nLab model structure on sSet-operads

model category

Model structures

for ∞-groupoids

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

Higher algebra

higher algebra

universal algebra

Contents

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 $f : P \to Q$

Theorem

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

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.

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.

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 $\mathcal{O}$-monoidal (∞,1)-categories (fourth table).

general pattern
$\downarrow$$\downarrow$
enriched (∞,1)-category$\hookrightarrow$internal (∞,1)-category
(∞,1)Cat
SimplicialCategories$-$homotopy coherent nerve$\to$SimplicialSets/quasi-categoriesRelativeSimplicialSets
$\downarrow$simplicial nerve$\downarrow$
SegalCategories$\hookrightarrow$CompleteSegalSpaces
SimplicialOperads$-$homotopy coherent dendroidal nerve$\to$DendroidalSetsRelativeDendroidalSets
$\downarrow$dendroidal nerve$\downarrow$
SegalOperads$\hookrightarrow$DendroidalCompleteSegalSpaces
$\mathcal{O}$Mon(∞,1)Cat