symmetric monoidal (∞,1)-category of spectra
Throughout, let be a symmetric monoidal model category equipped with an interval object as discussed at model structure on operads and at Boardman-Vogt resolution. We consider multi-coloured symmetric operads (symmetric multicategories) enriched in .
The BV resolution commutes with symmetrization: if , then
Therefore we describe in the following explicitly the BV-resolution of planar trees, that of non-planar trees then being the symmetrization of that construction.
For , and a tuple of colours (edges) of , notice that the set of operations is the set of those subtrees such that is the set of leaves and is the root of .
First regard as a topological operad (with a discrete space of operations in each degree). The corresponding Boardman-Vogt resolution of is the topological operad whose topological space of operations is the space of labeled trees as follows.
A point is a set of lengths , one for each inner edge of . (…)
where the coproduct ranges over the set of subtrees , as just discussed (which therefore is either the singleton set or is empty), and where is the set of inner edges of .
In general, when is regarded as an -operad, we have
where is the given interval object.
The composition operations in
correspond to grafting of trees and “assigning unit length to the new inner edge”. On the components as discussed above it is given by
on an inner face map the component of on a subtree of that contains the edge is the product of the inclusion
with the identity on
(meaning: if the label of an inner edge in a tree is 0, then the operations that it connects may be composed);
on a degenracy map that sends two given unary vertices to a single one, the component of on subtrees containing these removes one of the factors by the map
that sends to .
This is discussed in section 4.2 of (Cisinski-Moerdijk).
By the general discussion at nerve and realization, the functor
The dendroidal homotopy coherent nerve functor is the functor
Let be such that each object of operations is fibrant in . Then its homotopy coherent nerve is a dendroidal inner Kan complex.
Consider a tree and an inner edge of it. For each morphism we need to find a filler in
By the definition of dendroidal nerve, this is equivalently a diagram
The undetermined component to fill is that corresponding to the subtree of which is itself. According to prop. 2 on this the operad has the component
The map has to send this into while being compatible with the given faces. By prop. 4 this means that its precomposition with all the inclusions
is fixed. Moreover, the assignment needs to be compatible with the composition operations, which by prop. 3 means that also the precomposition with all the maps
is fixed. In total this means that the components of need to form an extension of the form
in , where
for the Yoneda extension of
By the general lore of nerve and realization we have
is left adjoint to
For , the counit
is essentially the Boardman-Vogt resolution of .
For a cofibrant and fibrant , the unit
may be viewed as a “strictification” of the (infinity,1)-operad given by , in that , being a simplicial operad, has strictly associative composition.
By the general properties of the Boardman-Vogt resolution (but also immediately checked directly) we have
There is a natural transformation
(natural in the tree ), which is a bijection on colors and is on the components of prop. 2 the canonical map
Each is hence a weak equivalence of simplicial operads. In particular
is an isomorphism.
This induces hence a natural transformation
to the left adjoint of the ordinary dendroidal nerve (the “fundamental operad” construction).
For every dendroidal set , the natural morphism
is an isomorphism of simplicial operads.
This appears as Cisinski-Moerdijk, prop. 4.4.
We discuss some input to this statement.
This appears as Cisinski-Moerdijk, prop. 4.5.
Observe that the morphism classes in question are, as discussed at dendroidal set, the saturated classes generated by the dendroidal boundary inclusions and by the dendroidal horn inclusions, respectively.
Since is left adjoint, it therefore suffices to check the statement on these generating inclusions. Moreover, by construction, on trees coincides with the Boardman-Vogt resolution of the operads free on these trees.
It follows that the generating inclusions are sent by to morphisms of simplicial operads which are
bijective on objects;
isomorphisms on all but one simplicial set of operations: that corresponding to the maximal subtree;
on this remaining simplicial set of operations a product of identities with cofibrations of simplicial sets (monomorphisms), and following through the combinatorics shows that these are acyclic for the case of anodyne extensions.
It follows that these morphisms of simplicial operads have the left lifting property again operation-object-wise Kan fibrations (there is no further composition to be respected, since the maximal subtree operation has no further non-trivial composites), and hence against the fibrations of the model structure on sSet-operads.
Prop.6 is, in turn, a direct consequence of this.
is isomorphic to the Boardman-Vogt resolution of .
In particular, therefore, there is a natural isomorphism
(Here we are using that on a discrete operad the homotopy coherent dendroidal nerve trivially coincides with the ordinary dendroidal nerve .)
By inspection of the relevant formulas.
For a cofibrant and fibrant dendroidal set , the -unit
is an equivalence.
Since composition of operations in a simplicial operad is strictly associative, this may be understood as producing a semi-strictification of the -operad .
The entries of the following table display models, model categories, and Quillen equivalences between these that present the (∞,1)-category of (∞,1)-categories (second table), of (∞,1)-operads (third table) and of -monoidal (∞,1)-categories (fourth table).
|enriched (∞,1)-category||internal (∞,1)-category|
|SimplicialCategories||homotopy coherent nerve||SimplicialSets/quasi-categories||RelativeSimplicialSets|
|SimplicialOperads||homotopy coherent dendroidal nerve||DendroidalSets||RelativeDendroidalSets|
The fact that the homotopy coherent nerve if a locally fibrant operad is inner Kan is shown in section 7 of
The Quillen adjunction properties of the homotopy coherent dendroidal nerve are discussed in section 4 of
Lecture notes on these two topics are in section 6 and 9 of