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I thought it might be good if somebody explained the relationship between decategorification and extended TQFT. My understanding from talking to physicists is that you should multiply your space by $S^1$; is this right in a mathematical sense? I've added a query box asking roughly the same thing.
Also, I attempted to add a sidebar, mostly just to try it out, and somehow it's not rendering right. Anyone want to explain what I did wrong?
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Anyone want to explain what I did wrong?
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<p>I can't really explain it, all I can say is that the problem went away after I added closing sharp-signs not just before but also after the headlines. Not sure why that makes a difference here but not elsewhere. Probably the parser gets mixed up somhow by the inclusion process.</p>
<p>Then I slightly expanded and slightly restructures the toc. Just a suggestion.</p>
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I replied at decategorification. But probably one can say something better. As I said, I should really go to bed now :-)
replied to David Roberts at horizontal categorification
I knew you couldn't resist! Get some sleep :)
Iām removing this query:
+ā{: .query} Ben Webster: Perhaps something could be said about an extended TQFT $F$? My understanding was that the decategorification of $F(X)$ was given by $F(X\times S^1)$; is this right?
Urs Schreiber: that process certainly makes an $n$-dimensional QFT becomes an $(n-1)$-dimensional one. It is pretty much exactly the mechanism of fiber integration.
So this certainly does have a flavor of decategorification. But the latter also has a precise sense in terms of taking equivalence classes in a category. So at face value fiber integration is something different. But perhaps there is some change of perspective that allows to regard it as decategorification in the systematic sense.
=ā
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