Resolution of coloured operads and rectification of homotopy algebras

Abstract

Abstract. We provide general conditions under which the algebras for a coloured operad in a monoidal model category carry a Quillen model structure, and prove a Comparison Theorem to the effect that a weak equivalence between suitable such operads induces a Quillen equivalence between their categories of algebras. We construct an explicit Boardman-Vogt style cofibrant resolution for coloured operads, thereby giving a uniform approach to algebraic structures up to homotopy over coloured operads. The Comparison Theorem implies that such structures can be rectified.Many algebraic stuctures are parametrised by operads, and deformations of such structures are then controlled by suitable resolutions of operads. Early examples in topology are the W-construction of Boardman and Vogt [7], and the explicit A ∞ -and E ∞ -operads of Stasheff [30] [3] for operads in general monoidal model categories. When such a homotopy theory is available, one can define for any operad P the notion of P -algebra up to homotopyor homotopy P -algebra -in a homotopy invariant way as an algebra over a cofibrant resolution of P . Indeed, all important instances of homotopy P -algebras occurring in the literature are of this form. For many operads P , it can moreover be proved that homotopy P -algebras, defined in this way, can be rectified, in the sense of being weakly homotopy equivalent to strict P -algebras. One of the first instances of this phenomenon is the well known fact that in topology, any A ∞ -space is weakly homotopy equivalent to a topological monoid (Stasheff [30], Boardman-Vogt [7]). The model theoretic framework of [3] provides a general rectification result which includes many of the known cases.In categories where the tensor product is the cartesian product, there is another approach to algebraic structures, based on Lawvere's notion of algebraic theory.1991 Mathematics Subject Classification. Primary 18D50; Secondary 18G55, 55U35. For such an algebraic theory T , this gives rise to a somewhat different notion of homotopy T -algebra (not based on a resolution of T ), for which a rectification result has been proved in the context of simplicial sets by Badzioch [1] and Bergner [6].One purpose in this paper is to prove a multi-sorted version of such a rectification theorem, based on the notion of coloured operad. This concept goes back to Boardman and Vogt [7], and also occurs in homotopy theory under the name multicategory (cf. e.g. ). The precise definition and some typical examples of coloured operads and their algebras will be reviewed in Section 1 of this paper. These examples include bimodules over monoids, enriched (e.g. topological, simplicial, or differential graded) categories, diagrams on a fixed such enriched category, morphisms between algebras over a given operad P , and many more. In particular, operads themselves are also algebras for a suitable coloured operad. We will discuss how the homotopy theory of operads developed in [3,4] extends to coloured operads. The results will be formulated and proved in a gener...

Polynomial Functors and Trees

“…Now we have described how all the edges and vertices are connected, and hence we have described a finite, oriented graph with boundary. Condition (4) implies that the graph is connected: every e = 1 has a 'next edge' σ(e) distinct from itself, and in a finite number of steps comes down to the root edge. There can be no loops because there is precisely one edge coming out of each vertex, and linear cycles are excluded by connectedness and existence of a root.…”

“…(This viewpoint on Coll (I) was also used in the appendix of Berger-Moerdijk [4], except that they did not formulate it in terms of monomial functors.) An object I ← n → 1 → I amounts to an (n + 1)-tuple (i 1 , .…”

“…j * X is in the essential image of W 0 ). 4. k * j * X is isomorphic to the nerve of a nonsymmetric collection (i.e.…”

“…For fixed set of colours I, the category Coll (I) has a monoidal structure given by the substitutional tensor product (cf. [11], see also the Appendix of [4]). …”

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“…For an introduction of the theory of operads we refer to [Markl et al 2002;Merkulov 2008b] and especially to [Berger and Moerdijk 2007;Kontsevich and Soibelman 2000;Longoni and Tradler 2003]. Some key ideas of this language can be grasped by looking at the basic example of the 2-coloured endomorphism operad, Ᏹnd g,h , canonically associated to an arbitrary pair of vector spaces g and h as follows: (a) as an ‫-ޓ‬module the operad Ᏹnd g,h is given, by definition, by a collection of vector spaces,…”