REVIEW 2 minor
Rigid Algebras and Cospans
T0 review · 0 major / 2 minor · reviewed 2026-05-25 · grok-4.3
Pith's one-line read The (∞,1)-category of rigid commutative algebras in the cospans of an (∞,1)-category C is canonically identified with C.
desk verdict The paper defines rigid algebras in (∞,2)-categories and proves they recover the base (∞,1)-category C exactly when applied to its cospan (∞,2)-category, plus an adjunction. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The canonical identification of the (∞,1)-category of rigid commutative algebras in cospans with the base (∞,1)-category C
What would settle it
A concrete (∞,1)-category C together with an explicit computation showing that the rigid commutative algebras in its cospan (∞,2)-category fail to be equivalent to C.
Extended reading notes
Core claim
For the (∞,2)-category of cospans in an (∞,1)-category C, the (∞,1)-category of rigid commutative algebras is canonically identified with C. This identification is used to construct an adjunction between the cospan construction and the functor assigning to a symmetric monoidal (∞,2)-category its (∞,1)-category of rigid commutative algebras.
Load-bearing premise
The a priori (∞,2)-category of rigid algebras is in fact an (∞,1)-category and the cospan (∞,2)-category has the properties needed for the identification to hold.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces rigid algebras as a generalization of rigid categories to arbitrary symmetric monoidal (∞,2)-categories. It develops their general theory, proving in particular that the a priori (∞,2)-category of rigid algebras is in fact an (∞,1)-category. For the (∞,2)-category of cospans in an (∞,1)-category C, the (∞,1)-category of rigid commutative algebras is canonically identified with C. This identification is used to construct an adjunction between the cospan construction and the functor assigning to a symmetric monoidal (∞,2)-category its (∞,1)-category of rigid commutative algebras.
Significance. If the results hold, the work supplies a structural identification between cospans and rigid commutative algebras that yields an adjunction relating the cospan functor to the rigid-algebra functor. The reduction of the rigid-algebra (∞,2)-category to an (∞,1)-category is a noteworthy coherence result. These constructions may prove useful in higher-categorical approaches to spans, cospans, and algebraic structures in homotopy theory and derived geometry.
minor comments (2)
- The abstract states the main theorems without numbering; adding theorem labels (e.g., Theorem 4.12 for the identification) would improve cross-referencing once the body is read.
- Notation for the (∞,2)-category of cospans and the rigid-algebra functor should be introduced with explicit symbols in the introduction to aid readability.
Simulated Author's Rebuttal
We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No major comments were provided in the report.
Circularity Check
No significant circularity; results presented as derived theorems
full rationale
The abstract states the core claims as theorems: the (∞,2)-category of rigid algebras reduces to an (∞,1)-category, and for cospans the rigid commutative algebras are canonically identified with C. These are presented as proved statements rather than definitional equivalences or fitted inputs. No self-citations, ansatzes smuggled via prior work, or uniqueness theorems imported from the same author appear in the provided text. The adjunction construction is described as using the identification, not presupposing it. The derivation chain is therefore self-contained against external benchmarks with no load-bearing step reducing to its own inputs by construction.
Assumptions & free parameters
Cite this review
Pith. "Pith review of Rigid Algebras and Cospans." pith.science (2026). https://pith.science/paper/MOT5OSF2
@misc{pith2026250622072,
author = {Pith},
title = {Pith review of: Rigid Algebras and Cospans},
year = {2026},
howpublished = {\url{https://pith.science/paper/MOT5OSF2}},
note = {Machine review of arXiv:2506.22072}
}
abstract
We introduce rigid algebras, a generalization of rigid categories to arbitrary symmetric monoidal $(\infty,2)$-categories. We develop their general theory, showing in particular that the a priori $(\infty,2)$-category of rigid algebras is in fact an $(\infty,1)$-category. For the $(\infty,2)$-category of cospans in an $(\infty,1)$-category $\mathcal{C}$, we show that the $(\infty,1)$-category of rigid commutative algebras is canonically identified with $\mathcal{C}$. This identification is used to construct an adjunction between the cospan construction and the functor assigning to a symmetric monoidal $(\infty,2)$-category its $(\infty,1)$-category of rigid commutative algebras.
Reviewed May 25, 2026 · model on record in the stance chip above.
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