Tangent bundles, monoidal theories, and Weil algebras

The construction of the tangent bundle of a manifold lies at the very foundations of differential geometry. There are various approaches to characterise the tangent bundle, and two such approaches are through Synthetic Differential Geometry (SDG) and Tangent Structures (in the sense of Cockett-Cruttwell). Here, we shall give a different perspective, that Tangent Structure can be viewed as a model of an appropriate theory. This theory arises as a certain full subcategory Weil1 of the category Weil of all Weil algebras. The connection between Weil algebras and SDG is well established, but their connection to Tangent Structure is not evident. In this thesis, we shall exhibit Weil1 as the universal tangent structure and in fact the axioms of tangent structure actually form a presentation for Weil1. We shall then continue by describing how this perspective allows us to extend this theory in a canonical manner.