symmetric monoidal (∞,1)-category of spectra
The dendroidal homotopy coherent nerve is an operadic generalization of the standard homotopy coherent nerve. It is a functor
from the category of Top-operads to that of dendroidal sets, given by
where $T$ is an object of the tree category, regarded as a free symmetric operad, and $W_H(T)$ is its Boardman-Vogt resolution.
Throughout, let $\mathcal{E}$ be a symmetric monoidal model category equipped with an interval object $H$ as discussed at model structure on operads and at Boardman-Vogt resolution. We consider multi-coloured symmetric operads (symmetric multicategories) enriched in $\mathcal{E}$.
Standard examples are $\mathcal{E} =$ Top, sSet, which yields topological operads and simplicial operads, respectively.
We discuss in detail what the Boardman-Vogt resolution of operads free on an object in the tree category $\Omega$ is like (see dendroidal set for details on trees as operads).
Let
be the symmetrization functor, the left adjoint to the forgetful functor from symmetric operads to planar operads.
The BV resolution commutes with symmetrization: if $T = Symm(\bar T)$, 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 $T \in \Omega_{planar}$, and $(e_1, \cdots, e_n; e)$ a tuple of colours (edges) of $T$, notice that the set of operations $T(e_1, \cdots, e_n, e)$ is the set of those subtrees $V \subset T$ such that $\{e_1, \cdots, e_n\}$ is the set of leaves and $e$ is the root of $V$.
First regard $T$ as a topological operad (with a discrete space of operations in each degree). The corresponding Boardman-Vogt resolution $W(T)$ of $T$ is the topological operad whose topological space of operations $W(T)(e_1, \cdots, e_n; e)$ is the space of labeled trees as follows.
A point is a set of lengths $\ell(e) \in [0,1]$, one for each inner edge $e \in I(T)$ of $T$. (…)
Hence
where the coproduct ranges over the set of subtrees $V$, as just discussed (which therefore is either the singleton set or is empty), and where $i(V)$ is the set of inner edges of $V$.
Regard then $T$ as a simplicial operad. The corresponding Boardman-Vogt resolution $W(T)$ of $T$ is the simplicial operad whose simplicial sets of operations are
In general, when $T$ is regarded as an $\mathcal{E}$-operad, we have
where $H$ is the given interval object.
The composition operations in $W(T)$
correspond to grafting of trees $T_\sigma, T_\rho \subset T$ and “assigning unit length to the new inner edge”. On the components as discussed above it is given by
The BV-resolution of trees extends to a functor on the category of simplicial operad
as follows
on an inner face map $\delta_e : \partial^e\Omega[T] \to \Omega[T]$ the component of $W(\delta)$ on a subtree $V$ of $T$ that contains the edge $e$ is the product of the inclusion
with the identity on $H^{i(V)-\{e\}}$
(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 $\sigma$ that sends two given unary vertices to a single one, the component of $W(\sigma)$ on subtrees containing these removes one of the factors $H$ by the map
given by the interval object $H$. For both simplicial operads and topological operads this may be taken to be the map
that sends $(x,y)$ to $max(x,y)$.
This is discussed in section 4.2 of (Cisinski-Moerdijk).
By the general discussion at nerve and realization, the functor
from prop. 4 induces a nerve functor as follows.
The dendroidal homotopy coherent nerve functor is the functor
given by
Its left adjoint (the corresponding “geometric realization”) we denote
When restricted to $\mathcal{E}$-enriched categories, the dendroidal homotopy coherent nerve reproduces the homotopy coherent nerve of enriched categories
In particular for $\mathcal{E} =$ Top / sSet it reproduces the original definition of homotopy coherent nerve.
Let $P \in \mathcal{E} Operad$ be such that each object of operations is fibrant in $\mathcal{E}$. Then its homotopy coherent nerve $hcN_d(P)$ is a dendroidal inner Kan complex.
This is (Moerdijk-Weiss, theorem 7.1). This statement will also follow as a corollary from prop. 10 below.
Consider a tree $T$ and an inner edge $e$ of it. For each morphism $\phi : \Lambda^e[T] \to X$ we need to find a filler $\psi$ in
Write $\Lambda^e[T] = \cup_{i \neq e}\partial^{i \neq e} \Omega[T]$.
