---
title: "Graded-fusion 2-categories and quantum homotopy invariants of 4-manifolds"
authors:
  - "Matthew Cellot"
year: 2026
venue: "arXiv:math.GT"
arxiv: "2608.20959"
doi: null
source: "https://arxiv.org/abs/2608.20959"
topics:
  - "curiosity"
  - "maths-foundations"
seed_rank: 1646
seed_batch: "curiosity-2026-09-09"
reviewed: "2026-09-09"
pool: "maths-foundations"
relevance_score: 10
lineage: fusion-tqft
cites:
  - title: "Graded-fusion 2-categories and quantum homotopy invariants of 4-manifolds"
    url: "https://arxiv.org/abs/2608.20959"
    year: 2026
    arxiv: "2608.20959"
    doi: null
see:
  - "1617-3-dimensional-tqfts-from-derived-categories-of-quantum-group-represent"
---

# Graded-fusion 2-categories and quantum homotopy invariants of 4-manifolds

## One-sentence takeaway

3-group-graded extensions of fusion 2-categories yield state-sum invariants of 4-manifolds with maps to a homotopy 3-type.

## Why it's lovely

Why you might love this: fusion categories already power 3D TQFTs; here the 2-categorical lift gets graded by a 3-group (modeled as a 2-crossed module) and pays out quantum-homotopy invariants of 4-manifolds. Pure "higher categories as physics machinery" delight, complementary to the derived-MTC 3D TQFT card (1617).

## Key ideas

- 3-group-graded extensions of fusion 2-categories via 2-crossed modules.
- State-sum invariants of 4-manifolds equipped with a map to a homotopy 3-type.
- Equivalently: invariants of flat 3-bundles over 4-manifolds.
- Lifts the Reshetikhin–Turaev / Turaev–Viro story one dimension.

## Caveats

Heavy higher-categorical machinery; computational examples may be sparse. Pair with 1617 (3D TQFTs from derived quantum-group categories) rather than treating as a drop-in engine technique.

## Links

- arXiv abs: https://arxiv.org/abs/2608.20959
- PDF: https://arxiv.org/pdf/2608.20959.pdf