homotopy theory, (∞,1)-category theory, homotopy type theory
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on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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symmetric monoidal (∞,1)-category of spectra
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
the W-construction on topological operads (BoardmanVogt);
the cobar-bar resolution of chain complex operads (GezlerJones, GinzburgKapranov);
the Godement simplicial resolution (Godement)
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
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 $P$.
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
where the tensor product runs over all internal edges of $T$. For $D \subset E(T)$ a subset of internal edges, let
The acyclic cofibration $0 \to H$ induces an acyclic cofibration
and, by the pushout-product axiom, an acyclic cofibration
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
is a cofibration. Consider for each $T$ the pushout
This induces a univesal morphism
and by the pushout-product axiom in the monoidal model category $\mathcal{E}$ this, too, is a cofibration.
Define $W(H,P)$ by induction. Start with setting
Assume that in each induction step we are given morphisms
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
Then in the induction step we define for each $k \in \mathbb{N}$ the object $W_n(H,P)$ by the pushout
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
One shows that this collection naturally carries the structure of an operad, etc. pp.
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.
The W-construction on topological operads is in
The cobar-bar resolution of chain complex operads is in
The Godement simplicial resolution is in
The generalization to operads enriched in any monoidal category with a suitable interval object is in