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Jon Beardsley
Jon Beardsley
02:18
Wonder if I'm the only homotopy theorist in West Virginia right now...
 
6 hours later…
Sean Tilson
Sean Tilson
08:11
@AaronMazel-Gee Yes, this is the type of thing that I mean. It is not a homology theory for $E_n$-algebras.
I did not mean to imply that it is not a bifunctor.
I was misremembering, the functoriality is that "it isn't sheafy on the manifold but sheafy on the Ran space of the manifold."
 
7 hours later…
Rune Haugseng
Rune Haugseng
15:32
@SeanTilson The analogue of TAQ for $E_n$-algebras would be the $E_n$-cotangent complex. This is related to the $S^{n-1}$-factorization homology via a fibre sequence of John Francis.
Sean Tilson
Sean Tilson
16:25
@RuneHaugseng: Yes, I am aware of this. Basterra-Mandell and Fresse prove the same result in the setting of EKMM spectra and chain complexes (respectively). I alluded to the fact you mention above (keep hitting the reply buttons to find the thread). ;)
 
5 hours later…
Rune Haugseng
Rune Haugseng
21:34
Basterra-Mandell consider $E_n$-algebras? I was definitely not aware of that. You seem to be pointing out that there is no proper functoriality for factorization homology in the algebra variable. Functoriality for extended TQFTs in general is a bit tricky, since there are no non-invertible natural transformations between them.
So you have to consider either TQFTs with singularities or lax natural transformations (I'm not sure what the expected relationship between these is) to get a notion of non-invertible morphism between TQFTs. But a bimodule between $E_n$-algebras should then determine such a 1-morphism, etc.
Not that I'm claiming anything like this would make it a "cohomology theory".

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