nLab
model structure on dendroidal sets

Idea

The Cisinski–Moerdijk model category structure on the category $\mathrm{dSet}$ of dendroidal sets models (∞,1)-operads in generalization of the way the Joyal model structure on simplicial sets models (∞,1)-categories.

Overview

We have the following diagram of model categories:

$$\begin{array}{cccccccccc}\mathrm{SSet}\text{-}\mathrm{Operad}& \stackrel{\simeq }{\to }& \mathrm{dSet}& \stackrel{\simeq }{\to }& \mathrm{dSpaces}& & & & & \mathrm{models}\mathrm{for}(\mathrm{\infty }\mathrm{,1})\text{-}\mathrm{operads}\\ \uparrow & & \uparrow & & \uparrow \\ \mathrm{SSet}\text{-}\mathrm{Cat}& \stackrel{\simeq }{\to }& \mathrm{sSet}& \stackrel{\simeq }{\to }& \mathrm{sSpaces}& & & & & \mathrm{models};\mathrm{for}(\mathrm{\infty }\mathrm{,1})\text{-}\mathrm{categories}\end{array},$$\array{ SSet\text{-}Operad &\stackrel{\simeq}{\to}& dSet &\stackrel{\simeq}{\to}& dSpaces &&&&& models\;for\;(\infty,1)\text{-}operads \\ \uparrow && \uparrow && \uparrow \\ SSet\text{-}Cat &\stackrel{\simeq}{\to}& sSet &\stackrel{\simeq}{\to}& sSpaces &&&&& models;for\;(\infty,1)\text{-}categories } \,,

where the entries are

• the category $\mathrm{SSet}\mathrm{Cat}$ of simplicially enriched categories equipped with the Bergner model structure;

• the category SSet of simplicial sets equipped with the Joyal model structure for quasi-categories;

• the category $\mathrm{dSet}$ of dendroidal sets

and where

• the horizontal morphisms are Quillen equivalences

• the vertical morphisms are homotopy full? embeddings.

Definition

Special morphisms

Recall frrom the entry on dendroidal sets the definition of inner and outer faces, boundaries and inner and outer horns.

The following definition are the obvious generalizations of the corresponding notions for the model structure on simplicial sets, in particular for the Joyal model structure.

Definition (inner anodyne extension)

The class of morphisms in $\mathrm{dSet}$ generated from the inner horn inclusions ${\Lambda }^e\Omega [T]\to \Omega [T]$ under

• pushout

• transfinite composition

• retracts

is called the inner anodyne extensions.

Definition (inner Kan fibration)

A morphism $A\to B$ in $\mathrm{dSet}$ is an inner Kan fibration if it has the right lifting property with respect to all inner horn inclusions.

$$\begin{array}{ccc}{\Lambda }^e[T]& \to & A\\ \downarrow & & \downarrow \\ \Omega [T]& \to & B\end{array}$$\array{ \Lambda^e[T] &\to& A \\ \downarrow && \downarrow \\ \Omega[T] &\to& B }

or equivalently with respect to the class of inner anodyne extensions.

Definition (inner Kan complex / quasi-operad)

A dendroidal set $X$ is an inner Kan complex or quasi-operad if the canonical morphism $X\to *$ to the terminal object is an inner Kan fibration.

Definition (trivial fibration)

A morphism $A\to B$ of dendroidal sets is an acyclic fibration if it has the right lifting property with respect to all monomorphisms (equivalently: with respect to all normal monomorphisms).

The model structure

A quasi-operad is a dendroidal set such that…

A trivial fibration of quasi-operads is a morphism of dendroidal sets such that…

An inner anodyne extension of dendroidal sets is a morphism such that…

Definition (model structure on dendroidal sets)

On the category of dendroidal sets let

• the cofibrations be the normal monomorphisms

• the fibrant objects are the weak Kan complexes/quasi-operads

• the fibrations between fibrant objects are the are the inner Kan fibrations $f:A\to B$ between quasi-operads such that the image ${\tau }_d(f)$ under the functor ${\tau }_d:\mathrm{dSet}\to \mathrm{Operads}$ is an operadic fibration

• the weak equivalences form the smallest class of maps that satisfy

  • 2-out-of-3

  • every inner anodyne extension is a weak equivalence

  • every trivial fibration between quasi-operads is a weak equivalence.

Properties

Theorem

The above choices of cofibrations, fibrations and weak equivalences equips the category $\mathrm{dSet}$ of dendroidal sets with the structure of a model category. This model structure is

• a left proper model category

• a cofibrantly generated model category

• a combinatorial model category

• a symmetric monoidal model category.

Proof

This is prop. 2.6 in CisMoe.

Together with the fact that $i^{*}:\mathrm{dSet}\to \mathrm{SSet}$ is a right Quillen functor (with respect to the Joyal model structure on simplicial sets) this also imples that $\mathrm{dSet}$ is an ${\mathrm{SSet}}_{\mathrm{Joyal}}$ enriched model category.

The generating cofibrations $I$ are the boundary inclusion of trees

$$I=\{\partial \Omega [T]\hookrightarrow \Omega [T]\}.$$I = \{\partial \Omega[T] \hookrightarrow \Omega[T]\} \,.

A set of generating cofibrations is guaranteed to exist, but no good explicit characterization is known to date.

Corollary

With this model structure and the standard model structure on operads the dendroidall nerve adjunction

$${\tau }_d:\mathrm{dSet}\overleftarrow{\to }\mathrm{Operad}:N$$\tau_d : dSet \stackrel{\leftarrow}{\to} Operad : N

is a Quillen adjunction and both functors preserve all weak equivalences. So in particular a morphism of operads is a weak equivalence precisely if the induced morphism between dendroidal nerves is a weak equivalence.

Proof

This is cor. 6.17 in CisMoe.

Proposition

A morphism $j:X\to Y$ between cofibrant objects in $\mathrm{dSet}$ is a weak equivalence precisely if for all fibrant objects $A$ the morphism

$$\tau \mathrm{dSet}(Y,A)\to \tau \mathrm{dSet}(X,A)$$\tau dSet(Y,A) \to \tau dSet(X,A)

is an equivalence of categories, where $\tau :\mathrm{SSet}\to \mathrm{Cat}$ is the left adjoint to the nerve.

Proof

This is in section 8.4 of the lecture notes cited below.

Proposition (compatibility with the Joyal model structure)

Let $\mid$ be the tree with a single leaf and no vertex. Then the overcategory $\mathrm{dSet}/\Omega [\mid ]$ is canonically isomorphic to SSet.

The model structure on SSet induced this way as the model structure on an overcategory from the model structure on $\mathrm{dSet}$ coincides with the Joyal model structure on simplicial sets (the one whose fibrant objects are weak Kan complexes).

Proof

This is for instance proposition 8.4.3 in the lecture notes cited below.

References

A useful discussion of of the model structure on dendroidal sets is section 8 of

• Ieke Moerdijk, Lectures on dendroidal sets , lectures given at the Barcelona workshop on Simplicial methods in higher categories (2008) (preliminary writeup)

An expanded and polished version has meanwhile been written up by Javier Guitiérrez and should appear in print soon. An electronic copy is probably available on request.

The model structure was originally given in

• Denis-Charles Cisinski, Ieke Moerdijk, Dendroidal sets as models for homotopy operads (arXiv)

A detailed discussion of the finbrant objects in the model structure is in

• Ieke Moerdijk Ittay Weiss, On inner Kan complexes in the category of dendroidal sets (web)