The formal theory of tangentads PART I

Abstract

Tangent categories offer a categorical context for differential geometry, by categorifying geometric notions like the tangent bundle functor, vector fields, Euclidean spaces, vector bundles, connections, etc. In the last decade, the theory has been extended in new directions, providing concepts such as tangent monads, tangent fibrations, tangent restriction categories, reverse tangent categories and many more. It is natural to wonder how these new flavours interact with the geometric constructions offered by the theory. How does a tangent monad or a tangent fibration lift to the tangent category of vector fields of a tangent category? What is the correct notion of vector bundles for a tangent restriction category? We answer these questions by adopting the formal approach of tangentads. Introduced in our previous work, tangentads provide a unifying context for capturing the different flavours of the theory and for extending constructions like the Grothendieck construction or the equivalence between split restriction categories and $\mathscr{M}$-categories, to the tangent-categorical context. In this paper, we construct the formal notion of vector fields for tangentads, by isolating the correct universal property enjoyed by vector fields in ordinary tangent categories. We show that vector fields form a Lie algebra and a $2$-monad and show how to construct vector fields using PIE limits. Finally, we compute vector fields for some examples of tangentads. In a forthcoming paper, we extend the theory to other constructions: differential objects, differential bundles, and connections.