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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?


    • 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 AA” (with axioms to be specified) we could say “a Lie bracket on B n1A\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 nE_n-algebras, and just observe that their “classical limit” given by decategorifying by passing to homology is a Poisson nn-algebra.

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

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