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\begin{document}

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\section*{Boardman-Vogt resolution}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory}

[[!include model category theory - contents]]

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{context_2}{}\section*{{Context}}\label{context_2}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{over_}{Over $Top$}\dotfill \pageref*{over_} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{Boardman-Vogt resolution} or \emph{W-construction} is a particular choice of cofibrant [[resolution]] of [[topological operads]] (or [[simplicial operads]] or similar). It is closely related to the operation of forming the [[dendroidal homotopy coherent nerve]]. Its restriction to [[Top]]-[[enriched categories]] / [[simplicial categories]] is accordingly closely related to the ordinary [[homotopy coherent nerve]].

The construction is an variant of the \emph{[[free operad]]} construction, where the free composites of operations are further labeled by ``distances'' such that for vanishing distance the free composite is replaced with the actual composite.

The original Boardman-Vogt resolution over [[Top]] generalizes naturally to a cofibrant [[resolution]] in the [[model structure on operads]] for operads enriched over any suitable [[monoidal model category]] that is equipped with a suitable [[comonoid|comonoidal]] [[interval object]]. In this general form the construction subsumes

\begin{itemize}%
\item the W-construction on topological operads (\hyperlink{BoardmanVogt}{BoardmanVogt});


\item the cobar-bar resolution of chain complex operads (\hyperlink{GetzlerJones}{GezlerJones}, \hyperlink{GinzburgKapranov}{GinzburgKapranov});


\item the Godement simplicial resolution (\hyperlink{Godement}{Godement})



\end{itemize}
\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

\hypertarget{over_}{}\subsubsection*{{Over $Top$}}\label{over_}

Over [[Top]] the BV-resolution works as follows:

for $P$ a [[topological operad]], its [[free operad]] $F_*(P)$ has as $n$-ary operations all [[trees]] with $n$ inputs, with each vertex of valence $k+1$ labeled by an element in $P(k)$. Composition is given by grafting of trees.

The operad $W(P)$ is obtained from this by in addition

\begin{itemize}%
\item labeling the inner edges of any tree by [[real number]]s $\ell \in [0,1]$;


\item identifying trees one of whose edges has length 0 with the tree with that edge removed and with the correspnding operad-operation labels composed.


\item equipping the set of labeled trees with the corresponding quotient of the product topology, to make it into a [[topological space]].



\end{itemize}
There is an evident operad morphism $F_*(P) \to W(P)$ obtained by regarding each edge of a tree as being of length 1, and there is an evident morphism $W(P) \to P$ obtained by forgetting all trees and sending their operad-operation-labels to their composite.

The composition

\begin{displaymath}
F_*(P) \to W(P) \to P
\end{displaymath}
is the [[unit of an adjunction|counit]] of the [[free functor|free]]/forgetful [[adjunction]] between operads and their underlying [[collection]]s and if $P$ is degreewise sufficiently nice, this factors that counit as a cofibration followed by a weak equivalence and exhibits $W(P)$ as a cofibrant [[resolution]] of $P$.

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Write $\mathbb{T}$ for the [[groupoid]] of planar [[trees]] and non-planar [[isomorphism]].

Fix a suitable [[interval object]] $H$, as described at [[model structure on operads]].

For $T$ a [[tree]], write

\begin{displaymath}
H(T) := \bigotimes_{e \in E(T)}H
  \,,
\end{displaymath}
where the tensor product runs over all \emph{internal} edges of $T$. For $D \subset E(T)$ a subset of internal edges, let

\begin{displaymath}
H_D(T) = \bigotimes_{ E(T)\setminus D} H
  \,.
\end{displaymath}
The acyclic cofibration $0 \to H$ induces an acyclic cofibration

\begin{displaymath}
H_D(T) \hookrightarrow H(T)
\end{displaymath}
and, by the [[pushout-product axiom]], an acyclic cofibration

\begin{displaymath}
H^-(T) := \coprod_{D \neq \emptyset} H_D(T) \hookrightarrow H(T)
  \,.
\end{displaymath}
In a similar fashion, for $P$ an operad, write $P(T)$ for the tensor product of one copy of its objects of $n$-ary operation for each $n$-ary vertex in $T$, and $P^-(T)$ for the coproduct over all such tensor products where at least one, maybe more, \emph{unary} vertices are omitted. Also the canonical

\begin{displaymath}
P^-(T) \hookrightarrow P(T)
\end{displaymath}
is a cofibration. Consider for each $T$ the [[pushout]]

\begin{displaymath}
\itexarray{
    H^-(T) \otimes P^-(T) &\to& H^-(T) \otimes P(T)
    \\
    \downarrow && \downarrow
    \\
    H(T) \otimes P^-(T) &\to&
    (H \otimes P)^-(T)
  }
  \,.
\end{displaymath}
This induces a univesal morphism

\begin{displaymath}
(H \otimes P)^-(T) \to H(T) \otimes P(T)
\end{displaymath}
and by the [[pushout-product axiom]] in the [[monoidal model category]] $\mathcal{E}$ this, too, is a cofibration.

