# Nikita Markarian’s mathblog

## Koszul duality for n-algebras

Posted in Mathematics by nikitamarkarian on February 19, 2011

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

This construction generalizes the Koszul duality, for $n=1$ this is the usual Koszul duality for algebras.

Another case of this duality is from topology. Take a topological space $X$. Then $\Omega^n X$  is a module over framed little $n$-discs.  The union of symmetric powers of $X$ is a comodule over the (trivial) operad $Com$ 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 $n$-manifold to $X$ 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”.