---
title: "A two-categorical Snake Lemma"
authors:
  - "Elena Caviglia"
  - "Luca Mesiti"
  - "Tim Van der Linden"
year: 2026
venue: "arXiv:math.CT"
arxiv: "2609.06428"
doi: null
source: "https://arxiv.org/abs/2609.06428"
topics:
  - "curiosity"
  - "maths-foundations"
seed_rank: 1782
seed_batch: "curiosity-2026-09-11"
reviewed: "2026-09-11"
pool: "maths-foundations"
relevance_score: 9
lineage: 2-homological-algebra
cites:
  - title: "A two-categorical Snake Lemma"
    url: "https://arxiv.org/abs/2609.06428"
    year: 2026
    arxiv: "2609.06428"
    doi: null
see:
  - "1616-koszul-duality-and-morita-categories"
---

# A two-categorical Snake Lemma

## One-sentence takeaway

The Snake Lemma lifts to 2-categories: 2-kernels/cokernels, a 2-dimensional Normal Short Five Lemma, and a characterisation of homologically self-dual 2-categories.

## Why it's lovely

Why you might love this: the Snake Lemma is the emblematic diagram chase of homological algebra; watching it survive — carefully — in a 2-category with bizero objects is pure foundational delight. Dinversion, Pure Snake, and Third Isomorphism as equivalent characterisations of homological self-duality is a tidy package.

## Key ideas

- 2-kernels/cokernels; 2-monos/epis as fully faithful / cofully faithful 1-cells.
- Short 2-exact sequences and normal image factorisations.
- Two-dimensional Normal Short Five Lemma.
- Dinversion of an antinormal pair; homologically self-dual 2-categories.
- Equivalence of: homology self-duality ↔ Pure Snake Lemma ↔ Third Isomorphism Property.

## Caveats

2-categorical foundations; concrete examples (e.g. Cat, abelian categories as a 2-category) need care with size and enrichment. Companion to exactness-of-2-categories work in the same orbit.

## Links

- arXiv abs: https://arxiv.org/abs/2609.06428
- PDF: https://arxiv.org/pdf/2609.06428.pdf