ScalingStacks

Remark 5.12. We interpret Theorem 5.11 in terms of Koszul duality, after [GiK] and [Pr]. Given the calculation of the Koszul dual operad $\mathbb{D}\mathcal{E}_{n}\simeq\mathcal{E}_{n}[-n]$, computed at the level of homology by Getzler and Jones [GJ] and computed in chain complexes by Fresse [Fre], these functors should be equivalent to restriction and induction along the Koszul dual of the map $\mathcal{E}_{n-1}\rightarrow\mathcal{E}_{n}$. However, Theorem 5.11 is more general: it holds unstably (for instance, when $\mathcal{V}$ is $\spaces$), whereas this operadic form of Koszul duality would require $\mathcal{V}$ to be stable.