## Koszul duality for n-algebras

Below there is a construction that from any augmented dg algebra over the operad of chains on framed little -discs produces a coalgebra over the same operad.

This construction generalizes the Koszul duality, for this is the usual Koszul duality for algebras.

Another case of this duality is from topology. Take a topological space . Then is a module over framed little -discs. The union of symmetric powers of is a comodule over the (trivial) operad and thus over framed discs (I am grateful to Victor Turchin for this observation). (Co)chains of this (co)modules over operads are dual to each other in our sence. This explains why one may calculate homology of mapping spaces from a -manifold to either by means of the generalization of the higher Hochschild homology or the spherical homology.

It seems that it is connected with work in progress of M. Ching and P. Salvatore “Geometric self-duality for the little discs operads” (I found it on the homepage of Michael Ching).

**UPD:** This Koszul duality is mentioned in the draft of J. Lurie’s “Higher algebra”(see e.g. example 7.3.6.7). Unfortunately, details are postponed for future work. See also “Moduli Problems for Ring Spectra”.

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