nLab
model structure on sSet-operads

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 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:PQf : P \to Q

Theorem

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

This is (Cisinski-Moerdijk, theorem 1.14).

Remark

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

Then sSetOperad C(Operad C) Δ opsSet Operad_C \simeq (Operad_C)^{\Delta^{op}} is the category of simplicial objects in CC-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 !:sSetCatsSetOperad 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 sSetOperadsSet 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 𝒪\mathcal{O}-monoidal (∞,1)-categories (fourth table).

general pattern
strict enrichment(∞,1)-category/(∞,1)-operad
\downarrow\downarrow
enriched (∞,1)-category\hookrightarrowinternal (∞,1)-category
(∞,1)Cat
SimplicialCategories-homotopy coherent nerve\toSimplicialSets/quasi-categoriesRelativeSimplicialSets
\downarrowsimplicial nerve\downarrow
SegalCategories\hookrightarrowCompleteSegalSpaces
(∞,1)Operad
SimplicialOperads-homotopy coherent dendroidal nerve\toDendroidalSetsRelativeDendroidalSets
\downarrowdendroidal nerve\downarrow
SegalOperads\hookrightarrowDendroidalCompleteSegalSpaces
𝒪\mathcal{O}Mon(∞,1)Cat
DendroidalCartesianFibrations

References

Revised on February 29, 2012 13:42:45 by Urs Schreiber (82.169.65.155)