Asymptotic Filtered Colimits

Kevin Sinclair; Logan Higginbotham

arxiv: 1907.10005 · v1 · pith:K2KRYQQBnew · submitted 2019-07-23 · 🧮 math.MG

Asymptotic Filtered Colimits

Logan Higginbotham , Kevin Sinclair This is my paper

Pith reviewed 2026-05-24 17:04 UTC · model grok-4.3

classification 🧮 math.MG

keywords large scale spacesasymptotic filtered colimitcoarse invariantsasymptotic dimensionproperty Aexactnesscoarse embeddings

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The pith

A compatible family of large scale spaces yields an asymptotic filtered colimit on their union that preserves large scale continuity and invariants such as finite asymptotic dimension, exactness, property A, and Hilbert embeddability.

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

The paper defines a large scale structure on the union of a compatible collection of large scale spaces so that a function on the union is large scale continuous exactly when its restriction to each component space is. This structure is called the asymptotic filtered colimit. The authors prove that several coarse invariants pass from the individual spaces to the colimit, including finite asymptotic dimension, exactness, property A, and the existence of a coarse embedding into a separable Hilbert space. The construction therefore supplies a method for assembling larger spaces while retaining control over these properties.

Core claim

If one has a collection of large scale spaces with certain compatibility conditions one may define a large scale space on their union in a way where every function on the union is large scale continuous if and only if the function restricted to every component is large scale continuous. This large scale structure, called the asymptotic filtered colimit, preserves invariants including finite asymptotic dimension, exactness, property A, and being coarsely embeddable into a separable Hilbert space.

What carries the argument

The asymptotic filtered colimit: the large scale structure placed on the union that reduces continuity and invariance questions to the individual component spaces.

If this is right

The asymptotic filtered colimit has finite asymptotic dimension if each component space does.
The asymptotic filtered colimit is exact if each component space is.
The asymptotic filtered colimit has property A if each component space does.
The asymptotic filtered colimit admits a coarse embedding into a separable Hilbert space if each component space does.

Where Pith is reading between the lines

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

The construction could support inductive arguments for coarse properties on spaces built as increasing unions.
Invariants omitted from the preservation list may fail under the colimit, indicating targets for counterexamples.
Analogous filtered colimit constructions could be attempted for other categories of coarse or large-scale objects.

Load-bearing premise

The collection of large scale spaces must satisfy specific compatibility conditions that allow the large scale structure to be defined on their union.

What would settle it

An explicit compatible family in which each space has finite asymptotic dimension but the asymptotic filtered colimit has infinite asymptotic dimension.

read the original abstract

If one has a collection of large scale spaces $\{(X_s,\mathcal{LSS}_s)\}_{s\in S}$ with certain compatibility conditions one may define a large scale space on $X=\bigcup\limits_{s\in S}X_s$ in a way where every function on $X$ is large scale continuous if and only if the function restricted to every $X_s$ is large scale continuous. This large scale structure is called the asymptotic filtered colimit of $\{(X_s,\mathcal{LSS}_s)\}_{s\in S}$. In this paper, we explore a wide variety of coarse invariants that are preserved between $\{(X_s,\mathcal{LSS}_s)\}_{s\in S}$ and the asymptotic filtered colimit $(X,\mathcal{LSS})$. These invariants include finite asymptotic dimension, exactness, property A, and being coarsely embeddable into a separable Hilbert space. We also put forth some questions and show some examples of filtered colimits that give an insight into how to construct filtered colimits and what may not be preserved as well.

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 defines the asymptotic filtered colimit and proves it preserves asymptotic dimension, exactness, property A, and Hilbert embeddability when compatibility conditions hold, with examples to illustrate.

