The formal theory of tangentads PART II
Marcello Lanfranchi
TL;DR
The paper develops a unified, higher-categorical theory for differential-geometric structures in tangent categories by embedding differential objects, differential bundles, and connections into the tangentad framework. It establishes robust universal properties, demonstrates 2-functoriality across Cartesian tangentads, and analyzes interactions with existing tangent notions such as tangent fibrations, tangent monads, and tangent restriction categories. A core contribution is showing how linear and affine connections, along with covariant derivatives, curvature, torsion, and linear sections, can be constructed via PIE limits and related universal objects. The framework yields corepresentable, universal characterizations that apply across a wide range of tangent-adapted settings, with concrete instantiations and applications to multiple tangentads. Overall, it provides a modular, abstract toolkit for differential-geometry-style structures in categorical contexts, enabling systematic transfer of geometric intuition to generalized environments.
Abstract
Tangent category theory is a well-established categorical framework for differential geometry. A long list of fundamental geometric constructions, such as the tangent bundle functor, vector fields, Euclidean spaces, and vector bundles have been successfully generalized and internalized within tangent categories. Over the past decade, the theory has also been extended in several directions, yielding concepts such as tangent monads, tangent fibrations, tangent restriction categories, and reverse tangent categories. It is natural to wonder how these new flavours of the theory interact with the geometric constructions. How does a tangent monad or a tangent fibration lift to the tangent category of differential bundles of a tangent category? What is the correct notion of connections for a tangent restriction category? In previous work, we introduced tangentads, a unifying framework that generalizes many tangent-like notions, and developed a formal theory of vector fields for tangentads. In this paper, we extend this formal theory to three further fundamental constructions. These are differential objects, which generalize Euclidean spaces, differential bundles, which represent vector bundles in tangent category theory, and connections on differential bundles, which are the analogue of Koszul connections. These notions are introduced in the general theory of tangentads via appropriate universal properties. We then extend some of the main results of tangent category theory, including the equivalence between differential objects and differential bundles over the terminal object, and show that connections admit well-defined notions of covariant derivative, curvature, and torsion. Finally, we construct connections using PIE limits and apply our framework to several concrete instances of tangentads.