Florian Schwarz

Date: October 3, 2025
Time: 2:00 pm - 2:30 pm
Location: ICT 616

Title: The dimension of the tangent bundle and the universality of the vertical lift
Abstract: Tangent categories are a categorical generalization of the differentiation structure in the category of smooth manifolds. Part of the definition of a tangent category is a condition called the universality of the vertical lift. It is often the most difficult and hardest to check condition when showing that something is a tangent category.
We will define a notion of monoid valued dimension to give an intuition for the consequences of the universality of the vertical lift. Examples for such dimensions are the dimension of smooth manifolds, the cardinality of sets (in the opposite category) and the Betti numbers of CW complexes (in the opposite category).
Using this notion of dimension we will see that the universality of the vertical lift restricts the possibilities for tangent structures on a given category.