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

Florian Schwarz

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.

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

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.
Paper Structure (20 sections, 38 theorems, 96 equations, 2 figures, 1 table)

This paper contains 20 sections, 38 theorems, 96 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Tangent categories
  3. Categorical dimensions of tangent bundles
  4. Monoid valued dimensions
  5. The tangent structure in the presence of a dimension
  6. Differential bundles in the presence of a dimension
  7. Cardinality as a non-trivial tangent structure on $\mathbb S \space \operatorname{et}^\mathrm{op}$
  8. Cardinality of groups as a dimension valued in a non-standard monoid
  9. Characteristic, Krull dimension and the lowest common multiple
  10. Characteristic as a dimension valued in the least common multiple monoid
  11. Krull dimension satisfying products but not pullbacks
  12. Transporting a dimension between modules and algebras with an adjunction
  13. Rank of a module over a ring
  14. Module rank of an Algebra
  15. Classification of Cartesian tangent structures on $\mathbb F\mathrm{inVect}$
  16. ...and 5 more sections

Key Result

Proposition 3.3

Let $\mathbb X$ and $\mathbb Y$ be categories, let $\mathbb X$ have an $R$-valued dimension $\dim_\mathbb X: \pi_0(\mathbb X) \to R$. Let $F: \mathbb Y \to \mathbb X$ be a functor that preserves finite limits. Then the assignment that sends $Y \in \pi_0(\mathbb Y)$ to is an $R$-valued dimension on $\mathbb Y$.

Figures (2)

  • Figure 1: The pullback of the interval with a 3-cell at the left and the interval with a 3-cell at the right is an interval with a 3-cell on either side.
  • Figure 2: On the left is the pushout of $A$ and $B$ from Example \ref{['ex:hausdorff_cylinder_not_pushout']}, on the right the double mapping cylinder $_AM_B$. They are not homotopy-equivalent. The intuitive reason is that every open neighbourhood around the blue point on the left contains infinitely many dots and circles, whereas on the right there is no such point.

Theorems & Definitions (104)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • Definition 2.6
  • Remark 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • ...and 94 more