nLab extended cobordism

Idea

In so far as a cobordism connects its boundary components with each other, the idea of an extended cobordism is that its (pieces of) boundary components are themselves extended cobordisms between their (pieces of) boundary components.

The idea arose in the context of extended quantum field theories – which were originally thought of as representations of ordinary cobordism categories – when it was realized that a decent description of these QFTs requires assigning data to pieces of arbitrary codimension to a manifold. In the context of QFT this is often thought of as an incarnation of the notion of locality in physics, which says that every data assigned to a chunk of space must already be fixed by what is assigned to all pieces of any of its decomopositions.

Indeed, the idea of extended cobordisms led to the generalized tangle hypothesis which takes this localization concept to its extreme by asserting, roughly, that the representation of fully extended cobordisms is already entirely determined by what is assigned to the point.

Formalizations

For a long time only partial progress was made with formalizing the idea of extended cobordisms, which are expected to form an infinity-category or omega-category of sorts. One approach using the notion of Trimble n-category as

• Eugenia Cheng and Nick Gurski, Towards an $n$-category of cobordisms (tac)

Another formalization is by Marco Grandis, using multi-cospans, see

• Cospans in Algebraic Topology

This is in principle very general, indicating that a general extended cobordisms should be a multi-cospan in Top. But details are worked out only in dimension up to 2.

In this dimension the resulting structure is closely related to the bicategory of cobordisms described in

• Jeffrey Morton, A Double Bicategory of Cobordisms With Corners (arXiv)

More recently

• Jacob Lurie has presented a complete formalization in terms of an (infinity,n)-category of cobordisms.

This generalizes an idea which for $n=2$ was for instance also applied by Chris Schommer-Pries, see

• Schommer Pries on Classification of 2-Dimensional Extended TFTs