on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
symmetric monoidal (∞,1)-category of spectra
The model structure for dendroidal left fibrations is an operadic analog of the model structure for left fibrations. Its fibrant objects over Assoc are A-∞ spaces, over Comm they are E-∞ spaces.
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For $f : S \to T$ any morphism of dendroidal sets, the induced adjunction (by Kan extension)
is a Quillen adjunction for the corresponding model structures for dendroidal left fibrations over $S$ and $T$. It is a Quillen equivalence if $f$ is a weak equivalences in the Cisinki-Moerdijk model structure on dendroidal sets.
This is (Heuts, prop. 2.4).
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For an overview of models for (∞,1)-operads see table - models for (infinity,1)-operads.
The model structure for dendroidal left fibrations is due to
The model structure for dendroidal Cartesian fibrations that it arises from by Bousfield localization is due to