---
title: "Crossed-module crossed braided categories"
authors:
  - "Azat M. Gainutdinov"
  - "Ingo Runkel"
  - "Bangxin Wang"
year: 2026
venue: "arXiv:math.QA"
arxiv: "2609.07626"
doi: null
source: "https://arxiv.org/abs/2609.07626"
topics:
  - "curiosity"
  - "maths-foundations"
seed_rank: 1780
seed_batch: "curiosity-2026-09-11"
reviewed: "2026-09-11"
pool: "maths-foundations"
relevance_score: 9
lineage: crossed-braided-tqft
cites:
  - title: "Crossed-module crossed braided categories"
    url: "https://arxiv.org/abs/2609.07626"
    year: 2026
    arxiv: "2609.07626"
    doi: null
see:
  - "1646-graded-fusion-2-categories-quantum-homotopy-4-manifolds"
  - "1617-3-dimensional-tqfts-from-derived-categories-of-quantum-group-represent"
  - "1546-all-once-extended-3d-tqfts-are-reshetikhin-turaev-theories"
---

# Crossed-module crossed braided categories

## One-sentence takeaway

For a crossed module χ: G→H, χ-crossed braided (and ribbon) categories unify G-crossed braided, G-graded braided, and G-acted braided categories in one package.

## Why it's lovely

Why you might love this: crossed modules are already the algebraic avatar of 2-groups; giving them a braided/ribbon categorical home is the natural next move after G-crossed braided categories. One definition that specialises to three classical notions — and sits next to the fusion-2-category / TQFT shelf (1646, 1617).

## Key ideas

- χ-crossed braided/ribbon categories: G-graded with H-action for χ: G→H.
- χ = id recovers classical G-crossed braided categories.
- χ: G→{∗} (abelian G) recovers G-graded braided categories.
- χ: {∗}→G recovers braided categories with a G-action.
- Equivalence relation is typically finer than for ordinary G-crossed braided categories.

## Caveats

Definitional/foundational; concrete TQFT constructions from χ-crossed ribbon categories are future work. Pair with graded fusion 2-categories (1646).

## Links

- arXiv abs: https://arxiv.org/abs/2609.07626
- PDF: https://arxiv.org/pdf/2609.07626.pdf