Schreiber disk-shaped cobordism

By a disk-shaped cobordism we mean a cobordism that – as a topological space – is a $k$-dimensional disk, for some $k \in \mathbb{N}$.

Whereas the collection of all cobordisms $\Sigma$ equipped with a map into some object $X$ arranges itself into the (∞,n)-category of cobordism $Bord(X)$, the collection of disk-shaped cobordisms equipped with maps to $X$ arranges itself to the path ∞-groupoid $\Pi(X)$ of $X$.

Notice that

• functors out of $\Pi(X)$ have the interpretation of (flat) parallel transport (“local system”) in $X$, as described at differential cohomology?

• whereas extension of such a functor to a functor on all of $Bord(X)$ amounts to enhancing the parallel transport by a notion of holonomy .

Whether these disk shaped cobordism objects are modeled

• as globes, as in [[BaSc](http://arxiv.org/abs/math/0511710), ScWaI ScWaII ScWaIII]

• as simplicies, as described at path ∞-groupoid

• or as cubes, as in [[MaPiI](http://arxiv.org/abs/0710.4310) MaPiII MaPiIII]

is a question of technical implementation and not of principle.