THE BOARDMAN-VOGT RESOLUTION OF OPERADS IN MONOIDAL MODEL CATEGORIES

By C. Berger, Ieke Moerdijk. 2005

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We extend the W-construction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for well-pointed �-cofibrant operads. The standard simplicial resolution of Godement as well as the cobar-bar chain res- olution are shown to be particular instances of this generalised W-construction. In (2), sufficient conditions are given for the existence of a model structure on operads in an arbitrary (symmetric) monoidal model category. These conditions imply in particular that each operad may be resolved by a cofibrant operad. This general existence result leaves open the relation to various explicit resolutions of operads in literature, like the W-construction of Boardman and Vogt (3) for topo- logical operads (special PROP's in their terminology), the cobar-bar resolution for chain operads (8), (9), and the standard simplicial resolution of Godement (10) aris- ing from the free-forgetful adjunction between pointed collections and operads. The purpose of this article is to introduce a general, inductively defined W- construction, and to prove that it provides a functorial cofibrant resolution for oper- ads whose underlying collection is cofibrant and well-pointed. Our W-construction applies in an arbitrary monoidal model category which comes equipped with a suitable interval, and specialises to each of the above mentioned resolutions by an appropriate choice of ambient model category and interval. It is completely uniform, and has remarkable functorial and homotopical properties. The main idea of Boardman and Vogt was to enrich the free operad construc- tion by assigning lengths to edges in trees. Surprisingly, all we need to carry out this topological idea in general, is an abstract interval with suitable algebraic and homotopical properties. We call the underlying algebraic object a segment (i.e. an augmented associative monoid with absorbing element), and use the term in- terval for segments which, in addition, induce cylinder-objects in Quillen's model- theoretic sense (17). The real unit-interval equipped with its maximum operation is an example of such an interval for compactly generated spaces, while the stan- dard representable 1-simplex is an interval for simplicial sets. Moreover, the model category of chain complexes has an interval with segment structure because the normalised chain functor transfers the required structure from simplicial sets to chain complexes. We show that for reduced chain operads, the W-construction is isomorphic to the cobar-bar resolution, cf. Kontsevich and Soibelman (13). The fact that such a

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

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Citations in the corpus

Dendroidal Sets (2007, methodology): “We briefly recall the basic definitions, and refer to Berger and Moerdijk [3] for a more extensive discussion.”
The rational homotopy of mapping spaces of E${}_n$ operads (2017, background): “The morphism FM Q n → K Q n which we get in this pushout construction is a weak-equivalence because the model category of Λ-operads in topological spaces satisfies good left properness properties, like the model category of ordinary symmetric operads (see [3]).”