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A while back I had a discussion here with Domenico on how the framed cobordim $(\infty,n)$-category $Bord^{fr}_n(X)$ of cobordisns in a topological space $X$ should be essentially the free symmetric monoidal $(\infty,n)$-category on the fundamental $\infty$-groupoid of $X$.
This can be read as saying
Every flat $\infty$-parallel transport of fully dualizable objects has a unique $\infty$-holonomy.
(!)
Some helpful discussion with Chris Schommer-Pries tonight revealed that this is (unsurprisingly) already a special case of what Jacob Lurie proves. He proves it in more generality, which makes the statement easy to miss on casual reading. So I made it explicit now at cobordism hypothesis in the new section For cobordisms in a manifold.
I can flesh this out tomorrow. Am too tired now.
Hi Urs,
this is very good news. In view of this, sooner or later we should also look back at what we have been discussing on Topological Quantum Field Theories from Compact Lie Groups somewhere here on nForum.
better sooner than later..
Urs’ first post above can be rephrased as follows: TQFTs over a given oo-groupoid $\Pi$ are the same thing as representations of $\Pi$ on fully dualizable objects in an oo-symmetric category. in particular (fully extended) TQFTs over a manifold $X$ are the same thing of flat oo-bundles over $X$. now, it would be interesting to have a push-forwarding of flat oo-bundles along morphisms $X\to Y$. by the above equivalence this would push TQFTs over $X$ forward to TQFTs over $Y$.
Now, let us come to the case of a finite group $G$. Here the relevant oo-groupoid $\Pi$ is the delooping $\mathbf{B}G$, and a flat higher vector bundle over $\mathbf{B}G$ is the same thing as an higher representation of $G$. So we are in Dijkgraaf-Witten theory playground: a $U(1)$-valued 3-cocycle on $G$, i.e. a morphism $\mathbf{B}G\to \mathbf{B}^3U(1)$ induces a “1-dimensional” 3-representation of $G$, and so a TQFT over $\mathbf{B}G$. now we want to push this TQFT forward to a TQFT over the point. to do this we just have to push forward our 3-vector bundle from $\mathbf{B}G$ to the point, i.e. to take its global sections.
If $\Sigma$ is a manifold of dimension $\leq 3$ with a morphism to $\mathbf{B}G$, then we have an induced flat 3-bundle over $\Sigma$ and so a TQFT over $\Sigma$. pushing this construction along $\mathbf{H}(\Pi(\Sigma),\mathbf{B}G)\to *$ should give the higher vector space associated to $\Sigma$ by push-forwarding the original TQFT on $\mathbf{B}G$ to the point.
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