$L'$-localization in an $\infty$-topos

Marco Vergura

arxiv: 1907.03837 · v1 · pith:ECJZCUOXnew · submitted 2019-07-08 · 🧮 math.AT · math.CT

L'-localization in an infty-topos

Marco Vergura This is my paper

Pith reviewed 2026-05-25 00:19 UTC · model grok-4.3

classification 🧮 math.AT math.CT

keywords ∞-toposreflective subfibrationL-separated mapsL-local mapslocalizationdiagonal maphomotopy theory

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The pith

Given any reflective subfibration L on an ∞-topos, there exists another L' whose local maps are the maps with L-local diagonals.

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

The paper proves that any reflective subfibration L on an ∞-topos E induces a new reflective subfibration L' on the same E. The local maps of this new subfibration are exactly the L-separated maps, meaning those maps whose diagonal maps are L-local. This construction works uniformly in the setting of ∞-topoi and supplies a way to produce localizations that enforce a separation condition on maps. A sympathetic reader would care because separation conditions frequently appear when one wants certain maps to behave like monomorphisms or embeddings after localization has been applied. The result therefore enlarges the set of available operations for building and combining localizations in higher categorical geometry and homotopy theory.

Core claim

We prove that, given any reflective subfibration L_• on an ∞-topos E, there exists a reflective subfibration L'_• on E whose local maps are the L-separated maps, that is, the maps whose diagonals are L-local.

What carries the argument

The L-separated maps (maps whose diagonals are L-local), which serve as the local maps of the induced reflective subfibration L'.

If this is right

The class of L-separated maps forms a reflective subfibration on the given ∞-topos.
The construction associates to each reflective subfibration a canonically induced 'separated' version.
The new subfibration L' can be substituted for L in further localization constructions.
The result supplies a uniform operation that can be iterated inside the theory of localizations in ∞-topoi.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Specializing the construction to the ∞-topos of spaces may recover classical separation axioms or sheaf separation conditions.
Iterating the passage from L to L' produces a sequence of increasingly separated localizations whose intersection might satisfy a full separation property.
The same mechanism could be applied to the terminal map to define L-separated objects inside the ∞-topos.

Load-bearing premise

The ∞-topos possesses all limits and colimits needed for diagonals of maps to exist and for the class of L-separated maps to form a reflective subfibration.

What would settle it

An explicit ∞-topos E together with a reflective subfibration L such that the maps whose diagonals are L-local do not form the local maps of any reflective subfibration on E.

read the original abstract

We prove that, given any reflective subfibration $L_\bullet$ on an $\infty$-topos $\mathcal{E}$, there exists a reflective subfibration $L'_\bullet$ on $\mathcal{E}$ whose local maps are the $L$-separated maps, that is, the maps whose diagonals are $L$-local. This is the companion paper to "Localization theory in an $\infty$-topos".

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows how to build a companion reflective subfibration L' whose local maps are exactly the L-separated ones.

read the letter

The punchline for this paper is the existence of the L' reflective subfibration defined by taking L-separated maps as its local maps. Given L, the maps whose diagonals are L-local become the local maps for this new L' on the same ∞-topos E. This is presented as the direct companion to the main localization paper, so the new content is the specific construction of L' via the separated condition. The paper does well by staying inside the standard framework: diagonals exist by definition of an ∞-topos, and the claim uses the reflective subfibration properties already developed for L without adding unrelated structure. The result is an existence statement built from a given L, which avoids circularity. The soft spots are limited and proportionate. The abstract states the existence but does not display the lemmas, so the detailed steps cannot be inspected from what is here; that keeps soundness from being fully checked. The stress-test finds no internal inconsistency, no hidden non-constructive step, and no missing closure property, and the ambient assumptions on limits and colimits are the usual ones for ∞-topoi. No other weaknesses stand out. This work is for specialists already reading the paired localization paper. A reader in that niche will see it as a natural extension and get concrete value from the construction. It is not aimed at a wider audience. It deserves a serious referee to verify the details of how the subfibration L' is assembled and shown to be reflective.

Referee Report

0 major / 1 minor

Summary. The manuscript proves that, given any reflective subfibration L_• on an ∞-topos E, there exists a reflective subfibration L'_• on E whose local maps are precisely the L-separated maps (those whose diagonals are L-local). The result is presented as a direct companion to the paper 'Localization theory in an ∞-topos'.

Significance. If the result holds, the construction supplies a canonical way to produce the separated localization associated to any given reflective subfibration. This is a standard and useful operation in the theory of localizations of ∞-topoi, enabling iterative constructions and the study of separated objects without additional data. The argument is asserted to be carried out in full within the manuscript and relies only on the standard limits, colimits, and internal structure of an ∞-topos.

minor comments (1)

[Abstract] The abstract is concise; a one-sentence indication of the main technical tool (e.g., the universal property of the reflector or the relevant factorization system) would help readers locate the argument quickly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and their recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; existence claim is a direct construction from given input

full rationale

The central result is an existence theorem: given any reflective subfibration L_• on an ∞-topos E (with standard limits/colimits for diagonals), there exists a reflective subfibration L'_• whose local maps are exactly the L-separated maps. This is a forward construction from the supplied L_• rather than any self-definition, fitted prediction, or reduction to a self-citation chain. The companion paper is referenced only for context; the present claim does not rely on an unverified uniqueness theorem or ansatz imported from the same authors. The derivation remains self-contained against the external structure of ∞-topoi.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The result rests on standard domain assumptions of ∞-topos theory rather than new free parameters or invented entities.

axioms (2)

domain assumption An ∞-topos admits all small limits and colimits and supports the formation of diagonals for any map.
Required for the definition of L-separated maps via diagonals.
domain assumption Reflective subfibrations are closed under the operations needed to produce a new subfibration from the separated condition.
Invoked to guarantee that L' exists and is reflective.

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