The Lie algebra of a Lie group can be defined in at least two different flavours: as the algebra of left-invariant vector fields or, as the tangent space at the unit.
A similar correspondence appears in algebraic geometry for algebraic groups. So, one could wonder, is there a more systematic theory of Lie groups that generalize these two geometric contexts?
Recently, Cockett and Schwarz studied group objects in tangent categories, a categorical context for differential geometry. In their paper, they constructed the external Lie algebra of left-invariant vector fields (LIVF) of a group object. However, their paper does not introduce an internal Lie algebra object: their Lie algebra does not live within the tangent category itself.
In this talk, we fill this gap by introducing tangent Lie groups, which are group objects which carry a natural notion of internal Lie algebra object. Surprisingly, we prove that a group object is a tangent Lie group object if and only if it admits the tangent space at the unit. In fact, the internal Lie algebra corresponds precisely to this tangent space.