  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.
