We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.

We define a notion of a strong homotopy BV algebra and apply it to deformation theory problems. Formality conjectures for Hochschild cochains are formulated. We prove several results supporting these conjectures.

Abstract. We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to Fulton-MacPherson and Axelrod-Singer in the setting of smooth manifolds, as well as a simplicial variant of this compactification. Our constructions are elementary and give simple global coordinates for the compactified configuration space of a general manifold embedded in Euclidean space. We stratify the canonical compactification, identifying the diffeomorphism types of the strata in terms of spaces of configurations in the tangent bundle, and give completely explicit local coordinates around the strata as needed to define a manifold with corners. We analyze the quotient map from the canonical to the simplicial compactification, showing it is a homotopy equivalence. We define projection maps and diagonal maps, which for the simplicial variant satisfy cosimplicial identities.

Let Em denote the space of embeddings of the interval I = [−1, 1] in the cube I m with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube, equipped with a homotopy through immersions to the unknot – see Definition 5.1. By Proposition 5.17, Em is homotopy equivalent to Emb(I, I m) × ΩImm(I, I m). In [28], McClure and Smith define a cosimplicial object O • associated

ABSTRACT. We show how the Hopf algebra of rooted trees encodes the combinatorics of Epstein-Glaser renormalization and coordinate space renormalization in general. In particular, we prove that the Epstein-Glaser time-ordered products can be obtained from the Hopf algebra by suitable Feynman rules, mapping trees to operator-valued distributions. Twisting the antipode with a renormalization map formally solves the Epstein-Glaser recursion and provides local counterterms due to the Hochschild 1-closedness of the grafting operator B+.

It is well known that the forgetful functor from symmetric operads to nonsymmetric operads has a left adjoint Sym1 given by product with the symmetric group operad. It is also well known that this functor does not affect the category of algebras of the operad. From the point of view of the author’s theory of higher operads, the nonsymmmetric operads are 1-operads and Sym1 is the first term of the infinite series of left adjoint functors Symn, called symmetrisation functors, from n-operads to symmetric operads with the property that the category of one object, one arrow,..., one (n − 1)-arrow algebras of an n-operad A is isomorphic to the category of algebras of Symn(A). In this paper we consider some geometrical and homotopical aspects of the symmetrisation of n-operads. We follow Getzler and Jones and consider their decomposition of the Fulton-Macpherson operad of compactified real configuration spaces. We construct an n-operadic counterpart of

Abstract. 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 Σ-cofibrant operads. The standard simplicial resolution of Godement as well as the cobar-bar chain resolution are shown to be particular instances of this generalised W-construction.

We outline an obstruction theory for deciding when a homotopy commutative and associative ring spectrum is actually an E∞ ring spectrum. The obstruction groups are André-Quillen cohomology groups of an algebra over an E∞ operad. The same cohomology theory is part of a spectral sequence for computing the homotopy type of mapping spaces between E∞ ring spectrum. The obstruction theory arises out of techniques of Dwyer, Kan, and Stover, and the main application here is to prove an analog of a theorem of Haynes Miller and the second author: the Lubin-Tate spectra En are E∞ and the space of E∞ self-maps has weakly contractible components.

Now if it was dusk outside, Inside it was dark.

Categorification is the process of finding category-theoretic analogs of set-theoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural isomorphisms between functors, which in turn should satisfy certain equations of their own, called ‘coherence laws’. Iterating this process requires a theory of ‘n-categories’, algebraic structures having objects, morphisms between objects, 2-morphisms between morphisms and so on up to n-morphisms. After a brief introduction to n-categories and their relation to homotopy theory, we discuss algebraic structures that can be seen as iterated categorifications of the natural numbers and integers. These include tangle n-categories, cobordism n-categories, and the homotopy n-types of the loop spaces Ω k S k. We conclude by describing a definition of weak n-categories based on the theory of operads. 1