The n-Category Café

Grandis on Collared Cobordisms and TQFT

Posted by Urs Schreiber

Marco Grandis has a new preprint

Marco Grandis
Collared cospans, cohomotopy and TQFT (Cospans in Algebraic Topology, II)
Dip. Mat. Univ. Genova, Preprint 555 (2007).
(pdf,ps)

As we know, quantum field theory is the study of representations of cobordism categories.

In most cases, in order to set up decent categories of cobordisms one needs to equip these with collars. A collar on a cobordism is certain extra data associated with its boundary which helps to avoid pathological gluing of cobordisms along their boundary components.

Grandis sets up the theory of cobordisms in full generality by considering categories of cospans.

A cospan $$\array{ && \Sigma \\ &\nearrow && \nwarrow \\ \partial_{\mathrm{in}}\Sigma &&&& \partial_{\mathrm{out}}\Sigma }$$ models a cobordism $\Sigma$ with its “incoming” and “outgoing” boundary components injected into it. (This way of putting it is much less general that what Grandis actually considers).

These cospans are naturally composed using pushouts, which models the idea of gluing along common boundary components. Grandis proposes a way how to incorporate the notion of a collared cobordism into his category-theoretic setup. Then he uses this to study 2-dimensional topological field theory.

Cobordism cospans, as well as their dual spans of configuration spaces of maps from these into some “target space”, have been discussed at the $n$-Café previously for instance in Slides for Freed’s Andrejewski Lecture or in QFT of charged $n$-Particle: Dynamics.

In Hopkins Lecture on TFT: Infinity-Category Definition I had begun listing some relevant literature, like

Cheng/Gurski: Towards an $n$-category of cobordisms

and

Jeffrey Morton: A double bicategory of cobordisms with corners.

So the gods will, a more detailed summary might appear here at some point…