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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeDec 3rd 2010
1. it seems the natural setting for Poisson n-algebras are symmetric monoidal stable (oo,1)-categories. should we attemp a definition in that direction?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeDec 5th 2010

it seems the natural setting for Poisson n-algebras are symmetric monoidal stable (oo,1)-categories.

That seems right. All these operad entries could do with a more intrinsic $\infty$-description.

should we attemp a definition in that direction?

Sure.

• CommentRowNumber4.
• CommentAuthordomenico_fiorenza
• CommentTimeDec 5th 2010
• (edited Dec 5th 2010)

fine. what suggested me that Poisson n-algebras should have a stable oo-category version is that instead of “degree 1-n bracket on $A$” (with axioms to be specified) we could say “a Lie bracket on $\mathbf B^{n-1} A$”.

the next step is to convince ourselves that the oo-versions are more natural than the truncated versions (well, we should be quite convinced of this, actually :) ), and concentrate our efforts on En-algebras (the entry little cubes operad is still stubby) rather than on Poisson n-algebras.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeDec 6th 2010

Yes, certainly we want $E_n$-algebras, and just observe that their “classical limit” given by decategorifying by passing to homology is a Poisson $n$-algebra.

I need to polish my notes at Hochschild cohomology on this point. There is some details missing where I describe the canonical $E_n$-action. I need to spend more time on that.