related by the Dold-Kan correspondence
symmetric monoidal (∞,1)-category of spectra
The Cisinski–Moerdijk model category structure on the category 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 of dendroidal sets
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 class of morphisms in generated from the inner horn inclusions under
is called the inner anodyne extensions.
The class of morphisms in generated from boundary inclusions under
is called the normal monomorphisms.
or equivalently with respect to the class of inner anodyne extensions.
A dendroidal set is an inner Kan complex or quasi-operad if the canonical morphism to the terminal object is an inner Kan fibration.
A morphism of dendroidal sets is an acyclic fibration if it has the right lifting property with respect to all normal monomorphisms.
A morphism of dendroidal sets is called an isofibration if it
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
The above choices of cofibrations, fibrations and weak equivalences equips the category of dendroidal sets with the structure of a model category . This model structure is
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 between cofibrant objects in is a weak equivalence precisely if for all fibrant objects the morphism
This is in (Moerdijk, section 8.4).
We state a list of lemmas to establish theorem 1.
The strategy is the following: we slice over a normal resolution (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 . This indeed exists, prop. 4 below, and the model structure on itself can then be transferred along the inverse image of the etale geometric morphism , prop. 5.
(with “” the Boardman-Vogt tensor product) is an inner anodyne extension.
This is (Cis-Moer, prop. 3.1).
Write for the codiscrete groupoid on two objects. We use the same symbol for its image along .
The -anodyne extensions in are the morphisms generated under pushouts, transfinite composition and retracts from the the inner anodyne extensions and from the pushout products of with the tree boundary inclusion
Call a morphism a -fibration if it has the right lifting property against -anodyne extensions.
This is (Cis-Moer, 3.2).
is a -anodyne extension.
This is (Cis-Moer, prop. 3.3).
For , the BV-tensor product with
defines an interval object
This is compatible with slicing, in that for any and a given morphism, we have an interval object in the over category given by a diagram
Say a morphism in is a -equivalence if it is a homotopy equivalence with respect to this notion of homotopy, hence if for all -fibrations the induced morphism
is a bijection.
the weak equivalences are the -equivalences;
the cofibrations are the monomorphisms;
the fibrant objects are the -fibrations into ;
a morphism between fibrant objects is a fibration precisely if its image in is a -fibration.
This is (Cis-Moer, prop. 3.5).
every monomorphism over a normal object is normal,
the over topos (see there) may be identified with presheaves on the slice site
Therefore for normal the slice , as opposed to itself, has a chance to carry a Cisinski model structure. And this is indeed the case: one checks that
is a functorial cyclinder in the sense of this definition;
the class of -anodyne extensions is a class of corresponding anodyne extensions in the sense of this definition
so that together these form a homotopical structure on in the sense of this definition.
Let be any normal dendroidal set such that the terminal morphism is an acyclic fibration in that it has the right lifting property against the tree boundary inclusions.
where simply forgets the map to and where forms the product with
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 -fibrant, def. 8, hence fibrant in , precisely if it is an inner Kan complex.
A morphism in between inner Kan complexes is a -fibration, hence a fibration in , precisely if
This is (Cisinski Moerdijk, theorem 5.10).
This is (Cis-Moer, prop. 3.17).
(compatibility with the Joyal model structure)
This is for instance (Moerdijk, proposition 8.4.3).
However, is not an enriched model category over , the standard model structure on simplicial sets (but see model structure for dendroidal Cartesian fibrations). But it comes close, as the following propositions show.
For normal and an inner Kan dendroidal set, write
for the underlying quasi-category, and write
This is (Cis-Moer, prop. 6.7).
If is the above is an anodyne extension (acyclic monomorphism) of simplicial sets, then
is an acyclic fibration in .
This is (Cis-Moer, cor. 6.9).
For normal and fibrant, the Kan complex
is the correct derived hom-space of .
By the tensoring-definition of 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.
By the proof of (Cisinski-Moerdijk, cor. 2.10) the model structure for quasi-categories is in fact the restriction, along , of the model structure on dendroidal sets. Therefore 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 in terms of the dendroidal interval as follows.
The class of acyclic cofibrations between normal dendroidal sets is the smallest class of morphisms between normal dendroidal sets
which contains the -anodyne extensions;
with left cancellation property: if a composite is in the class and is, then so is .
There exists an adjunction
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.
Moreover, detects and preserves weak equivalences, while preserves weak equivalences.
This is (Cisinski-Moerdijk 09, prop. 2.5).
There is a Quillen adjunction
See (Heuts, remark 18.104.22.168).
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 -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