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The model structure for dendroidal complete Segal spaces is an operadic generalization of the model structure for complete Segal spaces. It serves to present the (∞,1)-category of (∞,1)-operads.
A complete dendroidal Segal space is much like a dendroidal set, only that it has for each tree not just a set of dendrices, but a simplicial set , subject to some conditions. The model structure discussed here is defined on the category of all simplicial presheaves over the tree category, such that the fibrant objects are precisely the dendroidal complete Segal spaces.
Write for the tree category, the site for dendroidal sets
Write for the Boardman-Vogt tensor product on dendroidal sets (see there for details).
Let be the category of dendroidal simplicial sets, equipped with the generalized Reedy model structure induced from the generalized Reedy category .
for the left Bousfield localization at the set of dendroidal spine (“Segal core”) inclusions , to be called the model structure for dendroidal Segal spaces.
A fibrant object in this category is called a dendroidal Segal space.
for the further left Bousfield localization at the set of morphisms , where is the dendroidal groupoidal interval
Call this the model structure for complete dendroidal Segal spaces.
A fibrant object in here is called a complete dendroidal Segal space.
This is (Cisinski-Moerdijk, def. 5.4, def. 6.2).
This is Cisinski-Moerdijk, prop. 5.5, def. 6.2.
By the nature of left Bousfield localization, it is sufficient to show that one localizing set of morphisms is contained in the weak equivalences of the other.
In one direction, it is clear that every inner anodyne morphism of dendroidal sets is a weak equivalence in the localization at the horn inclusions. By the discussion at spine, the spine inclusions are indeed inner anodyne.
Conversely, one checks that the weak equivalences generated by the spine inclusions contain all inner anodyne morphisms (Cisinski-Moerdijk, prop. 2.8)
Write for the tree with a single edge and no vertex. For write for the -corolla, the tree with a single vertex and leaves (and the root).
For a dendroidal Segal space, and for , write for the pullback
The inclusion is a cofibration in . So in this simplicial model category the right vertical morphism in def. 3 are Kan fibrations. These are stable under ordinary pullback, and their ordinary pullback is a homotopy pullback (as discussed there).
A morphism between dendroidal Segal spaces is fully faithful if for all , for all the corresponding morphism
is a homotopy equivalence.
Basic technical properties
As for any category of simplicial presheaves we have
The category is canonically tensored, cotensored and enriched over sSet.
The tensoring is given by the degreewise cartesian product in sSet:
For a dendroidal set, the hom object functor restricted along
is the essentially unique limit-preserving functor such that for all
We will often write ”” also for the tensoring ””.
The essential uniqueness in the last clause follows, because by the co-Yoneda lemma every dendroidal set may be written as a colimit over its cells
For , and , the matching object of at (in the sense of generalized Reedy model structure) is
For a morphism, the relative matching morphism
is the universal morphism induced from the commutativity of the diagram
where the limit is over faces of . By remark 2 this is
By the discussion at dendroidal set, the exponent is the boundary of the tree .
Similarly one finds that the morphism is
We discuss a different set of morphisms, such that the model structure localized at it still coincides with the localization from def. 1. This different localization makes more immediate the Quillen equivalence to the model structure on dendroidal sets that we discuss below in in Relation to dendroidal sets.
Notice that, by the discussion there, the model structure on dendroidal sets, , is a cofibrantly generated model category. Accordingly, there is a set of generating acyclic cofibrations, which we will write
While its existence is known, no explicit description is presently available, but we do know that we may assume that
domain and codomain of all elements of are normal dendroidal sets, hence cofibrant;
it contains a morphism , where is the codiscrete groupoid on two objects, regarded as a unary operad, regarded as a dendroidal set.
The model structure coincides with the left Bousfield localization of at the set of pushout-product morphisms
(Cis-Moer, cor. 6.5).
For every normal dendroidal set , the morphism is a weak equivalence in .
Moreover, every ”-anodyne extension” is a weak cSegal-equivalence, meaning every morphism generated by pushout, transfinite composition and retracts from the pushout-products of with tree boundary inclusions.
(Cis-Moer, prop. 6.3)
For the first statement, it is sufficient to show that the morphism is a weak equivalence regarded in the slice model structure
The category of elements is a “regular skeletal category” in the sense of Cisinski model structure theory. By a lemma there, natural transformations between functors preserving colimits and monomorphisms are componentwise weak equivalences is they are so on representables.
Now does preserve colimits and monomorphisms, and on representables the transformation is a cSegal-equivalence by definition.
