Schreiber disk-shaped cobordism

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

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

Notice that

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

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

Whether these disk shaped cobordism objects are modeled

is a question of technical implementation and not of principle.

Last revised on August 22, 2009 at 11:18:13. See the history of this page for a list of all contributions to it.