Involution algebroids: a generalisation of Lie algebroids for tangent categories

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.