# Involution algebroids: a generalisation of Lie algebroids for tangent categories

## TL;DR

This paper introduces involution algebroids as a broad generalization of Lie algebroids within tangent categories, replacing the Jacobi identity with a Yang-Baxter-like condition, and explores their foundational properties and homotopy theory.

## Contribution

It defines involution algebroids in tangent categories, generalizes Lie algebroids, and initiates their homotopy theory development.

## Key findings

- Classical Lie algebroids are special cases of involution algebroids.
- Involution algebroids admit a Lie bracket on sections.
- The Jacobi identity is replaced by a Yang-Baxter-like condition.

## Abstract

We define involution algebroids which generalise Lie algebroids to the abstract setting of tangent categories. As a part of this generalisation the Jacobi identity which appears in classical Lie theory is replaced by an identity similar to the Yang-Baxter equation. Every classical Lie algebroid has the structure of an involution algebroid and every involution algebroid in a tangent category admits a Lie bracket on the sections of its underlying bundle. As an illustrative application we take the first steps in developing the homotopy theory of involution algebroids.