The Algebroid of a Groupoid in a Tangent Category

TL;DR

This paper extends the concept of Lie algebroids from Lie groupoids to a broader setting of tangent categories, providing a categorical framework that generalizes classical differential geometry results.

Contribution

It introduces a categorical construction of algebroids in tangent categories, generalizing classical Lie theory to a more abstract, algebraic setting.

Findings

Established a bijection between invariant vector fields and tangent vectors in tangent categories.

Constructed the algebroid of a groupoid within any tangent category.

Extended classical splitting results of tangent bundles to pregroupoids in this framework.

Abstract

We generalise the construction of the Lie algebroid of a Lie groupoid so that it can be carried out in any tangent category. First we reconstruct the bijection between left invariant vector fields and source constant tangent vectors based at an identity element for a groupoid in a category equipped with an endofunctor that has a retraction onto the identity functor. Second we use the full structure of a tangent category to construct the algebroid of a groupoid. Finally we show how the classical result concerning the splitting of the tangent bundle of a Lie group can be carried out for any pregroupoid.