The Algebroid of a Groupoid in a Tangent Category

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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.