Pullbacks in tangent categories and tangent display maps

TL;DR

This paper introduces tangent display maps in tangent categories, providing a new approach to handling pullbacks and simplifying the categorical framework of differential geometry, especially relating to smooth manifolds.

Contribution

It defines tangent display maps, shows their equivalence to submersions in smooth manifolds, and uses them to construct a canonical split restriction tangent category.

Findings

Tangent display maps are well-behaved with respect to pullbacks and tangent functor applications.

In smooth manifolds, tangent display maps coincide with submersions.

A canonical split restriction tangent category can be constructed using tangent display maps.

Abstract

In differential geometry, the existence of pullbacks is a delicate matter, since the category of smooth manifolds does not admit all of them. When pullbacks are required, often submersions are employed as an ideal class of maps which behaves well under this operation and the tangent bundle functor. This issue is reflected in tangent category theory, which aims to axiomatize the tangent bundle functor of differential geometry categorically. Key constructions such as connections, tangent fibrations, or reverse tangent categories require one to work with pullbacks preserved by the tangent bundle functor. In previous work, this issue has been left as a technicality and solved by introducing extra structure to carry around. This paper gives an alternative to this by focusing on a special class of maps in a tangent category called tangent display maps; such maps are well-behaved with respect to pullbacks and applications of the tangent functor. We develop some of the general theory of such maps, show how using them can simplify previous work in tangent categories, and show that in the tangent category of smooth manifolds, they are the same as the submersions. Finally, we consider a subclass of tangent display maps to define open subobjects in any tangent category, allowing one to build a canonical split restriction tangent category in which the original one naturally embeds.