Crossed-module crossed braided categories

@misc{gainutdinov2026crossedmodule,
  title = {{Crossed-module crossed braided categories}},
  author = {Azat M. Gainutdinov and Ingo Runkel and Bangxin Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07626},
  eprint = {2609.07626},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07626},
}

TY  - UNPB
TI  - Crossed-module crossed braided categories
AU  - Azat M. Gainutdinov
AU  - Ingo Runkel
AU  - Bangxin Wang
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07626
UR  - https://arxiv.org/abs/2609.07626
AB  - For a crossed module χ: G → H, we introduce the notion of χ-crossed braided (resp. ribbon) categories, where the categories are graded by group G and carry an H-action. Our definition unifies and generalises several familiar notions: taking χ= id: G → G with the conjugation action recovers G-crossed braided categories; taking χ: G → {*} for abelian G yields G-graded braided categories; taking χ: {*} → G leads to braided categories equipped with a G-action. The equivalence relation between χ-crossed braided categories is typically finer than that between G-crossed braided ones. We classify χ-crossed braided structures on the category of G-graded vector spaces in terms of cohomological data, and give explicit examples for cyclic groups. Given a doubly central algebra with G- and H-actions in a braided monoidal category, we define a notion of twisted-local modules and show how they give rise to χ-crossed braided categories. We furthermore give sufficient conditions so that these categories are additionally χ-crossed ribbon or admit an orthogonal G-decomposition.
ER  -