  We introduce a discrete cobordism category for nested manifolds and nested
cobordisms between them. A variation of stratified Morse theory applies in this
case, and yields generators for a general nested cobordism category.
Restricting to a low-dimensional example of the ``striped cylinder'' cobordism
category Cyl, we give a complete set of relations for the generators. With an
eye towards the study of TQFTs defined on a nested cobordism category, we
describe functors Cyl$\to\mathcal{C}$, which we call Cyl-objects in
$\mathcal{C}$, and show that they are related to known algebraic structures
such as Temperley-Lieb algebras and cyclic objects. We moreover define novel
algebraic constructions inspired by the structure of Cyl-objects, namely a
doubling construction on cyclic objects analogous to edgewise subdivision, and
a cylindrical bar construction on self-dual objects in a monoidal category.
