The dimension of the tangent bundle and the universality of the vertical lift

TL;DR

This paper introduces a categorification of the dimension of smooth manifolds within tangent categories, revealing constraints on tangent bundle dimensions and implications for tangent structures, including the absence of non-trivial structures on sets.

Contribution

It presents a novel categorical dimension concept for tangent categories and establishes key theorems relating tangent bundle dimensions to base dimensions.

Findings

Dimension of tangent bundles relates to base dimension via specific equations.

Strong tangent dimension implies tangent bundle dimension is either twice or equal to base.

No non-trivial tangent structures exist on sets.

Abstract

This paper explores a new perspective on the universality of the vertical lift in tangent categories by presenting a categorification of the dimension of smooth manifolds. The universality of the vertical lift is a key part of the axioms of a tangent category as presented in [4]. The categorical dimension presented in this paper provides insight into the nature of this property. The main result is Theorem 3.7, showing that if it exists, the dimension of the tangent bundle must fulfill an equation relating the dimension of the tangent bundle to the dimension of the base. In particular, when the dimension function is a strong tangent dimension, Theorem 3.8 shows that the dimension of the tangent bundles is either twice the dimension of the base, or equal to the dimension of the base. Many examples of dimension functions are provided to demonstrate the utility of the definition. In particular, a consequence of Theorem 3.7 is that there are limitations on which functors may be tangent bundle endofunctors for a category. We show that this means that there are no non-trivial tangent structures on sets, as an example.