Doubly weak double categories

Aaron David Fairbanks; Michael Shulman

arxiv: 2506.23651 · v2 · pith:4CORVKLEnew · submitted 2025-06-30 · 🧮 math.CT

Doubly weak double categories

Aaron David Fairbanks , Michael Shulman This is my paper

Pith reviewed 2026-05-25 08:07 UTC · model grok-4.3

classification 🧮 math.CT

keywords double categoriesweak compositiondouble computadsimplicit double categoriesrepresentability criteriontidiness condition

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The pith

Double categories can have weak composition of 1-cells in both directions when built from double computads with all compositions made coherent.

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

The paper defines doubly weak double categories as double computads equipped with coherent composition operations for all shapes. This matters for modeling structures where neither horizontal nor vertical 1-cell composition is required to be strict. The same objects arise by applying a representability criterion to implicit double categories, which have 2-cell compositions but lack 1-cell ones, or by adding a tidiness condition to existing double bicategories or cubical bicategories.

Core claim

A doubly weak double category is a double computad equipped with all possible composition operations, coherently. They are equivalently obtained from implicit double categories via a representability criterion on compositions of 1-cells, or from the double bicategories of Verity or cubical bicategories of Garner by imposing a tidiness condition.

What carries the argument

The double computad, a structure with 2-cells of all possible double-categorical shapes, which carries the argument by supporting the addition of all coherent composition operations.

If this is right

Double categories become available as models when weakness is needed simultaneously in both composition directions.
Implicit double categories become equivalent to doubly weak ones precisely when the representability criterion holds.
Tidiness supplies a direct translation between the new definition and prior notions of double bicategories.

Where Pith is reading between the lines

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

The approach may generalize to triple categories or other multi-directional weak structures by extending the computad shapes.
Comparison of the three presented definitions could identify which one is easiest to use for concrete examples such as spans or relations.
The tidiness condition might simplify calculations in applications where existing bicategory models are already available.

Load-bearing premise

The coherence conditions for all possible compositions in the double computad can be satisfied without contradiction.

What would settle it

An explicit double computad in which the required coherent horizontal and vertical compositions produce a contradiction in the associativity or unit laws would show the definition cannot be realized.

read the original abstract

We propose a definition of double categories whose composition of 1-cells is weak in both directions. Namely, a doubly weak double category is a double computad -- a structure with 2-cells of all possible double-categorical shapes -- equipped with all possible composition operations, coherently. We also characterize them using "implicit" double categories, which are double computads having all possible compositions of 2-cells, but no compositions of 1-cells; doubly weak double categories are then obtained by a simple representability criterion. Finally, they can also be defined by adding a "tidiness" condition to the double bicategories of Verity, or to the cubical bicategories of Garner.

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

Fairbanks and Shulman define doubly weak double categories as computads with coherent 1-cell compositions in both directions and give two clean characterizations via implicit structures and tidiness.

read the letter

The main point is that they define doubly weak double categories as double computads carrying 2-cells of every shape plus all possible coherent compositions for the 1-cells. They also show these are equivalent to implicit double categories satisfying a representability condition and to tidy versions of Verity's double bicategories or Garner's cubical bicategories. That triple framing is the actual new content and it sits cleanly on top of the cited prior work. The implicit characterization in particular looks like a practical way to avoid writing out every coherence axiom by hand. The connections to existing notions are direct and give immediate context without stretching. The citation pattern is standard and appropriate for the area. The soft spot is the coherence question raised in the stress-test note. The abstract does not display an explicit check that the representability step preserves weakness rather than forcing some compositions to become strict or identifying distinct 2-cells. If the full paper contains only the universal-property argument without a non-trivial example or a diagram chase confirming the associators and interchangers commute as intended, that would be a minor but real gap for a definition paper. Still, the overall setup follows the usual computad and representability patterns in the literature, so the risk looks manageable rather than load-bearing. This is for people already working with double categories or weak higher structures who need a handle on two-directional weakness. A reader in that niche would get concrete definitions to try out. It deserves peer review because the definitions are new, the equivalences are non-trivial, and the grounding in standard tools is clear enough for referees to verify the details.

