nLab model structure on dendroidal sets

Idea

The Cisinski–Moerdijk model category structure on the category 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:

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

where the entries are

• the category $SSet Cat$ 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 $dSet$ of dendroidal sets

and where

• the horizontal morphisms are Quillen equivalences

• the vertical morphisms are homotopy full embeddings.

Definition

Special morphisms

Recall from 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 model structure for quasi-categories.

Definition

The class of morphisms in $dSet$ 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

The class of morphisms in $dSet$ generated from boundary inclusions $\delta \Omega[T] \to \Omega[T]$ under

• pushout

• transfinite composition

• retracts

is called the normal monomorphisms.

Definition

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

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

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

Definition

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

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

Definition

A morphism $f : X \to Y$ of dendroidal sets is called an isofibration if it

1. it is an inner Kan fibration;

2. the morphism of operads $\tau_d(f) : \tau_d(X) \to \tau_d(Y)$ is a fibration in the canonical model structure on operads, hence the underlying functor of categories is an isofibration.

The model structure

Definition

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 precisely the isofibrations.

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

  • 2-out-of-3

  • every inner anodyne extension is a weak equivalence

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

Properties

Establishing the model structure

Statement

Theorem

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

• a left proper model category

• a cofibrantly generated model category

• a combinatorial model category

• a symmetric monoidal model category (with respect to the Boardman-Vogt tensor product).

This is (Cisinski Moerdijk, theorem 2.4 and prop. 2.6). We indicate the proof below.

Remark

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

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

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

This is (CisMoe, cor. 6.17).

Proposition

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

$$\tau : dSet(Y,A) \to \tau dSet(X,A)$$

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

This is in (Moerdijk, section 8.4).

Proof

We state a list of lemmas to establish theorem .

The strategy is the following: we slice $dSet$ over a normal resolution $E_\infty$ (a model of the E-∞ operad) of the terminal object. Since over a normal object all morphisms are normal, we have a chance to establish a Cisinski model structure on $dSet_{/E_\infty}$. This indeed exists, prop. below, and the model structure on $dSet$ itself can then be transferred along the inverse image of the etale geometric morphism $dSet_{/E_\infty} \to dSet$, prop. .

Proposition

For $A \to B$ an inner anodyne extension and $X \to Y$ a normal monomorphism of dendroidal sets, the pushout-product morphism

$$A \otimes Y \cup B \otimes X \to B\otimes Y$$

(with “$\otimes$” the Boardman-Vogt tensor product) is an inner anodyne extension.

This is (Cis-Moer, prop. 3.1).

Definition

Write $J := \{0 \stackrel{\simeq}{\to} 1\}$ for the codiscrete groupoid on two objects. We use the same symbol for its image along $N_d i_! : Cat \hookrightarrow dSet$.

The $J$-anodyne extensions in $dSet$ are the morphisms generated under pushouts, transfinite composition and retracts from the the inner anodyne extensions and from the pushout products of $\{e\} \to J$ with the tree boundary inclusion

$$\{ \partial \Omega[T] \otimes J \; \cup \; \Omega[T] \otimes \{0\} \to \Omega[T] \otimes J \}_{e \in \{0,1\}, T \in \Omega} \,.$$

Call a morphism a $J$-fibration if it has the right lifting property against $J$-anodyne extensions.

This is (Cis-Moer, 3.2).

Proposition

For $A \to B$ a $J$-anodyne extension and $X \to Y$ a normal monomorphism of dendroidal sets, the pushout-product morphism

$$A \otimes Y \cup B \otimes X \to B\otimes Y$$

is a $J$-anodyne extension.

This is (Cis-Moer, prop. 3.3).

Definition

For $X \in dSet$, the BV-tensor product with

$$\{0\} \coprod \{1\} \to J \to \eta$$

defines an interval object

$$X \coprod X \to J \otimes X \to X \,.$$

This is compatible with slicing, in that for any $B \in sSet$ and $a : X \to B$ a given morphism, we have an interval object in the over category $dSet_{/B}$ given by a diagram

$$\array{ X \coprod X &\to& J \otimes X &\to& X \\ & {}_{(a,a)}\searrow & \downarrow & \swarrow_{a} \\ && B } \,.$$

If in the above $B$ is normal and $a : X \to B$ is $J$-anodyne, then the left homotopies given by the above cylinder defines a notion of left homotopy in $dSet_{/B}$.

Say a morphism $f : A \to A'$ in $dSet_{/B}$ is a $B$-equivalence if it is a homotopy equivalence with respect to this notion of homotopy, hence if for all $J$-fibrations $X \to B$ the induced morphism

$$[A',X]_{\sim_B} \to [A,X]_{\sim B}$$

is a bijection.

Proposition

Let $B \in dSet$ be normal. Then $dSet_{/B}$ carries a left proper cofibrantly generated model category for which

• the weak equivalences are the $B$-equivalences;

• the cofibrations are the monomorphisms;

• the fibrant objects are the $J$-fibrations into $B$;

• a morphism between fibrant objects is a fibration precisely if its image in $dSet$ is a $J$-fibration.