\begin{udefn}
Define $W(H,P)$ by induction. Start with setting

\begin{displaymath}
W_0(H,P) := P
  \,.
\end{displaymath}
Assume that in each induction step we are given morphisms

\begin{displaymath}
(H(S) \otimes P(S)) \otimes_{Aut(S)} I[\Sigma_n] \to W_{k-1}(H,P)(n)
\end{displaymath}
for all trees $S$ with less than $k$ internal edges. Using the composition operation in the operad $P$ to compose two operation when the edge connecting them carries no $H$-label, we obtain from this a morphism

\begin{displaymath}
\alpha_T^- : (H \otimes P)^-(T) \otimes_{Aut(T)} I[\Sigma_n] \to W_{k-1}(H,P)(n)
\end{displaymath}
Then in the induction step we define for each $k \in \mathbb{N}$ the object $W_n(H,P)$ by the [[pushout]]

\begin{displaymath}
\itexarray{
     \coprod_{[T], T \in \mathbb{T}(n,k)}
      (H \otimes P)^-(T)
      \otimes_{Aut(T)}
       I[\Sigma_n]
     &\stackrel{\coprod \alpha_T^-}{\to}&
     W_{k-1}(H,P)(n)
     \\
     \downarrow && \downarrow
     \\
     \coprod_{[T], T \in \mathbb{T}(n,k)}
      (H \otimes P)(T)
      \otimes_{Aut(T)}
       I[\Sigma_n]
     &\stackrel{\coprod \alpha_T^-}{\to}&
     W_{k}(H,P)(n)
  }
 \,,
\end{displaymath}
where $\mathbb{T}$ is the subcategory of trees with precisely $n$ inputs and $k$ internal edges.

The bottom morphism we feed back into the induction procedure.

This gives a sequence of collections, and the W-resolution is its [[colimit]]

\begin{displaymath}
W(H,P) := {\lim_{\to}}_k W_k(H,P)
  \,.
\end{displaymath}
\end{udefn}
One shows that this collection naturally carries the structure of an operad, etc. pp.

\begin{uremark}
The object $(H \otimes P)(T)$ is to be thought of as the space whose points are tuples conisting of one operation in $P$ per vertex in $T$, of that arity, and of labels in $H$ assigned to the inner edges in $T$.

The object $(H \otimes P)^-(T)$ is a similar space, but where some of the labels on the inner edges are omitted.

The above pushout identifies points that contain lables of inner edges that are 0 with points in one $W$-stratum below where that edge (or rather its label) is simply omitted and the corresponding operations composed.

\end{uremark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[Jim Stasheff]]`s [[A-∞ operad]] is the relative Boardman-Vogt resolution $W([0,1], I_* \to Assoc)$ of [[Assoc]] in [[Top]] where $I_*$ is the operad for [[pointed object]]s (\hyperlink{BergerMoerdijkAlgebras}{Berger-Moerdijk}).

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The W-construction on topological operads is in

\begin{itemize}%
\item [[Michael Boardman]] and [[Rainer Vogt]], \emph{Homotopy invariant algebraic structures on topological spaces} , Lect. Notes Math. 347 (1973).

\end{itemize}
The cobar-bar resolution of chain complex operads is in

\begin{itemize}%
\item [[Ezra Getzler]], J.D.S. Jones, \emph{Operads, homotopy algebra, and iterated integrals for double loop spaces} , (1995)


\item [[Victor Ginzburg]], [[Mikhail Kapranov]], \emph{Koszul duality for operads} , Duke Math. J. 76 (1994) 203--272.



\end{itemize}
The Godement simplicial resolution is in

\begin{itemize}%
\item R. Godement, \emph{Topologie alg\'e{}brique et th\'e{}orie des faisceaux} , no. 13, Publ. Math. Univ. Strasbourg, Hermann, Paris, 1958.

\end{itemize}
The generalization to [[operad]]s enriched in any monoidal category with a suitable interval object is in

\emph{ [[Clemens Berger]], [[Ieke Moerdijk]], \emph{The Boardman-Vogt resolution of operads in monoidal model categories} , Topology 45 (2006), 807--849. (\href{http://math.unice.fr/~cberger/BV.pdf}{pdf})}

\begin{itemize}%
\item [[Clemens Berger]], [[Ieke Moerdijk]], \emph{Resolution of coloured operads and rectification of homotopy algebras} (\href{http://arxiv.org/abs/math/0512576}{arXiv:math/0512576})

\end{itemize}
The Boardman-Vogt-resolution for the operad for [[local nets of observables]] in [[homotopical algebraic quantum field theory]] (\href{homotopical+algebraic+quantum+field+theory#BeniniSchenkelWoike17}{Benini-Schenkel-Woike 17}) is discussed in

\begin{itemize}%
\item [[Donald Yau]], \emph{Homotopical Quantum Field Theory} (\href{https://arxiv.org/abs/1802.08101}{arXiv:1802.08101})

\end{itemize}
[[!redirects Boardman-Vogt resolutions]]

[[!redirects W-construction]] [[!redirects W-constructions]]

[[!redirects relative Boardman-Vogt resolution]]

[[!redirects relative Boardman-Vogt resolutions]]



\end{document}