Dotted $2$-limits

Joanna Ko

arxiv: 2306.01625 · v2 · submitted 2023-06-02 · 🧮 math.CT

Dotted 2-limits

Joanna Ko This is my paper

Pith reviewed 2026-05-24 08:10 UTC · model grok-4.3

classification 🧮 math.CT

keywords dotted 2-limitsmarked limitsF-weighted limitscodescent objects2-category theoryenhanced 2-categoriesCat-weighted limits

0 comments

The pith

Dotted 2-limits and F-weighted limits have the same expressive power in enhanced 2-categories.

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

The paper gives a new proof that marked limits equal Cat-weighted limits by introducing marked codescent objects of marked coherence data. It then defines dotted 2-limits as the direct generalization of marked limits into the enhanced 2-categorical setting. The central result is that these dotted 2-limits capture exactly the same information as F-weighted limits. A reader cares because the result lets 2-category theorists switch between the two descriptions of limits without changing what can be expressed.

Core claim

Dotted 2-limits, obtained by generalizing marked limits, have the same expressive power as F-weighted limits; this is shown by first proving the equivalence of marked limits and Cat-weighted limits via the new marked codescent objects of marked coherence data.

What carries the argument

Marked codescent objects of marked coherence data, which supply the new proof that marked limits match Cat-weighted limits and thereby support the definition of dotted 2-limits.

If this is right

Marked limits can replace Cat-weighted limits via the codescent construction.
Dotted 2-limits serve as a complete substitute for F-weighted limits in the enhanced setting.
Any construction or theorem stated with F-weights can be rewritten using dotted 2-limits.

Where Pith is reading between the lines

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

The same codescent technique might produce similar equivalences in higher-dimensional category theory.
Practical calculations of limits could choose the dotted or weighted form according to which data is easier to handle.

Load-bearing premise

The enhanced 2-categorical setting is defined so that generalizing marked limits preserves the relevant universal properties and coherence conditions.

What would settle it

An explicit F-weighted limit in an enhanced 2-category that cannot be realized as any dotted 2-limit.

read the original abstract

Marked limits, or Cartesian quasi-limits introduced by Gray, give an alternative approach to $\mathbf{Cat}$-weighted limits in $2$-category theory. This was first established by Street, and we aim to give a new approach to this result using marked codescent objects of marked coherence data which we introduce in this article. We then propose the notion of dotted $2$-limits, which is a natural generalisation of marked limits to the enhanced $2$-categorical setting. We establish that dotted $2$-limits and $\mathscr{F}$-weighted limits both have the same expressive power.

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 dotted 2-limits via a new marked codescent construction and claims they match the power of F-weighted limits, but the coherence carry-over in the enhanced setting is the spot that needs explicit checking.

read the letter

The main takeaway is that Ko defines dotted 2-limits as a direct generalization of marked limits to enhanced 2-categories and gives a fresh route to the known equivalence with F-weighted limits by introducing marked codescent objects of marked coherence data. That construction is the actual new piece, and it looks like a clean alternative to Street's original argument on marked limits. The definitions appear set up to preserve the relevant universal properties, which is the part that matters for the expressive-power claim. If the full text spells out the diagrams and the coherence data without hidden assumptions, that counts as useful technical work inside the subfield. The paper does not overclaim reach; it stays within 2-category theory and focuses on organizing existing notions differently. The soft spot is the generalization step itself. Moving from the ordinary marked case to the enhanced setting requires that the dotted version satisfies the same weighted-limit universal property and coherence conditions. The stress-test note is right to flag that this step can be implicit rather than verified separately, and any new obstructions from the extra structure would break the equivalence. I would check the relevant section to see whether the paper supplies an explicit argument or just relies on the ordinary case carrying over. This is narrow work aimed at people already deep in 2-categorical limits and weighted colimits. A reader who needs alternative presentations for specific constructions might get something out of the marked codescent objects, but most category theorists outside that niche will not. The paper shows clear engagement with the literature and states a precise claim, so it deserves a serious referee even if the coherence verification turns out to need tightening. I would send it to review rather than desk-reject.

Referee Report

1 major / 2 minor

Summary. The paper introduces marked codescent objects of marked coherence data to give a new approach to Street's result equating marked (Cartesian quasi-)limits with Cat-weighted limits in 2-category theory. It then defines dotted 2-limits as a natural generalization of marked limits to the enhanced 2-categorical setting and claims that dotted 2-limits and F-weighted limits have equivalent expressive power.