The second statement now follows using that is a left proper model category, being the left Bousfield localization of a left proper model category. Using this we have that with also its pushout is a weak equivalence, and so by two-out-of-three with the composite
also the pushout-product itself.
of prop. 3
By this proposition the acyclic cofibrations between normal dendroidal sets are generated from the -anodyne extensions and closure under left cancellation property. Therefore by lemma 1 and two-out-of-three, they are all complete weak equivalences.
Therefore by the pushout-product axiom in the simplicial model category , their powering into a fibration is an acyclic Kan fibration. By Joyal-Tierney calculus this means that all the pushout-products have the left lifting property against fibrations, hence that they are weak equivalences in .
Since therefore all the morphisms are weak equivalences in , it is now sufficient to show, conversely, that the morphisms that define the complete Segal localization are weak equivalence in the localization at these morphisms. For the tree horn inclusions this is clear, since they are among the localizing maps for . For the morphisms observe that
is -anodyne (see Cisinski model structure), hence by 2-out-of-3 its retraction is a weak equivalence.
Fibrations and Cofibrations
A morphism in is a fibration or acyclic fibration, precisely if for all trees , the morphism of hom objects
is a Kan fibration or acyclic Kan fibration, respectively.
The generalized Reedy model structure is a cofibrantly generated model category with set of generating cofibrations
and with set of acyclic generating cofibrations
The statement is (Cisinski-Moerdijk, prop. 5.2). The following proof proceeds in view of remark 5.3 there.
By prop. 4 we have that a morphism in is a fibration or acyclic fibration precisely if for all trees the canonical morphism
is a Kan fibration or acyclic Kan fibration, respectively.
This means equivalently that every diagram
has a lift. A little reflection shows (see Joyal-Tierney calculus) that this, in turn, is equivalent to that every diagram
has a lift.
The statement follows by using the small object argument.
The cofibrations in are precisely the simplicial-degree-wise normal monomorphisms of dendroidal sets (see here).
This is (Cisinski-Moerdijk, cor. 4.3).
The generating inclusions in prop. 5 are the boundary inclusions of representables in the product site , regarded as a Cisinski-generalized Reedy category. By the discussion there, these generate the normal monomorphisms on . But since contains no non-trivial automorphisms, this are just the degreewise dendroidal normal monomorphisms.
It is sufficient to check the pushout-product axiom for the tensoring operation. So for a monomorphism of simplicial sets and a degreewise normal monomorphisms in , we need to check, by prop 6, that the canonical morphism
is a simplicial-degreewise normal monomorphism, which is a weak equivalence if either of or is. Since this coproduct is computed objectwise, this morphism is over the pushout of simplicial sets
where now the tensoring is that of dendroidal sets over sets, which is given by coproduct of dendroidal sets, . It is clear that this is a monomorphism.
Moreover, the image of this morphism contains the image of , which for each summand is the image of . Therefore the dendrices not in this image are also summand-wise not in the image of , hence have trivial stabilizer groups, by the assumption that is a normal monomorphism.
Finally, to see that the above morphism out of the pushout is a weak equivalence if either of or is, use that in the weak equivalences are tree-wise those of simplicial sets. The statement then follows by being a monoidal model category with respect to its cartesian monoidal category structure.
Some of these properties are inherited by the actual model structure for dendroidal complete Segal spaces
The model structures and
Since cofibrations and simplicial enrichment are preserved by left Bousfield localization, this follows from the analogous statements for .
By prop. 4, using .
Let be fibrant. Then the following conditions are equivalent
is a dendroidal Segal space, hence fibrant in ;
for every spine inclusion , the induced morphism is an acyclic Kan fibration;
for every inner horn inclusion , the induced morphism is an acyclic Kan fibration.
This appears as (Cisinski-Moerdijk, cor. 5.6).
By prop. 1 Segal objects are equivalently spine-local and horn-local. By prop. 6 both the spine and the horn inclusion are morphisms between cofibrant objects in . By the general properties of left Bousfield localization and using that is a simplicial model category by prop. 7, it follows that a fibrant object is local with respect to the spine / horn inclusions precisely if powering these into this object, remark 2, is a weak equivalence of simplicial sets. Since moreover the horn and spine inclusions are cofibrations, by prop. 6, this will necessarily be an acyclic Kan fibration (by the dual of the pushout-product axiom in a simplicial model category).
Let be a set of generating acyclic cofibrations for the model structure on dendroidal sets, , chosen such that all domains and codomains are normal, hence cofibrant.
An object is fibrant precisely if
it is fibrant in ;
it has the right lifting property against the set
(Cis-Moer, cor. 6.5)
By prop. 3 and the basic nature of left Bousfield localization.
A morphism between dendroidal Segal spaces is a weak equivalence in , and hence in precisely if its components on the trees and for all , def. 2, are weak homotopy equivalences of simplicial sets.
