The formal theory of tangentads PART II

TL;DR

This paper extends the formal theory of tangentads within tangent category theory to include differential objects, bundles, and connections, providing universal properties and concrete constructions, thereby enriching the categorical framework for differential geometry.

Contribution

It introduces a formal theory of connections on differential bundles within tangentads, generalizing geometric notions and establishing their universal properties in tangent category theory.

Findings

Connections admit covariant derivatives, curvature, and torsion.

The equivalence between differential objects and bundles is extended.

Connections can be constructed via PIE limits.

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.