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 ${\pi }_0(f):\mathrm{Ho}(j^{*}P)\to \mathrm{Ho}(j^{*}Q)$ on the homotopy categories of the underlying simplicial categories is an isofibration.

Theorem

This defines on $\mathrm{sSet}\mathrm{Operad}$ 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 $\mathrm{sSet}{\mathrm{Operad}}_C\hookrightarrow \mathrm{sSet}\mathrm{Operad}$ be the full subcategory on operads with $C$ as their set of colours.

Then $\mathrm{sSet}{\mathrm{Operad}}_C\simeq ({\mathrm{Operad}}_C{)}^{{\Delta }^{\mathrm{op}}}$ 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

$$j_{!}:\mathrm{sSet}\mathrm{Cat}\hookrightarrow \mathrm{sSet}\mathrm{Operad}$$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 $\mathrm{sSet}\mathrm{Operad}$ is an acyclic fibration precisely if it is componentwise an acyclic Kan fibration.

Related concepts

The entries of the following table display models, model categories, and Quillen equivalences between these that present the (∞,1)-category of (∞,1)-categories (second table), of (∞,1)-operads (third table) and of $𝒪$-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
	$𝒪$Mon(∞,1)Cat
		DendroidalCartesianFibrations

References

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