Bubble-Free Criterion for Polygraphic and Nerve Homology

Definitions and mathematical statement

A two-dimensional polygraph, in the usual strict categorical sense, consists of a set $P_0$ of objects, a set $P_1$ of generating arrows with specified sources and targets in $P_0$, and a set $P_2$ of generating two-cells whose sources and targets are parallel finite paths in the graph $P_1\rightrightarrows P_0$. Empty paths are allowed. Write $P^*$ for the free strict two-category generated by this data. The supplied statement imposes no finiteness, connectivity, or termination hypotheses.

The intended conjecture concerns every two-dimensional polygraph $P$: its polygraphic homology coincides with the homology of its nerve if and only if $P$ is bubble-free. To express exactly the logical target without inventing missing conventions, let $\operatorname{BF}(P)$ denote the source's bubble-free condition and let $\operatorname{Agree}(P)$ denote the source's assertion that these two homology theories coincide. The supplied target is \[ \forall P\in\{\text{two-dimensional polygraphs}\},\qquad \operatorname{Agree}(P)\Longleftrightarrow\operatorname{BF}(P). \] Both implications are part of the conjecture.

Unresolved formulation qualifications. The supplied record does not define a bubble or the bubble-free condition. It also does not specify the nerve construction, the homology coefficients, the precise definition of polygraphic homology, or whether coincidence means degreewise abstract isomorphism, a natural isomorphism, or an isomorphism induced by a designated comparison map. The conventional association of $P$ with $P^*$ above does not resolve which construction the source intends when it says the nerve of a polygraph. Consequently, $\operatorname{BF}$ and $\operatorname{Agree}$ are explicitly unresolved placeholders, and the displayed equivalence is a faithful schematic statement rather than a fully specified mathematical formulation. Their definitions must be recovered from the cited primary material before this draft can be accepted.

Short English statement

The supplied conjecture asks whether bubble-freeness characterizes exactly those two-dimensional polygraphs whose polygraphic homology agrees with their nerve homology. The record labels the problem open, but its definitions and comparison conventions are missing. This statement-only draft requires human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004845$, exact source problem code catalogue entry, title "Bubble-Free Criterion for Polygraphic and Nerve Homology." The record gives proposed year 2021 and does not identify a proposer.
• Supplied report citation: Homotopical Algebra and Higher Structures (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/46, pp. 2497--2560. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. Supplied literature-source label: "Oberwolfach Report 46." https://doi.org/10.4171/owr/2021/46.
• Supplied literature reference: Homology of strict omega-categories. https://arxiv.org/abs/2104.12662. The record supplies no authors or specific theorem, conjecture, or page locator for this reference.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open, identifies it as the bubble-free conjecture, and reports that the primary work states the conjecture without proving it generally. This is supplied dated curation, not independently verified literature assessment. The background explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry supplies statement status corrected_verified and reports that its clean statement was checked against the official Oberwolfach report. That verification claim is part of the supplied record and has not been independently verified here. The supplied original-statement field is a clipped table-of-contents passage rather than the conjecture itself; it does not establish attribution to any of the people named there.