the canonical positive topology of a space

The functor from positive topologies to spaces is defined by restricting ordinary locale points to the chosen positivity structure. Since \(\Omega\) is the initial frame, \(\mathbf{\Delta}(\Omega)\) is terminal in \(\mathbf{PTop}\). A point of \((L,\Phi)\) is a morphism
\[
\mathbf{\Delta}(\Omega)\to (L,\Phi),
\]
which is equivalently a frame homomorphism \(f:L\to\Omega\) lying in \(\Phi\). Hence
\[
\mathbf{Pt}^+(L,\Phi)=\mathbf{PTop}(\mathbf{\Delta}(\Omega),(L,\Phi))
\]
is exactly the set of those frame homomorphisms \(L\to\Omega\) that belong to \(\Phi\), topologized as a subspace of the usual \(\mathbf{Pt}(L)\) [1812.09190].

In the opposite direction, a space \((X,\tau)\) is sent to a canonical positive topology built from closed subsets. Constructively, a subset \(C\subseteq X\) is closed if
\[
\forall I\in\tau\, \bigl(x\in I\ \Rightarrow\ C\between I\bigr) \;\Rightarrow\; x\in C
\]
for all \(x\in X\). Each closed set defines a suplattice homomorphism
\[
\varphi_C : \tau \to \Omega, \quad I\mapsto C\between I,
\]
and these maps form a sub-suplattice of \(\mathbf{SL}(\tau,\Omega)\). The canonical positive topology associated to \((X,\tau)\) is
\[
\mathbf{\Lambda}(X,\tau)=\bigl(\tau,\{\varphi_C \mid C\subseteq X \text{ closed}\}\bigr).
\]

For a continuous map \(f:(X,\tau_X)\to(Y,\tau_Y)\), the underlying frame map is \(f^{-1}:\tau_Y\to\tau_X\), and the relation
\[
\varphi_C\circ f^{-1}=\varphi_{\mathsf{cl}(f(C))}
\]
for closed \(C\subseteq X\) ensures that \(f^{-1}\) defines a morphism in \(\mathbf{PTop}\). This gives a functor
\[
\mathbf{\Lambda} : \mathbf{Top}\to\mathbf{PTop}.
\]

The adjunction
\[
\mathbf{\Lambda}\dashv\mathbf{Pt}^+
\]
is established by explicit natural bijections. Given \(f:\mathbf{\Lambda}(X,\tau)\to(L,\Phi)\), one defines
\[
\widetilde{f}(x)(y)=\mathsf{cl}\{x\}\between f(y).
\]
Given a continuous map \(g:(X,\tau)\to\mathbf{Pt}^+(L,\Phi)\), one defines
\[
\widehat{g}(y)
=
g^{-1}\bigl(\{\varphi\in \mathbf{Pt}(L)\cap\Phi \mid \varphi(y)=1\}\bigr)\in\tau.
\]
These operations are inverse and natural, yielding
\[
\mathbf{PTop}(\mathbf{\Lambda}(X,\tau),(L,\Phi))
\cong
\mathbf{Top}\bigl((X,\tau),\mathbf{Pt}^+(L,\Phi)\bigr).
\]

## 5. The factorization theorem: the content of FaR-Loc

The central result is that the classical adjunction between \(\mathbf{Top}\) and \(\mathbf{Loc}\) is the composite of the two adjunctions just described [1812.09190]. The relevant identities are
\[
\mathbf{Pt}=\mathbf{Pt}^+\circ \mathbf{\Delta},
\qquad
\Omega=\mathbf{U}\circ \mathbf{\Lambda}.
\]
The first states that if a locale \(L\) is embedded as
\[
\mathbf{\Delta}(L)=(L,\mathbf{SL}(L,\Omega)),
\]
then its positive points are exactly its usual points. The second states that forgetting the positivity structure of the canonical positive topology of a space recovers its frame of opens.

Consequently, the adjunction
\[
\Omega \dashv \mathbf{Pt} : \mathbf{Top} \rightleftarrows \mathbf{Loc}
\]
factors as
\[
\mathbf{Top} \overset{\mathbf{\Lambda}}{\longrightarrow} \mathbf{PTop} \overset{\mathbf{U}}{\longrightarrow} \mathbf{Loc}
\]
on the left, and
\[
\mathbf{Loc} \overset{\mathbf{\Delta}}{\longrightarrow} \mathbf{PTop} \overset{\mathbf{Pt}^+}{\longrightarrow} \mathbf{Top}
\]
on the right. Equivalently, the classical Top–Loc adjunction is the composite adjunction obtained from
\[
\mathbf{\Lambda}\dashv\mathbf{Pt}^+
\quad\text{and}\quad
\mathbf{U}\dashv\mathbf{\Delta}.
\]

This is the precise content of FaR-Loc: factorizing the Top–Loc adjunction through the intermediate category of positive topologies. The paper’s main theorem states that the adjunction between \(\mathbf{Top}\) and \(\mathbf{Loc}\) factors through an adjunction between \(\mathbf{Top}\) and \(\mathbf{PTop}\) and the reflection \(\mathbf{PTop}\to\mathbf{Loc}\) [1812.09190].

## 6. Constructive significance, weak sobriety, and classical collapse

FaR-Loc is not merely a formal decomposition; it isolates the constructive gap between closed-set positivity and arbitrary truth-valued homomorphisms. The unit of \(\mathbf{\Lambda}\dashv\mathbf{Pt}^+\),
\[
\eta_X : (X,\tau)\to \mathbf{Pt}^+(\mathbf{\Lambda}(X,\tau)),
\]
defines a notion of weak sobriety: a space is weakly sober when \(\eta_X\) is an isomorphism [1812.09190]. This differs from classical sobriety, which is defined using the unit of \(\Omega\dashv\mathbf{Pt}\),
\[
(X,\tau)\to \mathbf{Pt}(\Omega(X,\tau)).
\]

Constructively, weak sobriety is strictly weaker than classical sobriety. The reason is that the inclusion
\[
\mathsf{Closed}(X,\tau)\subseteq \mathbf{SL}(\tau,\Omega)
\]
may be proper, so the canonical positive topology \(\mathbf{\Lambda}(X,\tau)\) can be strictly smaller than \(\mathbf{\Delta}(\Omega(X,\tau))\). This suggests that \(\mathbf{PTop}\) records only those truth-valued maps arising from positivity associated with closed sets, rather than all frame homomorphisms.

Classically, the distinction disappears. In that setting every \(\varphi\in\mathbf{SL}(\tau,\Omega)\) arises from a closed set, so
\[
\mathbf{\Lambda}(X,\tau)=\mathbf{\Delta}(\Omega(X,\tau)),
\qquad
\mathbf{\Lambda}=\mathbf{\Delta}\circ\Omega.
\]
It follows that weak sobriety and classical sobriety coincide. A plausible implication is that FaR-Loc is most informative in predicative and constructive settings, where the intermediate category \(\mathbf{PTop}\) retains data that are invisible after passage directly from \(\mathbf{Top}\) to \(\mathbf{Loc}\).

Within this perspective, \(\mathbf{PTop}\) sits between point-set and pointfree topology as a category of frames equipped with a selected positivity structure. Locales appear as the special case in which all homomorphisms \(L\to\Omega\) are admitted, while spaces enter through the closed-set-generated positive topology. FaR-Loc therefore refines the relationship between \(\mathbf{Top}\) and \(\mathbf{Loc}\) by making explicit which points are present constructively and how they arise from positivity, overlap, and formal-topological data [1812.09190].

Source: https://www.emergentmind.com/topics/far-loc