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created Poisson n-algebra
it seems the natural setting for Poisson n-algebras are symmetric monoidal stable (oo,1)-categories. should we attemp a definition in that direction?
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.
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.
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.
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