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Hey everyone,
recently I came across the nLab entry on the general definition of the Hochschild complex. Personally I really like the example around “Tensoring with the simplicial circle”. Simple, straight forward and illuminating. (In particular since $\Delta[1] / \partial\Delta[1]$ is such a minimalistic model for the circle.)
http://ncatlab.org/nlab/show/Hochschild+cohomology#Pirashvili OK.
From Delignes “conjecture” we know, that the traditional Hochschild complex has an action of the chains of the little squares operad, say $E_2$. Can we make this action apparent in this simple model?
I mean maybe there is an action of the simplicial singular chains of $E_2$ on the simplicial circle, that leads to Delignes action on the classical Hochschild complex?
Even if it is not that simple, does anyone know if/how this appears in the previously mentioned simple model?
Maybe I’m shooting for the moon, when I ope for such an easy proof of Delignes “conjecture”.
Edit: Looks like I have some typesetting problems.
Notice that this is one of the big gains of understanding Hochschild (co-)homology in terms of funtions on derived loop spaces – and more generally on mapping spaces out of spheres etc: on these mapping spaces the corresponding $n$-disk operads canonically act, and hence so they do on the corresponding function algebras, by pullback.
That’s the content of this proposition, highlighted also at Deligne conjecture – Geometric interpretation.
This makes the statement “tautological” when speaking in a general abstract homotopy theoretic context.
The simplicity of what this comes down to when unwinding all structures and actions in terms of simplicial sets depends on ingenuity and on individual perception of “simplicity”.
“The simplicity of what this comes down to when unwinding all structures and actions in terms of simplicial sets depends on ingenuity and on individual perception of “simplicity”.”
So you say, that the prolog is the actual story! … No seriously, it means that you don’t know if there is any public place where the general abstract was unwinded to the particular situation I mentioned above?
That’s right, I don’t. But I wouldn’t be the one to have a good enough overview of this literature to say it’s not there.
Looks like the authors of “HIGHER HOCHSCHILD COHOMOLOGY, BRANE TOPOLOGY AND CENTRALIZERS” are a safe bet to ask…
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