---
title: "The Lie group-Lie algebra correspondence in tangent categories"
authors: ["Marcello Lanfranchi"]
year: 2026
venue: "arXiv:math.CT"
arxiv: "2609.03449"
doi: null
source: "https://arxiv.org/abs/2609.03449"
topics: ["curiosity", "maths-foundations"]
seed_rank: 1582
seed_batch: "curiosity-2026-09-05"
reviewed: "2026-09-05"
pool: "maths-foundations"
relevance_score: 10
lineage: tangent-categories
cites:
  - title: "The Lie group-Lie algebra correspondence in tangent categories"
    url: "https://arxiv.org/abs/2609.03449"
    year: 2026
    arxiv: "2609.03449"
    doi: null
  - title: "Reverse derivative categories"
    url: "https://arxiv.org/abs/1910.07065"
    year: 2020
    arxiv: "1910.07065"
    doi: "10.4230/LIPIcs.CSL.2020.18"
see:
  - "1056-reverse-derivative-categories"
---

# The Lie group-Lie algebra correspondence in tangent categories

## One-sentence takeaway

Lie theory rebuilt from tangent structure alone — manifolds and schemes as instances of one categorical machine.

## Why it's lovely

Why you might love this: the Lie correspondence is no longer a manifold theorem; it is a tangent-category theorem. Classic Lie theory matches groups in smooth manifolds to Lie algebras, and an analogous dictionary exists for group objects in affine schemes. Both settings are tangent categories, so Lanfranchi asks whether the whole correspondence can be constructed from the tangent structure and nothing else. Pure categorical differential geometry — the same shelf as reverse-derivative categories (1056).

## Key ideas

- Lie group objects: group objects in a tangent category that admit a tangent space at the unit.
- A Lie functor on any group object; it is tangentially representable exactly when the group is a Lie group.
- Representability yields an internal Lie algebra whose underlying object is the tangent space at the unit.
- That algebra is a differential Lie algebra, with bilinear bracket from the adjoint representation; the construction extends to a functor Lie groups → differential Lie algebras.
- Recovers the usual correspondences in both differential and algebraic geometry (building on Cockett–Schwarz).

## Caveats

Internal to tangent categories: you still need the tangent structure (not a free lunch from mere finite products). Recovery of classical Lie theory is an instance theorem, not a replacement for the analytic existence/uniqueness package on manifolds.

## Implications for Broadside

Bridges the reverse-derivative / tangent-category shelf (1056). Manifolds and schemes become two models of one machine — the same spirit as compiling differentiation to a categorical combinator rather than a chart-by-chart construction.

## Links

- arXiv abs: https://arxiv.org/abs/2609.03449
- PDF: https://arxiv.org/pdf/2609.03449.pdf