on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
symmetric monoidal (∞,1)-category of spectra
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.
We have the following diagram of model 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.
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.
The class of morphisms in $dSet$ generated from the inner horn inclusions $\Lambda^e \Omega[T] \to \Omega[T]$ under
is called the inner anodyne extensions.
The class of morphisms in $dSet$ generated from boundary inclusions $\delta \Omega[T] \to \Omega[T]$ under
is called the normal monomorphisms.
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.
or equivalently with respect to the class of inner anodyne extensions.
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.
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.
A morphism $f : X \to Y$ of dendroidal sets is called an isofibration if it
it is an inner Kan fibration;
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.
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
every inner anodyne extension is a weak equivalence
every acyclic fibration between quasi-operads is a weak equivalence.
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 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.
The generating cofibrations $I$ are the boundary inclusion of trees
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).
A morphism $j : X \to Y$ between cofibrant objects in $dSet$ is a weak equivalence precisely if for all fibrant objects $A$ the morphism
is an equivalence of categories, where $\tau : SSet \to Cat$ is the left adjoint to the nerve.
This is in (Moerdijk, section 8.4).
We state a list of lemmas to establish theorem 1.
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. 4 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. 5.
For $A \to B$ an inner anodyne extension and $X \to Y$ a normal monomorphism of dendroidal sets, the pushout-product morphism
(with “$\otimes$” the Boardman-Vogt tensor product) is an inner anodyne extension.
This is (Cis-Moer, prop. 3.1).
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
Call a morphism a $J$-fibration if it has the right lifting property against $J$-anodyne extensions.
This is (Cis-Moer, 3.2).
For $A \to B$ a $J$-anodyne extension and $X \to Y$ a normal monomorphism of dendroidal sets, the pushout-product morphism
is a $J$-anodyne extension.
This is (Cis-Moer, prop. 3.3).
For $X \in dSet$, the BV-tensor product with
defines an interval object
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
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
is a bijection.
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).
Notice that
every monomorphism over a normal object $B$ is normal,
the over topos (see there) $PSh(\Omega)_{/B}$ may be identified with presheaves on the slice site
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
where $p_!$ simply forgets the map to $E_\infty$ and where $p^*$ forms the product with $E_\infty$
The transferred model structure on $dSet$ along the right adjoint $p^*$ 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.
A dendroidal set is $J$-fibrant, def. 8, 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
it is an inner Kan fibration;
on homotopy categories $\tau i^* f$ is an isofibration.
This is (Cisinski Moerdijk, theorem 5.10).
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).
With respect to the internal hom corresponding to the Boardman-Vogt tensor product, $dSet_{CM}$ is a $dSet_{CM}$-enriched model category.
(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).
This model category is naturally an $sSet_{Joyal}$-enriched model category, where $sSet_{Joyal}$ is the model structure for quasi-categories.
This follows from the fact, cor. 1, that $dSet_{CM}$ is a monoidal model category and the fact that the functor $i^*: dSet \to sSet_{Joyal}$ is a right Quillen functor.
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.
Write
for the internal hom corresponding to the Boardman-Vogt tensor product.
For $A$ normal and $X$ an inner Kan dendroidal set, write
for the underlying quasi-category, and write
for the maximal Kan complex inside the quasi-category inside the internal hom.
Write
for the corresponding powering, characterized by the existence of a natural isomorphism
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
is a Kan fibration between Kan complexes.
This is (Cis-Moer, prop. 6.7).
If $A \to B$ is the above is an anodyne extension (acyclic monomorphism) of simplicial sets, then
is an acyclic fibration in $dSet_{CM}$.
This is (Cis-Moer, cor. 6.9).
For $A$ normal and $X$ fibrant, the Kan complex
is the correct derived hom-space of $dSet_{CM}$.
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
By the tensoring-definition of $X^{(\Delta[n])}$ this is isomorphic to
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.
The adjunction
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.
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.
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.
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 $j$.
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.
There exists an adjunction
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).
Under the inclusions (see the discussion at dendroidal set)
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.
Write Operad for the category of symmetric operads over Set.
The dendroidal nerve adjunction
is a Quillen adjunction between the model structure on dendroidal sets, def. 7, 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).
There is a Quillen adjunction
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).
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 $sSet$-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