By the definition of dendroidal nerve, this is equivalently a diagram
The undetermined component to fill is that corresponding to the subtree $\tau$ of $T$ which is $T$ itself. According to prop. 2 on this the operad $W(\Omega[T])$ has the component
The map $\hat \psi$ has to send this into $X$ 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 $\hat \psi$ need to form an extension of the form
in $\mathcal{E}$, where
One sees that the left vertical morphism is an acyclic cofibration, by the pushout-product axiom in the monoidal model category $\mathcal{E}$. Therefore by the assumption that $X(\tau)$ is fibrant, such a lift does exist.
Write
for the Yoneda extension of
hence for the functor from dendroidal sets to simplicial operads, which
preserves colimits;
on trees $\Omega \hookrightarrow Operad \hookrightarrow sSet Operad$ is given by the Boardman-Vogt resolution as discussed above.
By the general lore of nerve and realization we have
$W_!$ is left adjoint to $hcN_d$
For $P \in sSet Operad$, the counit
is essentially the Boardman-Vogt resolution of $P$.
For a cofibrant and fibrant $X \in dSet$, the unit
may be viewed as a “strictification” of the (infinity,1)-operad given by $X$, in that $W_!(X)$, 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 $T \in \Omega$), which is a bijection on colors and is on the components of prop. 2 the canonical map
Each $\epsilon_T$ is hence a weak equivalence of simplicial operads. In particular
is an isomorphism.
This induces hence a natural transformation
to the left adjoint $\tau_d$ of the ordinary dendroidal nerve (the “fundamental operad” construction).
For every dendroidal set $X$, the natural morphism
is an isomorphism of simplicial operads.
This appears as Cisinski-Moerdijk, prop. 4.4.
The functors $\pi_0 W_!$ and $\tau_d$, being left adjoints, both preserve small colimits. Therefore it is sufficient to check the statement for $X = \Omega[T]$ a tree. There it is prop. 8.
The adjunction $(W_! \dashv hcN_d)$ from above is a Quillen equivalence between the model structure on operads over Top/sSet and the model structure on dendroidal sets.
We discuss some input to this statement.
The functor $W_! : dSet \to sSet Operad$ sends normal monomorphisms to cofibrations, and inner anodyne extensions to acyclic cofibrations in the model structure on sSet-operads.
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 $W_!$ is left adjoint, it therefore suffices to check the statement on these generating inclusions. Moreover, by construction, on trees $W_!$ coincides with the Boardman-Vogt resolution of the operads free on these trees.
It follows that the generating inclusions are sent by $W_!$ 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.
Let $P$ be an operad in Set, regarded as an $\mathcal{E}$-operad. Then the $(W_! \dashv hcN_d)$-counit
is isomorphic to the Boardman-Vogt resolution $W_H(P)$ of $P$.
In particular, therefore, there is a natural isomorphism
(Here we are using that on a discrete operad $P$ the homotopy coherent dendroidal nerve trivially coincides with the ordinary dendroidal nerve $N_d$.)
By inspection of the relevant formulas.
For a cofibrant and fibrant dendroidal set $X$, the $(W_! \dashv hcN_d)$-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 $\infty$-operad $X$.
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 $\mathcal{O}$-monoidal (∞,1)-categories (fourth table).
general pattern | ||||
---|---|---|---|---|
strict enrichment | (∞,1)-category/(∞,1)-operad | |||
$\downarrow$ | $\downarrow$ | |||
enriched (∞,1)-category | $\hookrightarrow$ | internal (∞,1)-category | ||
(∞,1)Cat | ||||
SimplicialCategories | $-$homotopy coherent nerve$\to$ | SimplicialSets/quasi-categories | RelativeSimplicialSets | |
$\downarrow$simplicial nerve | $\downarrow$ | |||
SegalCategories | $\hookrightarrow$ | CompleteSegalSpaces | ||
(∞,1)Operad | ||||
SimplicialOperads | $-$homotopy coherent dendroidal nerve$\to$ | DendroidalSets | RelativeDendroidalSets | |
$\downarrow$dendroidal nerve | $\downarrow$ | |||
SegalOperads | $\hookrightarrow$ | DendroidalCompleteSegalSpaces | ||
$\mathcal{O}$Mon(∞,1)Cat | ||||
DendroidalCartesianFibrations |
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