nLab
model structure for dendroidal left fibrations

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Higher algebra

Contents

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

(…)

Properties

Proposition

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

(f !f *):dSet/Tf *f !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 SS and TT. It is a Quillen equivalence if ff is a weak equivalences in the Cisinki-Moerdijk model structure on dendroidal sets.

This is (Heuts, prop. 2.4).

Relation to other model structures

(…)

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

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

Revised on March 7, 2012 10:42:57 by Urs Schreiber (82.169.65.155)