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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 :)
I’m removing this query:
+–{: .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.
=–
Isn’t decategorification of an $n$-truncated object just $(n-1)$-truncation?
Broadly that’s the right idea, but:
the term “truncation” is typically used for higher groupoids (only)
(de)categorification is meant to apply not so much to objects but to their theories.
Decategorification can also mean looping: the decategorification of a symmetric monoidal (∞,n)-category is the endomorphism symmetric monoidal (∞,n-1)-category of its monoidal unit.
Isn’t truncation the dual concept in this context? I.e. the truncation of an object is the universal way to map into truncated objects. So if we take sets to be the truncated objects of Cat, the truncation would be the set of connected components of the category, not the set of isomorphism classes.
At least, that’s what I would expect you to mean if you told me out of the blue you were going to truncate a category to make it a set.
Edit: I guess it’s not the dual either, since that would be the set-of-objects functor. I guess the issue here is that the example here is a mix of both directions: you’re colocalizing from Cat to Gpd and localizing from Gpd to Set, and the second half is a truncation.
In the infinity category case it would just be colocalization, turning an $(\infty, n)$-category to an $(\infty, n - 1)$-category.
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