This appears as (Cisinski-Moerdijk, prop. 5.7).
By general properties of left Bousfield localization, a morphism between local objects is a weak equivalence precisely if it is so already in the unlocalized model structure . There the weak equivalences are the morphisms that are so over every tree. But by prop. 8 these are already implied by weak equivalences over the spines. These are, finally, colimits which happen to be homotopy colimits of and of corollas, and hence it suffices to have weak equivalences over these components in order to have them over all components.
A morphism of dendroidal Segal spaces is a weak equivalence in precisely if it is
fully faithful, def. 4;
essentially surjective in that is a weak equivalence of simplicial sets.
(See also equivalence of categories.)
This appears as (Cisinski-Moerdijk, cor. 5.10).
Being essentially surjective is equivalent to being an equivalence. By prop. 10 it only remains to check that in this situation being fully faithful is equivalent to being an equivalence, for all .
By remark 1, of is a weak equivalence for all then is fully faithful, since weak equivalence are preserved by homotopy pullback.
For the converse, consider for each the inclusion of all input and output colors
and similarly for . Since this evidently hits all connected components of , it is an effective epimorphism in an (∞,1)-category in ∞Grpd. These are stable under homotopy pullback, and so also
is an effective epimorphism, and similarly for . If now is fully faithful, then by the definition of effective epimorphism in an (∞,1)-category, this exhibits as the homotopy colimit of a diagram of equivalences. Hence is itself a weak equivalence.
Relation to other model structures
We discuss the relation to various other model structures for operads. For an overview see table - models for (infinity,1)-operads.
To complete Segal spaces
Write for the tree with a single edge and no non-trivial vertex.
Then slice category of over is evidently equivalent to that of bisimplicial sets
By restriction along this inclusion, the above model structure reproduces the model structure for complete Segal spaces.
To dendroidal sets / quasi-operads
The model structure for dendroidal complete Segal spaces is Quillen equivalent to the model structure on dendroidal sets, whose fibrant objects are the “quasi-operads” (the operadic generalization of quasi-categories).
We discuss in fact two Quillen equivalences, with right adjoints going in both directions:
From quasi-operads to dendroidal complete Segal spaces
From dendroidal complete Segal spaces to quasi-operads
Quasi-operads to dendroidal complete Segal spaces
Recall from complete Segal space the basic example Categories as complete Segal spaces which shows how an ordinary small category is regarded as a complete Segal space by setting
Recall also that this and its generalization to Complete Segal spaces of quasi-categories, amounts to simply forming a double-nerve with respect to the invertible interval object. We consider here the operadic generalization of this construction.
for the cosimplicial simplicial set that in degree is the nerve of the free groupoid on
We use the same symbol for the further prolongation to a cosimplicial dendroidal set
Moreover, we use the same symbol also for
(where is the Boardman-Vogt tensor product on dendroidal sets).
The induced nerve and realization adjunction we denote
This appears as (Cis-Moer, 6.10).
a small category, we have
The nerve and realization adjunction, def. 5 constitutes a Quillen equivalence to the model structure on dendroidal sets.
This appears as (Cis-Moer, prop. 6.11).
First we show that is a left Quillen functor. Since is a monoidal model category, it follows from the pushout-product axiom in that sends the generating (acyclic) cofibrations of from prop. 5 to (acyclic) cofibrations in . Since the cofibrations of are the same as those of , it is sufficient to see that sends the morphisms that define the localization, def. 1, to weak equivalences in . But since these moprhisms are in the image of the inclusion , the functor indeed sends them to themselves, and they are indeed weak equibalences in (since all inner anodyne morphisms are – this gives that is a weak equivalence – and all equivalences in the canonical model structure on operads are – this gives that is).
So far this shows that is left Quillen. To see that it is a Quillen equivalence, use that its composition with the left Quillen functor discussed in the companion section is evidently a Quillen equivalence.
If we write (as here), for normal and fibrant
for the maximax Kan complex inside the internal hom of , then, still for fibrant, we have
Complete dendroidal Segal spaces to quasi-operads
for the evident full subcategory inclusion of dendroidal sets into dendroidal simplicial sets induced by regarding a set as a discrete object in simplicial sets.
This is (Cisinski-Moerdijk, prop. 4.8, theorem 6.6).
The following proof proceeds by passing through another Bousfield localization of a global model structure on dendroidal simplicial sets.
Let be the Reedy model structure on simplicial objects in the model structure on dendroidal sets.
for its left Bousfield localization at the set
We call this the locally constant model structure on simplicial dendroidal sets.
(Cis-Moer, def. 4.6)
constitute a Quillen equivalence.
