FaR-Loc: Factorizing the Top–Loc Adjunction

• FaR-Loc is a factorization framework that decomposes the classical Top–Loc adjunction through an intermediate category of positive topologies, capturing constructive positivity data.
• It employs the Grothendieck construction to integrate positivity structures into locale theory, bridging formal topology and point-set methods.
• It establishes an adjunction factorization that distinguishes between classical and weak sobriety by using closed-set generated positivity in a categorical setting.

Searching arXiv for the cited paper and closely related work on positive topologies, locales, and formal topology. arXiv search query: (Ciraulo et al., 2018) 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 $\mathbf{Pt} = \mathbf{Pt}^+\circ \mathbf{\Delta}$ (Ciraulo et al., 2018).

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{Top}$0 of locales, defined as $\mathbf{Top}$1, where $\mathbf{Top}$2 is the category of frames and frame homomorphisms (Ciraulo et al., 2018). A frame is a suplattice $\mathbf{Top}$3 in which finite meets distribute over arbitrary joins: $\mathbf{Top}$4 A frame homomorphism preserves arbitrary joins and finite meets.

The functor

$\mathbf{Top}$5

sends a space $\mathbf{Top}$6 to its frame of opens $\mathbf{Top}$7, regarded contravariantly as a locale. A continuous map $\mathbf{Top}$8 is sent to the inverse-image frame homomorphism $\mathbf{Top}$9. In the opposite direction, the points functor

$\mathbf{PTop}$0

sends a locale $\mathbf{PTop}$1 to the space of its points, where a point is a frame homomorphism $\mathbf{PTop}$2.

The classical adjunction is expressed by the hom-set isomorphisms

$\mathbf{PTop}$3

Its unit characterizes sobriety: a space is sober exactly when the canonical map $\mathbf{PTop}$4 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 $\mathbf{PTop}$5, where $\mathbf{PTop}$6 is a set of generators and $\mathbf{PTop}$7 is a cover relation $\mathbf{PTop}$8 between $\mathbf{PTop}$9 and $\mathbf{PTop}$0. The quotient $\mathbf{PTop}$1 yields the associated suplattice, and a formal cover is a basic cover whose associated suplattice is a frame (Ciraulo et al., 2018).

A basic topology is a triple $\mathbf{PTop}$2 where $\mathbf{PTop}$3 is a basic cover and $\mathbf{PTop}$4 is a positivity relation satisfying three conditions: soundness of positivity, monotonicity in the target, and compatibility with cover. In particular, if $\mathbf{PTop}$5, then $\mathbf{PTop}$6; and if $\mathbf{PTop}$7 and $\mathbf{PTop}$8, then some $\mathbf{PTop}$9 satisfies $\mathbf{Loc}$0.

Using Sambin’s overlap notation $\mathbf{Loc}$1, each positivity relation induces suplattice homomorphisms

$\mathbf{Loc}$2

Ciraulo–Vickers show that positivity relations on $\mathbf{Loc}$3 are in bijection with sub-suplattices

$\mathbf{Loc}$4

where $\mathbf{Loc}$5 and $\mathbf{Loc}$6 denotes the hom-suplattice of suplattice homomorphisms $\mathbf{Loc}$7 (Ciraulo et al., 2018).

This yields an equivalent description of a basic topology as a pair $\mathbf{Loc}$8, with $\mathbf{Loc}$9 a suplattice and $\mathbf{Loc}$0 a sub-suplattice of $\mathbf{Loc}$1. A positive topology is the special case in which $\mathbf{Loc}$2 is a frame. Thus an object of $\mathbf{Loc}$3 is a pair

$\mathbf{Loc}$4

with $\mathbf{Loc}$5 and $\mathbf{Loc}$6 a sub-suplattice. A morphism

$\mathbf{Loc}$7

is a frame homomorphism $\mathbf{Loc}$8 such that

$\mathbf{Loc}$9

Conceptually, a locale supplies all frame homomorphisms $$\mathbf{\Lambda} \dashv \mathbf{Pt}^+ : \mathbf{Top} \rightleftarrows \mathbf{PTop}, \qquad \mathbf{U} \dashv \mathbf{\Delta} : \mathbf{PTop} \rightleftarrows \mathbf{Loc}$$0, 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 $$\mathbf{\Lambda} \dashv \mathbf{Pt}^+ : \mathbf{Top} \rightleftarrows \mathbf{PTop}, \qquad \mathbf{U} \dashv \mathbf{\Delta} : \mathbf{PTop} \rightleftarrows \mathbf{Loc}$$1 and the Grothendieck construction

