nLab model structure on sSet-operads

Idea

The operadic generalization of the model structure on sSet-categories: a presentation of (∞,1)Operad.

Definition

All operads considered here are multi-coloured symmetric operads (symmetric multicategories).

Definition

Call a morphism of simplicial operads $f : P \to Q$

• fully faithful if it is a weak homotopy equivalence on all component simplicial sets.

• essentially surjective if the induced morphism $\pi_0(f)$ on homotopy categories is an essentially surjective functor.

• a weak equivalence if it is fully faithful and essentially surjective.

• a local fibration if it is componentwise a Kan fibration.

• a fibration if it is a local fibration and the underlying functor $\pi_0(f) : Ho(j^* P) \to Ho(j^* Q)$ on the homotopy categories of the underlying simplicial categories is an isofibration.

Theorem

This defines on $sSet Operad$ the structure of a model category which is

• cofibrantly generated;

• right proper.

This is (Cisinski-Moerdijk, theorem 1.14).

Remark

For $C \in$ Set, let $sSet Operad_C \hookrightarrow sSet Operad$ be the full subcategory on operads with $C$ as their set of colours.

Then $sSet Operad_C \simeq (Operad_C)^{\Delta^{op}}$ is the category of simplicial objects in $C$-coloured symmetric operads, and restricted to this the above model category structure is corresponding the model structure on simplicial algebras.

See (Cisinski-Moerdijk, remark 1.9).

Remark

Restricted along the inclusion

$$j_! : sSet Cat \hookrightarrow sSet Operad$$

the above model structure restricts to the model structure on sSet-categories by Julie Bergner.

Properties

Remark

A morphism in $sSet Operad$ is an acyclic fibration precisely if it is componentwise an acyclic Kan fibration.

Related concepts

The entries of the following table display model categories and Quillen equivalences between these that present the (∞,1)-category of (∞,1)-categories (second table), of (∞,1)-operads (third table) and of $\mathcal{O}$-monoidal (∞,1)-categories (fourth table).

	general pattern
strict enrichment		(∞,1)-category/(∞,1)-operad
$\downarrow$		$\downarrow$
enriched (∞,1)-category	$\hookrightarrow$	internal (∞,1)-category
	(∞,1)Cat
SimplicialCategories	$-$homotopy coherent nerve$\to$	SimplicialSets/quasi-categories		RelativeSimplicialSets
$\downarrow$simplicial nerve		$\downarrow$
SegalCategories	$\hookrightarrow$	CompleteSegalSpaces
	(∞,1)Operad
SimplicialOperads	$-$homotopy coherent dendroidal nerve$\to$	DendroidalSets		RelativeDendroidalSets
$\downarrow$dendroidal nerve		$\downarrow$
SegalOperads	$\hookrightarrow$	DendroidalCompleteSegalSpaces
	$\mathcal{O}$Mon(∞,1)Cat
		DendroidalCartesianFibrations

References

• Denis-Charles Cisinski, Ieke Moerdijk, Dendroidal sets and simplicial operads (arXiv:1109.1004)