Referee Report

1 major / 1 minor

Summary. The paper proposes a definition of doubly weak double categories as double computads equipped with all possible coherent composition operations for 1-cells in both directions. It provides two characterizations: one via implicit double categories (double computads with all 2-cell compositions but no 1-cell compositions) using a representability criterion, and another by imposing a tidiness condition on Verity's double bicategories or Garner's cubical bicategories.

Significance. If the coherence conditions for the compositions are shown to be consistent without forcing strictness or contradictions, the work would offer a new, flexible notion of double categories with bidirectional weakness, extending existing frameworks in higher category theory. The multiple characterizations provide independent perspectives that strengthen the proposal and could facilitate comparisons or applications in related areas such as bicategorical structures.

major comments (1)

[Characterization section] Characterization section: the claim that the representability criterion on implicit double categories produces exactly the desired weak 1-cell compositions lacks an explicit verification that the resulting coherence diagrams (pasting associators, unitors, and interchangers in both horizontal and vertical directions) commute without identifying distinct 2-cells or forcing unintended strictness; this is load-bearing for the central definition as it is presented as automatically delivering the weak structure.

minor comments (1)

The abstract would benefit from a concrete example of a doubly weak double category or a brief motivation for why existing notions (e.g., Verity or Garner) are insufficient.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of verifying the coherence conditions in the characterization via implicit double categories. We address the major comment below and will strengthen the exposition accordingly.

read point-by-point responses

Referee: [Characterization section] Characterization section: the claim that the representability criterion on implicit double categories produces exactly the desired weak 1-cell compositions lacks an explicit verification that the resulting coherence diagrams (pasting associators, unitors, and interchangers in both horizontal and vertical directions) commute without identifying distinct 2-cells or forcing unintended strictness; this is load-bearing for the central definition as it is presented as automatically delivering the weak structure.

Authors: We agree that the manuscript would benefit from an explicit verification of the coherence diagrams. The representability criterion is intended to induce the weak 1-cell compositions (and their coherences) via the universal property of the implicit double category, but a direct check that the induced associators, unitors, and interchangers in both directions satisfy all required pasting diagrams without collapse or unintended strictness is indeed load-bearing. In the revised version we will add a dedicated subsection that constructs these data explicitly from the representability functor and verifies the relevant diagrams commute, using only the double computad axioms and the implicit structure. revision: yes

Circularity Check

0 steps flagged

Minor self-citation risk; definition and representability criterion are independently grounded

full rationale

The paper proposes a definition of doubly weak double categories as double computads with coherent 1-cell compositions in both directions, plus a characterization via representability on implicit double categories (double computads with 2-cell compositions but no 1-cell compositions). This is a standard categorical construction (representability criterion) with no reduction to fitted parameters, self-definitional equations, or load-bearing self-citations that force the result. Citations to Verity and Garner provide context but the central claim does not collapse to them by construction. No steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on standard higher-categorical coherence axioms and the new representability/tidiness conditions; no free parameters or invented physical entities, only the new mathematical structure itself.

axioms (1)

standard math Standard axioms of categories and double categories for objects, 1-cells, and 2-cells.
The double computad and composition operations presuppose the usual category-theoretic background.

invented entities (2)

doubly weak double category no independent evidence
purpose: To model double categories with weak 1-cell composition in both directions.
The paper introduces this as the primary new structure.
implicit double category no independent evidence
purpose: Auxiliary structure for the representability characterization.
Mentioned as a tool to obtain the main definition.

pith-pipeline@v0.9.0 · 5628 in / 1413 out tokens · 59234 ms · 2026-05-25T08:07:29.759309+00:00 · methodology

Doubly weak double categories

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)