The fibrational core of FaR-Loc is a doctrine built from truth-valued morphisms. For a suplattice $$\mathbf{\Lambda} \dashv \mathbf{Pt}^+ : \mathbf{Top} \rightleftarrows \mathbf{PTop}, \qquad \mathbf{U} \dashv \mathbf{\Delta} : \mathbf{PTop} \rightleftarrows \mathbf{Loc}$$2, the hom-set $$\mathbf{\Lambda} \dashv \mathbf{Pt}^+ : \mathbf{Top} \rightleftarrows \mathbf{PTop}, \qquad \mathbf{U} \dashv \mathbf{\Delta} : \mathbf{PTop} \rightleftarrows \mathbf{Loc}$$3 has a pointwise join structure, and the subobject functor assigns to a suplattice the preorder of its sub-suplattices. Composing these constructions gives

$$\mathbf{\Lambda} \dashv \mathbf{Pt}^+ : \mathbf{Top} \rightleftarrows \mathbf{PTop}, \qquad \mathbf{U} \dashv \mathbf{\Delta} : \mathbf{PTop} \rightleftarrows \mathbf{Loc}$$4

For a homomorphism $$\mathbf{\Lambda} \dashv \mathbf{Pt}^+ : \mathbf{Top} \rightleftarrows \mathbf{PTop}, \qquad \mathbf{U} \dashv \mathbf{\Delta} : \mathbf{PTop} \rightleftarrows \mathbf{Loc}$$5,

$$\mathbf{\Lambda} \dashv \mathbf{Pt}^+ : \mathbf{Top} \rightleftarrows \mathbf{PTop}, \qquad \mathbf{U} \dashv \mathbf{\Delta} : \mathbf{PTop} \rightleftarrows \mathbf{Loc}$$6

again a sub-suplattice (Ciraulo et al., 2018).

Applying the Grothendieck construction to $$\mathbf{\Lambda} \dashv \mathbf{Pt}^+ : \mathbf{Top} \rightleftarrows \mathbf{PTop}, \qquad \mathbf{U} \dashv \mathbf{\Delta} : \mathbf{PTop} \rightleftarrows \mathbf{Loc}$$7 produces a total category $$\mathbf{\Lambda} \dashv \mathbf{Pt}^+ : \mathbf{Top} \rightleftarrows \mathbf{PTop}, \qquad \mathbf{U} \dashv \mathbf{\Delta} : \mathbf{PTop} \rightleftarrows \mathbf{Loc}$$8 whose objects are pairs $$\mathbf{\Lambda} \dashv \mathbf{Pt}^+ : \mathbf{Top} \rightleftarrows \mathbf{PTop}, \qquad \mathbf{U} \dashv \mathbf{\Delta} : \mathbf{PTop} \rightleftarrows \mathbf{Loc}$$9 with $\Omega = \mathbf{U}\circ \mathbf{\Lambda}$0, and whose morphisms are precisely those homomorphisms satisfying the positivity-preservation condition $\Omega = \mathbf{U}\circ \mathbf{\Lambda}$1. This is exactly the category $\Omega = \mathbf{U}\circ \mathbf{\Lambda}$2 of basic topologies.

Restricting from suplattices to frames yields a doctrine over $\Omega = \mathbf{U}\circ \mathbf{\Lambda}$3, equivalently over $\Omega = \mathbf{U}\circ \mathbf{\Lambda}$4, whose Grothendieck construction is the category $\Omega = \mathbf{U}\circ \mathbf{\Lambda}$5 of positive topologies. The forgetful functor

$\Omega = \mathbf{U}\circ \mathbf{\Lambda}$6

sends $\Omega = \mathbf{U}\circ \mathbf{\Lambda}$7 to the underlying locale $\Omega = \mathbf{U}\circ \mathbf{\Lambda}$8. It has a right adjoint

$\Omega = \mathbf{U}\circ \mathbf{\Lambda}$9

called the constant object functor. Since $\mathbf{Pt} = \mathbf{Pt}^+\circ \mathbf{\Delta}$0 is full and faithful, $\mathbf{Pt} = \mathbf{Pt}^+\circ \mathbf{\Delta}$1 is a reflective subcategory of $\mathbf{Pt} = \mathbf{Pt}^+\circ \mathbf{\Delta}$2 (Ciraulo et al., 2018).

