Whitney categories and the Tangle Hypothesis

TL;DR

This paper introduces Whitney n-categories, a new framework for understanding stratified spaces and their relation to the Tangle Hypothesis, by connecting homotopy theory with categories with duals.

Contribution

It defines Whitney n-categories and shows how they relate to the Tangle Hypothesis and stratified spaces, providing a functorial construction of fundamental Whitney n-categories.

Findings

Whitney n-categories generalize categories with duals for stratified spaces.

Fundamental Whitney n-categories can be constructed from smooth stratified spaces.

The approach links homotopy theory of stratified spaces to categorical structures.

Abstract

We propose a new notion of `n-category with duals', which we call a Whitney n-category. There are two motivations. The first is that Baez and Dolan's Tangle Hypothesis is (almost) tautological when interpreted as a statement about Whitney categories. The second is that we can functorially construct `fundamental Whitney n-categories' from each smooth stratified space X. These are obtained by considering the homotopy theory of smooth maps into X which are transversal to all strata. This makes concrete another idea of Baez and Dolan's which is that a suitable version of homotopy theory for stratified spaces should allow one to generalise the relationship between spaces and groupoids to one between stratified spaces and categories with duals.