---
title: "Cohomology for solutions of polygon equations"
authors: ["Serban Matei Mihalache", "Tomoro Mochida"]
year: 2026
venue: "arXiv:math-ph"
arxiv: "2608.17263"
doi: null
source: "https://arxiv.org/abs/2608.17263"
topics: [curiosity, category-theory-plt]
seed_rank: 1454
seed_batch: "curiosity-2026-08-29"
reviewed: "2026-08-29"
pool: "maths-foundations"
relevance_score: 9
lineage: higher-categories
cites:
  - title: "Higher Segal spaces I"
    url: "https://arxiv.org/abs/1212.3563"
    year: 2012
    arxiv: "1212.3563"
    doi: null
  - title: "Quasi-categories vs Segal spaces"
    url: "https://arxiv.org/abs/math/0607820"
    year: 2006
    arxiv: "math/0607820"
    doi: null
see:
  - "1239-quasi-categories-vs-segal-spaces"
  - "1373-the-infinity-category-of-infinity-categories-in-simplicial-type-theory"
---

# Cohomology for solutions of polygon equations

## One-sentence takeaway

Set-theoretic solutions of polygon equations (pentagon and kin) are equivalent to certain higher Segal semi-simplicial sets, with a matching (co)homology theory of permitted colorings.

## Why it's lovely

Why you might love this: the pentagon equation grows a whole family, and their solutions are literally higher Segal spaces — associativity as combinatorial cohomology.

## Problem

Polygon equations generalize the pentagon equation; set-theoretic solutions lacked a uniform homological home and a clean dictionary to higher Segal geometry.

## Design

- Build semi-simplicial sets of permitted colorings from solutions; define (co)homology.
- Prove equivalence of categories: set-theoretic polygon solutions ↔ higher Segal semi-simplicial sets (with stated conditions).
- Recover Dyckerhoff–Kapranov: bijective pentagon solutions ↔ 2-Segal spaces.

## Evidence

21 pages, 2 figures; special-case recovery of the known 2-Segal correspondence.

## Limitations

Set-theoretic solutions only; continuous/quantum polygon equations untouched. Not a new cluster-algebra construction (see 1371).

## Implications for Broadside

Bridge from combinatorial associativity into the Segal/∞-category shelf already on Broadside (1239, 1373).

## Bottom line

Mint polygon cohomology. Associativity equations wear higher Segal clothes.

## Links

- arXiv abs: https://arxiv.org/abs/2608.17263
- PDF: https://arxiv.org/pdf/2608.17263.pdf