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{Pt} = \mathbf{Pt}^+\circ \mathbf{\Delta}$3 and $\mathbf{Pt} = \mathbf{Pt}^+\circ \mathbf{\Delta}$4 are complete and cocomplete, and that the adjunction $\mathbf{Pt} = \mathbf{Pt}^+\circ \mathbf{\Delta}$5 gives an idempotent monad with $\mathbf{Pt} = \mathbf{Pt}^+\circ \mathbf{\Delta}$6 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 $\mathbf{Pt} = \mathbf{Pt}^+\circ \mathbf{\Delta}$7 is the initial frame, $\mathbf{Pt} = \mathbf{Pt}^+\circ \mathbf{\Delta}$8 is terminal in $\mathbf{Pt} = \mathbf{Pt}^+\circ \mathbf{\Delta}$9. A point of $\mathbf{Top}$0 is a morphism

$\mathbf{Top}$1

which is equivalently a frame homomorphism $\mathbf{Top}$2 lying in $\mathbf{Top}$3. Hence

$\mathbf{Top}$4

is exactly the set of those frame homomorphisms $\mathbf{Top}$5 that belong to $\mathbf{Top}$6, topologized as a subspace of the usual $\mathbf{Top}$7 (Ciraulo et al., 2018).

In the opposite direction, a space $\mathbf{Top}$8 is sent to a canonical positive topology built from closed subsets. Constructively, a subset $\mathbf{Top}$9 is closed if

$\mathbf{Top}$00

for all $\mathbf{Top}$01. Each closed set defines a suplattice homomorphism

$\mathbf{Top}$02

and these maps form a sub-suplattice of $\mathbf{Top}$03. The canonical positive topology associated to $\mathbf{Top}$04 is

$\mathbf{Top}$05

For a continuous map $\mathbf{Top}$06, the underlying frame map is $\mathbf{Top}$07, and the relation

$\mathbf{Top}$08

for closed $\mathbf{Top}$09 ensures that $\mathbf{Top}$10 defines a morphism in $\mathbf{Top}$11. This gives a functor

$\mathbf{Top}$12

The adjunction

$\mathbf{Top}$13

is established by explicit natural bijections. Given $\mathbf{Top}$14, one defines

$\mathbf{Top}$15

Given a continuous map $\mathbf{Top}$16, one defines

$\mathbf{Top}$17

These operations are inverse and natural, yielding

$\mathbf{Top}$18

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

The central result is that the classical adjunction between $\mathbf{Top}$19 and $\mathbf{Top}$20 is the composite of the two adjunctions just described (Ciraulo et al., 2018). The relevant identities are

$\mathbf{Top}$21

The first states that if a locale $\mathbf{Top}$22 is embedded as

$\mathbf{Top}$23

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

$\mathbf{Top}$24

factors as

$\mathbf{Top}$25

on the left, and

$\mathbf{Top}$26

on the right. Equivalently, the classical Top–Loc adjunction is the composite adjunction obtained from

$\mathbf{Top}$27

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}$28 and $\mathbf{Top}$29 factors through an adjunction between $\mathbf{Top}$30 and $\mathbf{Top}$31 and the reflection $\mathbf{Top}$32 (Ciraulo et al., 2018).

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{Top}$33,

$\mathbf{Top}$34

defines a notion of weak sobriety: a space is weakly sober when $\mathbf{Top}$35 is an isomorphism (Ciraulo et al., 2018). This differs from classical sobriety, which is defined using the unit of $\mathbf{Top}$36,

$\mathbf{Top}$37

Constructively, weak sobriety is strictly weaker than classical sobriety. The reason is that the inclusion

$\mathbf{Top}$38

may be proper, so the canonical positive topology $\mathbf{Top}$39 can be strictly smaller than $\mathbf{Top}$40. This suggests that $\mathbf{Top}$41 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 $\mathbf{Top}$42 arises from a closed set, so

$\mathbf{Top}$43

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{Top}$44 retains data that are invisible after passage directly from $\mathbf{Top}$45 to $\mathbf{Top}$46.

Within this perspective, $\mathbf{Top}$47 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 $\mathbf{Top}$48 are admitted, while spaces enter through the closed-set-generated positive topology. FaR-Loc therefore refines the relationship between $\mathbf{Top}$49 and $\mathbf{Top}$50 by making explicit which points are present constructively and how they arise from positivity, overlap, and formal-topological data (Ciraulo et al., 2018).