This is (Cis-Moer, prop. 3.5).

Proof

Notice that

1. every monomorphism over a normal object $B$ is normal,

2. the over topos (see there) $PSh(\Omega)_{/B}$ may be identified with presheaves on the slice site

   $$PSh(\Omega)_{/B} \simeq PSh(\Omega_{/B}) \,.$$

Therefore for normal $B$ the slice $dSet_{/B}$, as opposed to $dSet$ itself, has a chance to carry a Cisinski model structure. And this is indeed the case: one checks that

• $J \otimes (-)$ is a functorial cyclinder in the sense of this definition;

• the class of $J$-anodyne extensions is a class of corresponding anodyne extensions in the sense of this definition

• so that together these form a homotopical structure on $dSet_{/B}$ in the sense of this definition.

The statement then is a special case of this theorem at Cisinski model structure.

Let $E_\infty \in dSet$ be any normal dendroidal set such that the terminal morphism $E_\infty \to *$ is an acyclic fibration in that it has the right lifting property against the tree boundary inclusions.

By the general discussion at over topos we have an adjunction

$$(p_! \dashv p^*) \,. dSet \stackrel{\overset{p_!}{\leftarrow}}{\underset{p^*}{\to}} dSet_{/E_\infty} \,,$$

where $p_!$ simply forgets the map to $E_\infty$ and where $p^*$ forms the product with $E_\infty$

Proposition

The transferred model structure on $dSet$ along the right adjoint $p^*$ of the model structure from prop. exists.

This is the model structure characterized in theorem .

This appears as (Cis-Moer, prop. 3.12).

So far this establishes the existence of the model structure and that every dendroidal inner Kan complex is fibrant. Below in characterization of the fibrant objects we consider the converse statement: that the fibrant objects are precisely the inner Kan complexes.

Characterization of the fibrations

Proposition

A dendroidal set is $J$-fibrant, def. , hence fibrant in $dSet_{CM}$, precisely if it is an inner Kan complex.

A morphism $f : X \to Y$ in $dSet$ between inner Kan complexes is a $J$-fibration, hence a fibration in $dSet_{CM}$, precisely if

1. it is an inner Kan fibration;

2. on homotopy categories $\tau i^* f$ is an isofibration.

This is (Cisinski Moerdijk, theorem 5.10).

Monoidal model category structure

Proposition

With respect to the Boardman-Vogt tensor product on dendroidal sets, the model structure $dSet_{CM}$ is a symmetric monoidal model category.

This is (Cis-Moer, prop. 3.17).

Corollary

With respect to the internal hom corresponding to the Boardman-Vogt tensor product, $dSet_{CM}$ is a $dSet_{CM}$-enriched model category.

Other enrichments of the underlying category

Proposition

(compatibility with the Joyal model structure)

Let $|$ be the tree with a single leaf and no vertex. Then the overcategory $dSet/\Omega[|]$ 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 $dSet$ coincides with the model structure for quasi-categories.

This is for instance (Moerdijk, proposition 8.4.3).

Proposition

This model category is naturally an $sSet_{Joyal}$-enriched model category, where $sSet_{Joyal}$ is the model structure for quasi-categories.

Proof

This follows from the fact, cor. , that $dSet_{CM}$ is a monoidal model category and the fact that the functor $i^*: dSet \to sSet_{Joyal}$ is a right Quillen functor.

Remark

However, $dSet_{CM}$ is not an enriched model category over $sSet_{Quillen}$, the standard model structure on simplicial sets (but see model structure for dendroidal Cartesian fibrations). But it comes close, as the following propositions show.

Definition

Write

$$\mathcal{Hom} : dSet^{op} \times dSet \to dSet$$

for the internal hom corresponding to the Boardman-Vogt tensor product.

For $A$ normal and $X$ an inner Kan dendroidal set, write

$$\mathcal{hom}(A,X) := i^* \mathcal{Hom}(A,X)$$

for the underlying quasi-category, and write

$$k(A,X) := Core(\mathcal{hom}(A,X)) \in KanCplx$$

for the maximal Kan complex inside the quasi-category inside the internal hom.

Write

$$-^{(-)} : sSet^{op} \times dSet \to dSet$$

for the corresponding powering, characterized by the existence of a natural isomorphism

$$Hom_{sSet}(K, k(A,X)) \simeq Hom_{dSet}(A, X^{(K)}) \,.$$

Proposition

For $p : X \to Y$ a fibration between fibrant dendroidal sets (hence an inner Kan fibration and an isofibration on the underlying homotopy category), and for $A \to B$ a normal monomorphism, the induced morphism

$$k(B,X) \to k(B,Y) \times_{k(A,Y)} k(A,X)$$

is a Kan fibration between Kan complexes.

This is (Cis-Moer, prop. 6.7).

Proposition

If $A \to B$ is the above is an anodyne extension (acyclic monomorphism) of simplicial sets, then

$$X^{(B)} \to Y^{(B)} \times_{Y^{(A)}} X^{(A)}$$

is an acyclic fibration in $dSet_{CM}$.

This is (Cis-Moer, cor. 6.9).

