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Computads with invertible generators for weak {\omega}-categories

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Pith's one-line read Marking certain generators as invertible in computads for weak ω-categories yields an inductive free construction, a coreflection to ordinary computads, and cofibrant generated categories.

desk verdict The paper adds invertible generators to computads, gives a finite handle on walking equivalences, and uses a coreflection to get cofibrancy plus a topos structure. read the letter →

arxiv 2606.30254 v1 pith:6BOOOCWZ submitted 2026-06-29 math.CT

Thibaut Benjamin , Camil Champin , Ioannis Markakis This is my paper · ORCID
classification math.CT
keywords computadsweakω-categoriesinvertiblegeneratorswalkingequivalencescofibrantobjectspresheaftoposcoreflection
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 extends computads for weak ω-categories so that some generators can be marked invertible. It gives an inductive description of the free ω-categories these generalised computads generate, including a finite presentation of the walking equivalences that classify invertible cells. A coreflection is built from the generalised computads to ordinary ones that preserves the generated ω-categories. This coreflection is used to conclude that ω-categories generated by generalised computads are cofibrant. The subcategory of generalised computads equipped with generator-preserving morphisms is shown to be a presheaf topos.

What carries the argument

Generalised computads with marked invertible generators, together with their inductive free ω-category construction and the coreflection to ordinary computads.

What would settle it

An explicit dimension at which the inductive construction of the free ω-category on a generalised computad fails to be well-defined, or a generalised computad whose coreflection image does not generate an isomorphic ω-category, would falsify the claims.

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Extended reading notes

Core claim

Generalised computads mark selected generators as invertible and generate free weak ω-categories through an inductive construction that remains finite at each dimension. The construction admits a coreflection into ordinary computads that preserves the generated ω-category, from which it follows that the generated ω-categories are cofibrant. The category whose objects are generalised computads and whose morphisms preserve the generators is a presheaf topos.

Load-bearing premise

The inductive description of the free ω-categories generated by the generalised computads is well-defined at every dimension and admits a coreflection to ordinary computads that preserves the generated ω-category.

Editorial extensions

If this is right

  • Walking equivalences receive a simple finite description as free ω-categories on generalised computads.
  • ω-categories generated by generalised computads are cofibrant in the model structure under consideration.
  • The category of generalised computads with generator-preserving morphisms is a presheaf topos.
  • Properties of ordinary computads that are preserved by the coreflection transfer directly to the generalised setting.

Reading between the lines

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

  • The finite description of walking equivalences may simplify explicit calculations of invertible cells in higher-dimensional pasting diagrams.
  • If the same marking technique extends to other presentations of higher categories, cofibrancy results could transfer across different model structures.
  • The presheaf topos structure suggests that generalised computads support a well-behaved internal logic for reasoning about invertible cells.
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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 / 3 minor

Summary. The paper extends the notion of computads for weak ω-categories to generalised computads that allow certain generators to be marked as invertible. It gives an inductive description of the free ω-categories generated by these objects (including a finite presentation of the walking equivalences), constructs a coreflection from generalised computads to ordinary computads that preserves the generated ω-category, and concludes that the resulting ω-categories are cofibrant. It further shows that the subcategory of generalised computads equipped with generator-preserving morphisms forms a presheaf topos.

Significance. If the inductive constructions and coreflection are correct, the work supplies a concrete, finite description of walking equivalences and establishes cofibrancy results that are useful for model-categorical approaches to weak ω-categories. The demonstration that the relevant subcategory is a presheaf topos is a clean categorical property that parallels known results for ordinary computads and may facilitate further structural investigations.

minor comments (3)
  1. The introduction refers to the 'corresponding result for ordinary computads' when stating the presheaf-topos property but does not cite the specific reference; adding this citation would improve traceability.
  2. Notation distinguishing marked invertible generators from ordinary ones (introduced early in the definitions) would benefit from a short illustrative example immediately after the definition to clarify the marking convention before the inductive clauses begin.
  3. The statement that the coreflection preserves the generated ω-category is central; a brief remark on how the preservation interacts with the dimension-wise inductive steps would aid readability even if the full verification appears later.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the paper, the significance statement, and the recommendation of minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained

full rationale

The paper extends ordinary computads by marking invertible generators, gives an inductive free-generation process, constructs an explicit coreflection preserving the generated ω-category, deduces cofibrancy, and shows the generator-preserving subcategory is a presheaf topos. None of these steps reduce by definition, by fitted parameters renamed as predictions, or by load-bearing self-citation to the paper's own inputs; the inductive description and coreflection are presented as new constructions whose well-definedness is asserted directly rather than derived from the target conclusions. The presheaf-topos claim is a standard categorical consequence once the subcategory is defined. No equations or self-referential reductions appear in the stated claims.

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

Abstract only; no concrete free parameters, background axioms, or newly postulated entities are identifiable from the provided text.

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Cite this review

Pith. "Pith review of Computads with invertible generators for weak {\omega}-categories." pith.science (2026). https://pith.science/paper/6BOOOCWZ

@misc{pith2026260630254,
  author       = {Pith},
  title        = {Pith review of: Computads with invertible generators for weak \omega-categories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6BOOOCWZ}},
  note         = {Machine review of arXiv:2606.30254}
}
abstract

We extend the notion of computads for weak \(\omega\)-categories to allow marking certain generators as invertible, and describe inductively the free \(\omega\)-categories they generate. This gives a simple, finite description of the walking equivalences, the \(\omega\)-categories classifying invertible cells. We then construct a coreflection from generalised to ordinary computads, preserving the generated \(\omega\)-categories, and conclude that \(\omega\)-categories generated by generalised computads are cofibrant. Finally, we study the subcategory of generalised computads and generator-preserving morphisms, and show that it is a presheaf topos, similarly to the case of ordinary computads.

Figures

Figures reproduced from arXiv: 2606.30254 by the authors.

Figure 1
Figure 1. A Batanin tree and its boundary Definition 2.7. The dimension of a Batanin tree is the natural number defined by the recursive formula dim(br[B1, . . . , Bn]) = sup{dim(B1) + 1, . . . , dim(Bn) + 1} (14) The k-boundary of a Batanin tree for k ∈ N is the tree defined recursively by ∂0(br[B1, . . . , Bn]) = br[] ∂k+1(br[B1, . . . , Bn]) = br[∂kB1, . . . , ∂kBn] (15) The dimension of a Batanin tree is the height of the… view at source ↗
Figure 2
Figure 2. The pasting diagrams P 1 0 , P 2 0 and P 2 1 3. Computads with invertible generators Computads, or polygraphs, were originally introduced in the context of 2-cat￾egories by Street [22], and then generalised to strict ω-categories independently by Street [23] and Burroni [24]. For a recent exposition of computads for strict ω-categories, we refer to the book by Ara et al. [25]. The theory of computads was further gen… view at source ↗

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