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Locales in presheaf toposes vs. presheaves of locales

T0 review · 0 major / 2 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read Internal suplattices and frames in presheaf toposes can be freely generated from any presheaf of such structures.

desk verdict The left adjoint for free internal suplattices/frames on presheaves is already in Henry-Townsend, but the paper's transfer results for internal local compactness and related properties are the actual new observations. read the letter →

arxiv 2606.21273 v1 pith:HZM5CKEV submitted 2026-06-19 math.CT

Vasileios Aravantinos-Sotiropoulos , Panagis Karazeris , Joshua Wrigley This is my paper · ORCID
classification math.CT
keywords presheaftoposesinternalsuplatticesframeslocalesleftadjointslocalcompactnessway-belowrelationtransitionmaps
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper constructs the free internal suplattice on a presheaf of suplattices, and likewise for frames, yielding a left adjoint to the forgetful functor from internal structures to presheaves of structures. This construction is then used to translate properties of internal frames, including local compactness, compactness and stable local compactness, into conditions that must hold for the sections of the underlying presheaf. For local compactness the transition maps must additionally preserve the way-below relation. The Hausdorff property for an internal frame does not require the same property on its sections.

What carries the argument

the left adjoint assigning to each presheaf of suplattices its free internal suplattice

What would settle it

An explicit calculation in a small presheaf topos showing that the constructed object fails the universal property of being free or fails to satisfy the internal suplattice axioms.

Watch

Extended reading notes

Core claim

We construct the free internal suplattice/frame on a presheaf of suplattices/frames, yielding a left adjoint to the forgetful functor from the respective internal structures to presheaves of structures. As an application of our construction, we investigate conditions on frames internal to a presheaf topos, such as being locally compact, compact, stably locally compact or Hausdorff, in terms of properties of their sections in the base topos. In the first three cases, it is necessary that all the sections have the respective properties, while the Hausdorff property is not transferred to the sections. Moreover for local compactness it is necessary that the transition maps preserve the way-below

Load-bearing premise

That not every presheaf of suplattices is already an internal suplattice, so a separate free construction is required to produce an internal one.

Editorial extensions

If this is right

  • An internal locally compact frame requires every section to be locally compact and every transition map to preserve the way-below relation.
  • An internal compact frame or stably locally compact frame requires every section to be compact or stably locally compact.
  • An internal Hausdorff frame need not have Hausdorff sections.
  • The way-below relation on an internal locally compact frame is determined by the way-below relations on its sections together with the transition maps.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same free-construction technique might be applied to produce internal versions of other poset-enriched structures inside presheaf toposes.
  • The distinction between internal and presheaf versions could be used to define a completion operation that turns arbitrary presheaves into internal ones in more general toposes.
  • Further comparison of the Hausdorff case with the compactness cases might clarify which properties are preserved under the forgetful functor.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript constructs the free internal suplattice (resp. frame) on an arbitrary presheaf of suplattices (resp. frames) in a presheaf topos, yielding an explicit left adjoint to the forgetful functor from internal structures to presheaves of structures. The construction is noted to coincide with one appearing in Henry–Townsend for the universal property of strictifying lax natural transformations. As an application, necessary conditions are derived for an internal frame to be locally compact, compact, stably locally compact or Hausdorff, expressed in terms of the corresponding properties of its sections together with preservation of the way-below relation by transition maps; the Hausdorff property does not transfer to sections.

Significance. If the explicit construction and the necessity claims hold, the work supplies a concrete tool for building free internal algebraic structures in presheaf toposes and clarifies when internal local-compactness-type properties descend to (or require) the same properties on sections. The alignment with the independent Henry–Townsend description is a strength, as is the analysis of the internal way-below relation in terms of the sections’ way-below relations.

minor comments (2)
  1. [Abstract] Abstract, line 3: 'on an presheaf' should read 'on a presheaf'.
  2. The title refers to locales while the body works throughout with frames (and suplattices); a brief sentence relating the two notions would help readers outside locale theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work, the assessment of its significance, and the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; explicit construction rests on standard external facts

full rationale

The paper's central result is an explicit construction of the free internal suplattice/frame on a presheaf of such structures in a presheaf topos, producing a left adjoint to the forgetful functor. This rests on the stated 'well-known characterisation' that internal suplattices/frames are precisely those presheaves whose transitions preserve the operations—an external fact in topos theory, not derived here. The description is noted to coincide with an independent construction in Henry–Townsend (different universal property), but the paper supplies its own derivation and applies it to derive conditions on internal local compactness etc. by direct examination of sections and transitions. No step reduces by definition, fitted parameter, or self-citation chain to the target claim; the adjunction and transfer results are independently verifiable from the construction.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Abstract only; no explicit free parameters, invented entities or additional axioms beyond the well-known characterisation are visible.

assumptions (1)
  • domain assumption Every internal suplattice is a presheaf of suplattices (and similarly for frames)
    Invoked at the start of the abstract as the well-known characterisation that motivates the construction.

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Cite this review

Pith. "Pith review of Locales in presheaf toposes vs. presheaves of locales." pith.science (2026). https://pith.science/paper/HZM5CKEV

@misc{pith2026260621273,
  author       = {Pith},
  title        = {Pith review of: Locales in presheaf toposes vs. presheaves of locales},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HZM5CKEV}},
  note         = {Machine review of arXiv:2606.21273}
}
read the original abstract

By a well-known characterisation, in a presheaf topos every internal suplattice is a presheaf of suplattices, but not every presheaf of suplattices is an internal suplattice (and similarly for frames). In this paper, we construct the free internal suplattice/frame on an presheaf of suplattices/frames, yielding a left adjoint to the forgetful functor from the respective internal structures to presheaves of structures. The description of this left adjoint has also appeared in recent work of Henry and Townsend, in connection to a different universal property, namely that of turning a lax natural transformation between poset-enriched functors to a strict one. As an application of our construction, we investigate conditions on frames internal to a presheaf topos, such as being locally compact, compact, stably locally compact or Hausdorff, in terms of properties of their sections in the base topos. In the first three cases, it is necessary that all the sections have the respective properties, while the Hausdorff property is not transferred to the sections. Moreover for local compactness it is necessary that the transition maps preserve the way-below relation. Finally, for an internal locally compact frame in presheaves we analyse the connection of its way-below relation to the respective relations of its sections.

Discussion (0). Continue with ORCID to comment.

Reference graph

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