Tangent categories provide an axiomatization for the notion of a tangent bundle. We will show that the category of groups has a tangent structure for which the tangent space at any point is the abelianization of the given group. This observation led us to study the properties of the abelianization procedure. We will introduce the notion of a linear projector, which captures the essential properties of the abelianization functor which allow us to build our tangent structure.
We then study the properties of the category of groups from which this abelianization arises. We recall the notion of a linear reflective subcategory, from which one can often build a linear projector. We describe a general abelianization procedure proposed by Borceux and Bourn, and which, for unital categories, yields a linear reflector, and in turn, a linear projector. To conclude, we review many examples of unital categories and their associated projectors and tangent structures, including the categories of pointed magmas, non-unital rings, Lie algebras, and cocommutative Hopf algebras.
This is joint work with JS Lemay and Tim Van der Linden.