nLab Boardman-Vogt resolution

Idea

A Boardman-Vogt resolution or 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 a variant of the 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 comonoidal interval object. In this general form the construction subsumes

• the W-construction on topological operads (BoardmanVogt);

• the cobar-bar resolution of chain complex operads (GezlerJones, GinzburgKapranov);

• the Godement simplicial resolution (Godement)

Definition

Over $Top$

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

• labeling the inner edges of any tree by real numbers $\ell \in [0,1]$;

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

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

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

$$F_*(P) \to W(P) \to P$$

is the counit of the free/forgetful adjunction between operads and their underlying collections 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 $P$.

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

$$H(T) := \bigotimes_{e \in E(T)}H \,,$$

where the tensor product runs over all internal edges of $T$. For $D \subset E(T)$ a subset of internal edges, let

$$H_D(T) = \bigotimes_{ E(T)\setminus D} H \,.$$

The acyclic cofibration $0 \to H$ induces an acyclic cofibration

$$H_D(T) \hookrightarrow H(T)$$

and, by the pushout-product axiom, an acyclic cofibration

$$H^-(T) := \coprod_{D \neq \emptyset} H_D(T) \hookrightarrow H(T) \,.$$

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, unary vertices are omitted. Also the canonical

$$P^-(T) \hookrightarrow P(T)$$

is a cofibration. Consider for each $T$ the pushout

$$\array{ H^-(T) \otimes P^-(T) &\to& H^-(T) \otimes P(T) \\ \downarrow && \downarrow \\ H(T) \otimes P^-(T) &\to& (H \otimes P)^-(T) } \,.$$

This induces a univesal morphism

$$(H \otimes P)^-(T) \to H(T) \otimes P(T)$$

and by the pushout-product axiom in the monoidal model category $\mathcal{E}$ this, too, is a cofibration.

Definition

Define $W(H,P)$ by induction. Start with setting

$$W_0(H,P) := P \,.$$

Assume that in each induction step we are given morphisms

$$(H(S) \otimes P(S)) \otimes_{Aut(S)} I[\Sigma_n] \to W_{k-1}(H,P)(n)$$

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

$$\alpha_T^- : (H \otimes P)^-(T) \otimes_{Aut(T)} I[\Sigma_n] \to W_{k-1}(H,P)(n)$$

Then in the induction step we define for each $k \in \mathbb{N}$ the object $W_n(H,P)$ by the pushout

$$\array{ \coprod_{[T], T \in \mathbb{T}(n,k)} (H \otimes P)^-(T) \otimes_{Aut(T)} I[\Sigma_n] &\stackrel{\coprod \alpha_T^-}{\to}& W_{k-1}(H,P)(n) \\ \downarrow && \downarrow \\ \coprod_{[T], T \in \mathbb{T}(n,k)} (H \otimes P)(T) \otimes_{Aut(T)} I[\Sigma_n] &\stackrel{\coprod \alpha_T^-}{\to}& W_{k}(H,P)(n) } \,,$$

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

$$W(H,P) := {\lim_{\to}}_k W_k(H,P) \,.$$

One shows that this collection naturally carries the structure of an operad, etc. pp.

Remark

The object $(H \otimes P)(T)$ is to be thought of as the space whose points are tuples consisting 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.

Examples

• 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 objects (Berger-Moerdijk).

References

The W-construction on topological operads is in

• Michael Boardman and Rainer Vogt, Homotopy invariant algebraic structures on topological spaces , Lect. Notes Math. 347 (1973).

The cobar-bar resolution of chain complex operads is in

• Ezra Getzler, J.D.S. Jones, Operads, homotopy algebra, and iterated integrals for double loop spaces , (1995)

• Victor Ginzburg, Mikhail Kapranov, Koszul duality for operads , Duke Math. J. 76 (1994) 203–272.

The Godement simplicial resolution is in

• R. Godement, Topologie algébrique et théorie des faisceaux , no. 13, Publ. Math. Univ. Strasbourg, Hermann, Paris, 1958.

The generalization to operads enriched in any monoidal category with a suitable interval object is in

Clemens Berger, Ieke Moerdijk, The Boardman-Vogt resolution of operads in monoidal model categories , Topology 45 (2006), 807–849. (pdf)

• Clemens Berger, Ieke Moerdijk, Resolution of coloured operads and rectification of homotopy algebras (arXiv:math/0512576)

The Boardman-Vogt-resolution for the operad for local nets of observables in homotopical algebraic quantum field theory (Benini-Schenkel-Woike 17) is discussed in

• Donald Yau, Homotopical Quantum Field Theory (arXiv:1802.08101)