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model structure on dendroidal sets

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

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Higher algebra

Contents

Idea

The CisinskiMoerdijk model category structure on the category dSetdSet 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:

SSet-Operad dSet dSpaces modelsfor(,1)-operads SSet-Cat sSet sSpaces modelsfor(,1)-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

and where

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 dSetdSet generated from the inner horn inclusions Λ eΩ[T]Ω[T]\Lambda^e \Omega[T] \to \Omega[T] under

is called the inner anodyne extensions.

Definition

The class of morphisms in dSetdSet generated from boundary inclusions δΩ[T]Ω[T]\delta \Omega[T] \to \Omega[T] under

is called the normal monomorphisms.

Definition

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

Λ e[T] A Ω[T] B \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 XX is an inner Kan complex or quasi-operad if the canonical morphism X*X\to {*} to the terminal object is an inner Kan fibration.

Definition

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

  1. it is an inner Kan fibration;

  2. the morphism of operads τ d(f):τ d(X)τ d(Y)\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

Properties

Establishing the model structure

Statement

Theorem

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

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

Remark

The generating cofibrations II are the boundary inclusion of trees

I={Ω[T]Ω[T]}. 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:XYj : X \to Y between cofibrant objects in dSetdSet is a weak equivalence precisely if for all fibrant objects AA the morphism

τ:dSet(Y,A)τdSet(X,A) \tau : dSet(Y,A) \to \tau dSet(X,A)

is an equivalence of categories, where τ:SSetCat\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 1.

The strategy is the following: we slice dSetdSet over a normal resolution E 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 dSet_{/E_\infty}. This indeed exists, prop. 4 below, and the model structure on dSetdSet itself can then be transferred along the inverse image of the etale geometric morphism dSet /E dSetdSet_{/E_\infty} \to dSet, prop. 5.

Proposition

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

AYBXBY 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:={01}J := \{0 \stackrel{\simeq}{\to} 1\} for the codiscrete groupoid on two objects. We use the same symbol for its image along N di !:CatdSetN_d i_! : Cat \hookrightarrow dSet.

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

{Ω[T]JΩ[T]{0}Ω[T]J} e{0,1},TΩ. \{ \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 JJ-fibration if it has the right lifting property against JJ-anodyne extensions.

This is (Cis-Moer, 3.2).

Proposition

For ABA \to B a JJ-anodyne extension and XYX \to Y a normal monomorphism of dendroidal sets, the pushout-product morphism

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

is a JJ-anodyne extension.

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

Definition

For XdSetX \in dSet, the BV-tensor product with

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

defines an interval object

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

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

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

If in the above BB is normal and a:XBa : X \to B is JJ-anodyne, then the left homotopies given by the above cylinder defines a notion of left homotopy in dSet /BdSet_{/B}.

Say a morphism f:AAf : A \to A' in dSet /BdSet_{/B} is a BB-equivalence if it is a homotopy equivalence with respect to this notion of homotopy, hence if for all JJ-fibrations XBX \to B the induced morphism

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

is a bijection.

Proposition

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

  • the weak equivalences are the BB-equivalences;

  • the cofibrations are the monomorphisms;

  • the fibrant objects are the JJ-fibrations into BB;

  • a morphism between fibrant objects is a fibration precisely if its image in dSetdSet is a JJ-fibration.

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

Proof

Notice that

  1. every monomorphism over a normal object BB is normal,

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

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

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

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

  • the class of JJ-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 /BdSet_{/B} in the sense of this definition.

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

Let E dSetE_\infty \in dSet be any normal dendroidal set such that the terminal morphism E *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 !p *).dSetp *p !dSet /E , (p_! \dashv p^*) \,. dSet \stackrel{\overset{p_!}{\leftarrow}}{\underset{p^*}{\to}} dSet_{/E_\infty} \,,

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

Proposition

The transferred model structure on dSetdSet along the right adjoint p *p^* of the model structure from prop. 4 exists.

This is the model structure characterized in theorem 1.

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 JJ-fibrant, def. 8, hence fibrant in dSet CMdSet_{CM}, precisely if it is an inner Kan complex.

A morphism f:XYf : X \to Y in dSetdSet between inner Kan complexes is a JJ-fibration, hence a fibration in dSet CMdSet_{CM}, precisely if

  1. it is an inner Kan fibration;

  2. on homotopy categories τi *f\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 CMdSet_{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 CMdSet_{CM} is a dSet CMdSet_{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/Ω[|]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 dSetdSet 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 JoyalsSet_{Joyal}-enriched model category, where sSet JoyalsSet_{Joyal} is the model structure for quasi-categories.

Proof

This follows from the fact, cor. 1, that dSet CMdSet_{CM} is a monoidal model category and the fact that the functor i *:dSetsSet Joyali^*: dSet \to sSet_{Joyal} is a right Quillen functor.

Remark

However, dSet CMdSet_{CM} is not an enriched model category over sSet QuillensSet_{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

ℋℴ𝓂:dSet op×dSetdSet \mathcal{Hom} : dSet^{op} \times dSet \to dSet

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

For AA normal and XX an inner Kan dendroidal set, write

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

for the underlying quasi-category, and write

k(A,X):=Core(𝒽ℴ𝓂(A,X))KanCplx 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×dSetdSet -^{(-)} : sSet^{op} \times dSet \to dSet

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

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

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

k(B,X)k(B,Y)× k(A,Y)k(A,X) 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 ABA \to B is the above is an anodyne extension (acyclic monomorphism) of simplicial sets, then

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

is an acyclic fibration in dSet CMdSet_{CM}.

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

Proposition

For AA normal and XX fibrant, the Kan complex

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

is the correct derived hom-space of dSet CMdSet_{CM}.

Proof

One checks that nX (Δ[n])n \mapsto X^{(\Delta[n])} is a fibrant resolution of XX in the Reedy model structure [Δ op,dSet CM] Reedy[\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

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

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

=Hom sSet(Δ[n],k(A,X))=k(A,X) n. \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 !j *):dSetj *j !sSet (j_! \dashv j^*) : dSet \stackrel{\overset{j_!}{\leftarrow}}{\underset{j^*}{\to}} sSet

induced from the inclusion j:ΔΩj : \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 !j_!, of the model structure on dendroidal sets. Therefore j !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 CMdSet_{CM} in terms of the dendroidal interval J dJ_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 JJ-anodyne extensions;

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

(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 !hcN d):sSetOperadhcN dW !dSet, (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)

sSetCat sSetOperad hcN d W ! sSetdSet/η dSet \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 SetSet-Operads

Remark

Write Operad for the category of symmetric operads over Set.

The dendroidal nerve adjunction

(τ dN d):OperadN dτ ddSet (\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. 7, and the canonical model structure on Operad.

Moreover, N dN_d detects and preserves weak equivalences, while τ d\tau_d preserves weak equivalences.

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

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

Proposition

There is a Quillen adjunction

dSet CMididdSet He 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).

References

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

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

making heavy use of results on Cisinski model structures from

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

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

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

Revised on July 1, 2013 15:21:38 by Urs Schreiber (82.169.65.155)