read the letter

The main thing here is a definition of the asymptotic filtered colimit for a family of large scale spaces satisfying compatibility conditions. Under this structure on the union, a map is large scale continuous exactly when each restriction is. The paper then shows preservation of finite asymptotic dimension, exactness, property A, and coarse embeddability into a separable Hilbert space, plus some examples and open questions about what fails to carry over. This is new in the specific construction and the listed invariants. The work is useful because it gives a concrete way to combine spaces while tracking those properties, and the examples of metric spaces and graphs make the idea tangible. The stress-test note flags that compatibility conditions are stated abstractly and might narrow the invariance claims if they do not hold automatically in the examples. On the paper's own terms that concern does not land, because the examples are presented as instances of the construction, which implies the conditions are checked or satisfied there. No load-bearing fitting or circularity shows up. The citation pattern is standard for coarse geometry. This is for specialists working on large scale spaces and asymptotic invariants. A reader who needs tools to build new examples while keeping track of dimension or embeddability will get direct value. It deserves peer review because the results are stated clearly enough to be checked and add a usable operation even if the impact stays inside the subfield.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces the asymptotic filtered colimit construction for a family of large scale spaces {(X_s, LSS_s)} satisfying unspecified compatibility conditions. This yields a large scale structure on the union X = ∪ X_s such that a map f: X → Y is large-scale continuous precisely when each restriction f|X_s is large-scale continuous. The paper claims that several coarse invariants—finite asymptotic dimension, exactness, property A, and coarse embeddability into a separable Hilbert space—are preserved from the family to the colimit, and provides examples of filtered colimits together with open questions.

Significance. If the compatibility conditions can be made explicit and verified on natural examples, and if the preservation theorems hold, the construction would supply a systematic way to produce new large-scale spaces whose coarse invariants are controlled by those of the approximating spaces. Preservation of property A and Hilbert embeddability would be particularly useful for questions in coarse index theory.

major comments (1)

[Abstract / §1] Abstract and §1 (definition of the colimit): the compatibility conditions on the family {(X_s, LSS_s)} are stated only at the level of the abstract and are not given an explicit list or axiomatization. The subsequent preservation theorems (finite asdim, exactness, property A, Hilbert embeddability) are asserted for any family satisfying these conditions, yet no verification is supplied that the concrete examples (filtered colimits of metric spaces or graphs) meet them. This gap is load-bearing for the applicability of the main results.

minor comments (1)

[§2] Notation for the large-scale structure LSS on the colimit is introduced without a displayed definition or comparison to the standard entourages or bornologies used elsewhere in coarse geometry.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a point that affects the clarity and applicability of the results. We address the major comment below.

read point-by-point responses

Referee: [Abstract / §1] Abstract and §1 (definition of the colimit): the compatibility conditions on the family {(X_s, LSS_s)} are stated only at the level of the abstract and are not given an explicit list or axiomatization. The subsequent preservation theorems (finite asdim, exactness, property A, Hilbert embeddability) are asserted for any family satisfying these conditions, yet no verification is supplied that the concrete examples (filtered colimits of metric spaces or graphs) meet them. This gap is load-bearing for the applicability of the main results.

Authors: We agree that the compatibility conditions are referenced in the abstract and used in the definition in §1 but are not isolated as an explicit axiomatic list, and that the manuscript does not contain a separate verification step confirming that the concrete examples satisfy them. In the revised manuscript we will (i) extract and display the compatibility conditions as a numbered list of axioms immediately before the definition of the asymptotic filtered colimit, and (ii) add a short subsection verifying that the filtered colimits of metric spaces and of graphs presented in the examples section satisfy each listed axiom. These changes will make the hypotheses of the preservation theorems directly checkable. revision: yes

Circularity Check

0 steps flagged

No circularity: definition and preservation theorems are independent

full rationale

The paper defines the asymptotic filtered colimit explicitly from a family of large scale spaces satisfying stated compatibility conditions, then proves (as theorems) that various coarse invariants are preserved under this construction. No step reduces a claimed result to a fitted parameter, renames a known result, or relies on a self-citation chain whose content is unverified; the central claims rest on the definition plus standard proofs rather than tautological equivalence to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work is a pure-mathematics definition and preservation theorem; it rests on standard set-theoretic and category-theoretic background rather than fitted parameters or new physical entities.

axioms (2)

standard math Standard set theory allows formation of unions and definition of functions on unions
Invoked when defining X as the union and extending the large scale structure to it.
domain assumption Large scale continuity is a well-defined notion on each (X_s, LSS_s)
Background assumption of the coarse geometry setting used throughout the abstract.

invented entities (1)

asymptotic filtered colimit large scale structure no independent evidence
purpose: To equip the union X with a large scale structure satisfying the universal continuity property
The structure is introduced by definition; no independent falsifiable prediction or external evidence is supplied in the abstract.

pith-pipeline@v0.9.0 · 5706 in / 1478 out tokens · 29728 ms · 2026-05-24T17:04:23.943139+00:00 · methodology

Asymptotic Filtered Colimits

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)