Proposition

For $A$ normal and $X$ fibrant, the Kan complex

$$k(A,X) \simeq \mathbb{R}Hom(A,X)$$

is the correct derived hom-space of $dSet_{CM}$.

Proof

One checks that $n \mapsto X^{(\Delta[n])}$ is a fibrant resolution of $X$ in the Reedy model structure $[\Delta^{op}, dSet_{CM}]_{Reedy}$. By the discussion at simplicial model category and derived hom-space the latter is therefore given by the simplicial set

$$n \mapsto Hom_{dSet}(A, X^{(\Delta[n])}) \,.$$

By the tensoring-definition of $X^{(\Delta[n])}$ this is isomorphic to

$$\cdots = Hom_{sSet}(\Delta[n], k(A,X)) = k(A,X)_n \,.$$

Relation to other model structures

We discuss the relation of the model structure on dendroidal sets to other model category structures for operads.

See the table - models for (infinity,1)-operads for an overview.

Model structure for quasi-categories

Proposition

The adjunction

$$(j_! \dashv j^*) : dSet \stackrel{\overset{j_!}{\leftarrow}}{\underset{j^*}{\to}} sSet$$

induced from the inclusion $j : \Delta \hookrightarrow \Omega$ constitutes a Quillen adjunction between the above model structure on dendroidal sets, and the model structure for quasi-categories.

Proof

By the proof of (Cisinski-Moerdijk, cor. 2.10) the model structure for quasi-categories is in fact the restriction, along $j_!$, of the model structure on dendroidal sets. Therefore $j_!$ is left Quillen.

Model structure for dendroidal complete Segal spaces

There is a Quillen equivalence to the model structure for dendroidal complete Segal spaces (see there). A crucial step in the proof is the following expression of the acyclic cofibrations on $dSet_{CM}$ in terms of the dendroidal interval $J_d$ as follows.

Proposition

The class of acyclic cofibrations between normal dendroidal sets is the smallest class of morphisms between normal dendroidal sets

• which contains the $J$-anodyne extensions;

• with left cancellation property: if a composite $\stackrel{i}{\to} \stackrel{j}{\to}$ is in the class and $i$ is, then so is $j$.

(Cis-Moer 09, prop. 3.16)

Model structure on simplicial operads

There exists also a model structure on simplicial operads, which is Quillen equivalent to the model structure on dendroidal sets.

This Quillen equivalence is an operadic generalization of the Quillen equivalence between the model structure on sSet-categories and the model structure for quasi-categories.

Theorem

There exists an adjunction

$$(W_! \dashv hcN_d) : sSet Operad \stackrel{\overset{W_!}{\leftarrow}}{\underset{hcN_d}{\to}} dSet \,,$$

Whise right adjoint is the dendroidal homotopy coherent nerve.

This is a Quillen equivalence between the model structure on dendroidal sets, and the model structure on simplicial operads.

This is (Cisinski-Moerdijk 11, theorem 815).

Remark

Under the inclusions (see the discussion at dendroidal set)

$$\array{ sSet Cat &\hookrightarrow & sSet Operad \\ && {}^{\mathllap{hcN_d}}\downarrow \uparrow^{\mathrlap{W_!}} \\ sSet \simeq dSet/\eta &\hookrightarrow & dSet }$$

this restricts to the Quillen equivalence between the model structure on sSet-categories and the model structure for quasi-categories discussed at relation between quasi-categories and simplicial categories.

Model structure on $Set$-Operads

Remark

Write Operad for the category of symmetric operads over Set.

The dendroidal nerve adjunction

$$(\tau_d \dashv N_d) : Operad \stackrel{\overset{\tau_d}{\leftarrow}}{\underset{N_d}{\to}} dSet$$

is a Quillen adjunction between the model structure on dendroidal sets, def. , and the canonical model structure on Operad.

Moreover, $N_d$ detects and preserves weak equivalences, while $\tau_d$ preserves weak equivalences.

This is (Cisinski-Moerdijk 09, prop. 2.5).

Model structure for symmetric monoidal $(\infty,1)$-categories

Proposition

There is a Quillen adjunction

$$dSet_{CM} \stackrel{\overset{id}{\leftarrow}}{\underset{id}{\to}} dSet_{He}$$

which exhibits the model structure for dendroidal left fibrations as a left Bousfield localization of the Cisinski-Moerdijk model structure on dendroidal sets.

See (Heuts, remark 6.8.0.2).

Related concepts

• canonical model structure on Operad,

• model structure on operads

• model structure for dendroidal complete Segal spaces

• table - models for (infinity,1)-operads

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:0902.1954)

making heavy use of results on Cisinski model structures from

• Denis-Charles Cisinski, Les préfaisceaux comme type d'homotopie, Astérisque, Volume 308, Soc. Math. France (2006), 392 pages (pdf)

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

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

The proof of the Quillen equivalence between the model structure on dendroidal sets and that on $sSet$-operads is given in

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

The relation to the model structure for dendroidal Cartesian fibrations and the model structure for dendroidal left fibrations is discussed in

• Gijs Heuts, Algebras over infinity-Operads (arXiv:1110.1776)