nLab
model structure for dendroidal left fibrations

Idea

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.

Definition

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Properties

Proposition

For $f:S\to T$ any morphism of dendroidal sets, the induced adjunction (by Kan extension)

$$(f_{!}⊣f^{*}):\mathrm{dSet}/T\stackrel{\stackrel{f_{!}}{\leftarrow }}{\underset{f^{*}}{\to }}\mathrm{dSet}/S$$(f_! \dashv f^* ) : dSet/T \stackrel{\overset{f_!}{\leftarrow}}{\underset{f^*}{\to}} dSet/S

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).

Relation to other model structures

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Related concepts

• model structure for left fibrations

• model structure for dendroidal Cartesian fibrations

• model structure on spectra

For an overview of models for (∞,1)-operads see table - models for (infinity,1)-operads.

References

The model structure for dendroidal left fibrations is due to

• Gijs Heuts, Algebras over infinity-Operads (arXiv:1110.1776)

The model structure for dendroidal Cartesian fibrations that it arises from by Bousfield localization is due to

• Gijs Heuts, Algebras over infinity-Operads (arXiv:1110.1776)