Title: The Lie Algebra of a group object
Abstract: At a previous presentation Geoff Cruttwell asked me, if the vertical bundle of a principal bundle in a join restriction category is trivial. This was not just an interesting question on its own, it also pointed towards more going on with group objects in tangent categories.
In differential geometry, given a Lie-group G, its tangent space TuG is known as its Lie-Algebra. Generalizing this perspective, I will prove that the tangent bundle of a group object in a tangent category is trivial (isomorphic to a product). Then I will continue discussing how the addition from the tangent bundle structure interacts with the group multiplication and show that there is a negation, even if the tangent category was not required to have negatives. From this I will conclude that there is a Lie-Algebra structure on the elements of the tangent space.