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Boardman-Vogt resolution

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Higher algebra

Context

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 an 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

Definition

Over TopTop

Over Top the BV-resolution works as follows:

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

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

  • labeling the inner edges of any tree by real numbers [0,1]\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)W(P)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)PW(P) \to P obtained by forgetting all trees and sending their operad-operation-labels to their composite.

The composition

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

is the counit of the free/forgetful adjunction between operads and their underlying collections and if PP is degreewise sufficiently nice, this factors that counit as a cofibration followed by a weak equivalence and exhibits W(P)W(P) as a cofibrant resolution of PP.

General

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

Fix a suitable interval object HH, as described at model structure on operads.

For TT a tree, write

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

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

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

The acyclic cofibration 0H0 \to H induces an acyclic cofibration

H D(T)H(T) H_D(T) \hookrightarrow H(T)

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

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

In a similar fashion, for PP an operad, write P(T)P(T) for the tensor product of one copy of its objects of nn-ary operation for each nn-ary vertex in TT, and P (T)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)P(T) P^-(T) \hookrightarrow P(T)

is a cofibration. Consider for each TT the pushout

H (T)P (T) H (T)P(T) H(T)P (T) (HP) (T). \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

(HP) (T)H(T)P(T) (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)W(H,P) by induction. Start with setting

W 0(H,P):=P. W_0(H,P) := P \,.

Assume that in each induction step we are given morphisms

(H(S)P(S)) Aut(S)I[Σ n]W k1(H,P)(n) (H(S) \otimes P(S)) \otimes_{Aut(S)} I[\Sigma_n] \to W_{k-1}(H,P)(n)

for all trees SS with less than kk internal edges. Using the composition operation in the operad PP to compose two operation when the edge connecting them carries no HH-label, we obtain from this a morphism

α T :(HP) (T) Aut(T)I[Σ n]W k1(H,P)(n) \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 kk \in \mathbb{N} the object W n(H,P)W_n(H,P) by the pushout

[T],T𝕋(n,k)(HP) (T) Aut(T)I[Σ n] α T W k1(H,P)(n) [T],T𝕋(n,k)(HP) (T) Aut(T)I[Σ n] α T W k(H,P)(n), \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 𝕋\mathbb{T} is the subcategory of trees with precisely nn inputs and kk 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 kW k(H,P). 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 (HP)(T)(H \otimes P)(T) is to be thought of as the space whose points are tuples conisting of one operation in PP per vertex in TT, of that arity, and of labels in HH assigned to the inner edges in TT.

The object (HP) (T)(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 WW-stratum below where that edge (or rather its label) is simply omitted and the corresponding operations composed.

Examples

References

The W-construction on topological operads is in

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)

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

Revised on March 6, 2012 18:23:51 by Urs Schreiber (82.169.65.155)