Significance. If the equivalence is established with explicit verification of universal properties, the work would supply an alternative construction for weighted limits via codescent objects and extend the theory to enhanced 2-categories; this could be useful for coherence questions in higher-dimensional category theory. The introduction of the new marked codescent objects is noted as a potential technical contribution.

major comments (1)

[Section defining dotted 2-limits and the main equivalence theorem] The central claim that every F-weighted limit can be realized as a dotted 2-limit (and conversely) requires that the dotted version inherits the same weighted-limit universal property and coherence conditions as in the ordinary 2-categorical case. The generalization step from the marked case (via codescent objects of marked coherence data) to the enhanced setting is described but lacks an explicit check that the additional structure does not introduce new coherence obstructions or alter the representing object up to equivalence; this verification is load-bearing for the expressive-power equivalence.

minor comments (2)

Notation for the enhanced 2-categorical setting and the functor F could be introduced with a short preliminary subsection to improve readability for readers unfamiliar with the enhanced context.
The abstract states the main result but does not name the precise theorem or section where the equivalence is proved.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful report and for identifying the need for a more explicit verification in the generalization from marked limits to dotted 2-limits. We address the major comment below and will revise the manuscript to strengthen the exposition of the equivalence.

read point-by-point responses

Referee: [Section defining dotted 2-limits and the main equivalence theorem] The central claim that every F-weighted limit can be realized as a dotted 2-limit (and conversely) requires that the dotted version inherits the same weighted-limit universal property and coherence conditions as in the ordinary 2-categorical case. The generalization step from the marked case (via codescent objects of marked coherence data) to the enhanced setting is described but lacks an explicit check that the additional structure does not introduce new coherence obstructions or alter the representing object up to equivalence; this verification is load-bearing for the expressive-power equivalence.

Authors: We agree that the manuscript would benefit from a more explicit verification of the universal property in the enhanced setting. The construction proceeds by first realizing the F-weighted limit via the marked codescent object of the corresponding marked coherence data (as developed for the ordinary 2-categorical case), then lifting this to the dotted 2-limit by adjoining the additional 2-categorical data encoded in the dotted notation. In the revised version we will insert a new subsection immediately after the definition of dotted 2-limits that carries out the following steps in detail: (i) construct the candidate representing object explicitly from the codescent object; (ii) verify that its universal property in the enhanced 2-category coincides with the F-weighted universal property by checking the relevant 2-natural transformations and modifications; (iii) confirm that the extra coherence data required by the enhanced setting is already accounted for by the marking and does not impose additional obstructions or change the representing object up to equivalence. This will make the load-bearing step fully explicit while preserving the overall argument structure. revision: yes

Circularity Check

0 steps flagged

No circularity: new notions introduced and equivalence proved via explicit constructions

full rationale

The paper introduces marked codescent objects of marked coherence data as a new tool, then defines dotted 2-limits as a generalization to the enhanced 2-categorical setting, and separately establishes that these have the same expressive power as F-weighted limits. No step reduces the central claim to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain. The equivalence is presented as a theorem proved from the new definitions rather than assumed by construction. External citations (e.g., to Street) are to prior independent work and do not create circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claims rest on newly introduced definitions (marked codescent objects and dotted 2-limits) together with background results from 2-category theory; no free parameters or invented entities with independent evidence are visible.

axioms (1)

standard math Standard axioms and coherence conditions of 2-category theory as developed by Gray and Street
The paper explicitly builds on these established foundations.

invented entities (2)

marked codescent objects of marked coherence data no independent evidence
purpose: To supply a new route to the equivalence between marked limits and Cat-weighted limits
Newly defined in the paper; no external falsifiable handle supplied.
dotted 2-limits no independent evidence
purpose: To generalize marked limits to the enhanced 2-categorical setting
Newly proposed in the paper; no external falsifiable handle supplied.

pith-pipeline@v0.9.0 · 5603 in / 1240 out tokens · 67466 ms · 2026-05-24T08:10:02.560734+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Enhanced $2$-categories of models of sketches as enhanced $2$-categories of algebras over monads
math.CT 2026-05 unverdicted novelty 7.0

Models of enhanced limit 2-sketches are equivalent to algebras over enhanced 2-monads, including lax morphisms, and inherit w-rigged limits.