Pullbacks in tangent categories and tangent display maps
Geoffrey Cruttwell, Marcello Lanfranchi
TL;DR
The paper addresses the challenge that pullbacks are not guaranteed to exist or be preserved by the tangent bundle functor in general tangent categories. It introduces tangent display maps as the maximal, well-behaved class of morphisms compatible with pullbacks and all iterates of the tangent functor $T$, enabling a unified treatment of pullbacks across tangent-category constructions. A comprehensive theory is developed, including tangent display systems, their stability properties, Cauchy completion, and the equivalence of tangent display maps with submersions in the category of smooth manifolds; the framework is then leveraged to streamline differential bundles, tangent fibrations, slicing, and open-subobject constructions, and to illuminate open pathways for reverse tangent categories and restriction structures. The results offer a principled, scalable approach to handling pullbacks in differential-geometric-like settings within tangent categories, with practical implications for simplifying and unifying foundational structures such as connections, fibrations, and slice theories.
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.