(Cis-Moer, prop. 4.8)
The set is a set of generators, in that a morphism in is a weak equivalence precisely if under the derived hom space functor is a weak equivalence, for all . Therefore the localization in def. 6 is of the general kind discussed at simplicial model category in the section Simplicial Quillen equivalent models. The above statement is thus a special case of the general theorem discussed there.
The fibrant objects in are precisely
the Reedy fibrant simplicial dendroidal sets ,
such that for every the morphism is a weak equivalence in the model structure on dendroidal sets;
(Cis-Moer, 4.7 ii)+iii)).
The proof is again a special case of the general discussion at Simplicial Quillen equivalent models. Here is a self-contained proof, for completeness.
By standard facts of left Bousfield localization a simplicial dendroidal set is fibrant in the locally constant model structure, def. 6, precisely if it is fibrant in and moreover the derived hom-space functor is a weak equivalence for all .
We compute this derived hom space now in a maybe slightly non-obvious way, in order to get the result in a form that we can compare to the derived hom in . First of all, since the derived hom space only depends on the weak equivalences, we may compute it working with the projective model structure on functors . Here in turn we use as framing of that:
Since is cofibrant in (because is representable and is normal), also is cofibrant in , and so we have that
We claim now that such a resolution is given by (using the notation for core simplicial enrichement of here)
To see that this is indeed Reedy fibrant, notice that this is so precisely if for all the morphism
is fibrant in , which is the case precisely if for all the morphism
is a fibration in . But using that the Reedy fibrant is in particular projectively fibrant (see Reedy model structure), hence that is fibrant (is a quasi-operad) for all , this is indeed the case, by discussion here at model structure on dendroidal sets.
So finally we find that we may compute the derived hom as
The right hand here is now manifestly the derived hom in , from to , computed itself by a framing resolution.
Therefore we have found that is fibrant in the locally constant model structure, def. 6, precisely if for all and the morphisms
are weak equivalences. Since the form a set of generators, this is the case precisely if is already a weak equivalence in .
Now for the equivalence to the second item.
By Joyal-Tierney calculus the morphisms in question are of the form
Since the horn inclusions generate the acyclic monomorphisms, a morphism that has right lifting against this set also has right lifting against
This in turn means that has the right lifting property against the tree boundary inclusions. Since these are the generating cofibrations in the model structure on dendroidal sets, this implies that is an equivalence.
For the converse, it is sufficient to see that all the morphisms in the localizing set are acyclic cofibrations in the locally constant model structure. This follows with the discussion here at model structure on dendroidal sets.
The fibrant objects in are also precisely
the Reedy fibrant simplicial dendroidal sets ,
such that the morphism has the right lifting property against the set of pushout product morphisms
(Cis-Moer, 4.7 i)+iii)).
Since the simplicial horn inclusions generate all acyclic cofibrations in , it follows that a (Reedy fibrant) object which has right lifting against also has right lifting against . This means that is an acyclic fibration for all , in particular a weak equivalence, hence is fibrant in the locally constant structure by 15.
Conversely, one finds with … and … that the morphisms in the above set are acyclic cofibrations in the locally constant model structure, hence if an object is locally constant fibrant, it lifts against these.
Under the canonical identification of categories
the two model structures , def. 6 and , def. 1, coincide.
By a standard fact (see at model category the section Redundancy of the axioms) it is sufficient to show that the cofibrations and the fibrant objects coincide.
By prop. 5 we know the generating cofibrations of . By the same kind of argument we find the cofibrations of , and hence of :
by definition of Reedy model structure, a morphism here is an acyclic fibration if for all the morphism
is an acyclic fibration in . Since has generating cofibrations given by the set of tree boundary inclusions , one finds as in the proof of prop. 5 that is an acyclic fibration precisely if it has the right lifting property against the morphisms in the set
Therefore the cofibrations in the two model structures do coincide.
(Notice that a similar statement holds for the acyclic cofibrations, only that the generating set of acyclic cofibrations in is, while known to exist, not known explicitly.)
Next, to see that the fibrant objects also coincide, write again for a choice of set of generating acyclic cofibrations for between normal dendroidal sets.
By prop. 16 the fibrant objects of are those that
are Reedy fibrant over , meaning that they have the right lifting property against
are local, meaning, by prop. 16, that they have the right lifting property against
On the other hand, the fibrant objects in are those
that are Reedy fibrant over , meaning that they have the right lifting property against
are Segal local, meaning by prop. 1 that they have right lifting against
are complete Segal local, meaning by prop. 9 that they have right lifting property against
The union of the three respective sets coincides in both cases.
of theorem 1
Combining prop. 14 and prop. 17 we have a total Quillen equivalence
To Segal operads
(…) model structure for Segal operads
The model structure for dendroidal complete Segal spaces was introduced in