The formal theory of tangentads PART I
Marcello Lanfranchi
TL;DR
This work develops a comprehensive, model‑independent theory of vector fields within tangentads, a 2‑categorical framework for tangent structures. It identifies a universal property for vector fields, proves that VF(X,T) forms a Lie algebra and a $2$-monad, and constructs vector fields via PIE limits, with explicit liftings to tangent monads, tangent fibrations, tangent indexed categories, and tangent restriction categories. The results provide a unified, 2‑categorical toolkit for transporting differential-geometric constructions across diverse tangent flavours, and lay groundwork for extending to differential objects, differential bundles, and connections. By establishing corepresentability and functoriality of the VF construction, the paper offers a robust, model‑independent mechanism to study vector fields in broad categorical contexts and to relate tangent‑categorical notions through structured limits and extensions.
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.