In this paper we show that if a cosimplicial space has a certain kind of combinatorial structure then its total space has an action of an operad weakly equivalent to the little n-cubes operad. Our results are also valid for cosimplicial spectra. 1 Introduction. The little n-cubes operad Cn was introduced by Boardman and Vogt in [6] (except that they used the terminology of theories rather than that of operads) as a tool for understanding n-fold loop spaces. They showed that for any topological space Y the n-fold loop space Ω n Y has an action of Cn. In the other direction, May showed in [19] that if Z is a space with

Abstract. Generalising Segal’s approach to 1-fold loop spaces, the homotopy theory of n-fold loop spaces is shown to be equivalent to the homotopy theory of reduced Θn-spaces, where Θn is an iterated wreath product of the simplex category ∆. A sequence of functors from Θn to Γ allows for an alternative description of the Segal spectrum associated to a Γ-space. In particular, each Eilenberg-MacLane space has a canonical reduced Θnset model. The number of (n + d)-dimensional cells of the resulting CWcomplex of type K(Z/2Z, n) is a generalised Fibonacci number.

Abstract. We present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the n th polytope in the sequence known as the multiplihedra. This answers the open question of whether the multiplihedra could be realized as convex polytopes. Contents

Abstract. In this paper we introduce a notion of generalized operad containing as special cases various kinds of operad–like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories (and categories of algebras over them). We argue that they provide an approach to symmetry and moduli objects in non-commutative geometries based upon these “ring–like ” structures. We give a unified axiomatic treatment of generalized operads as functors on categories of abstract labeled graphs. Finally, we extend inner cohomomorphism constructions to more general categorical contexts. This version differs from the previous ones by several local changes (including the title) and two extra references. 0.1. Inner cohomomorphisms of associative algebras. Let k be a field. Consider pairs A = (A, A1) consisting of an associative k–algebra A and a finite dimensional subspace A1 generating A. For two such pairs A = (A, A1) and B =

We review definitions and basic properties of operads, PROPs and algebras over these structures.

We proved in a previous article that the bar complex of an E ∞-algebra inherits a natural E ∞-algebra structure. As a consequence, a well-defined iterated bar construction B n (A) can be associated to any algebra over an E ∞-operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E ∞-algebras. We use this effective definition to prove that the n-fold bar construction admits an extension to categories of algebras over En-operads. Then we prove that the n-fold bar complex determines the homology theory associated to the category of algebras over an En-operad. In the case n = ∞, we obtain an isomorphism between the homology of an infinite bar construction and the usual Γ-homology with trivial coefficients. 57T30; 55P48, 18G55, 55P35

It is well known since Stasheff’s work that 1-fold loop spaces can be described in terms of the existence of higher homotopies for associativity (coherence conditions) or equivalently as algebras of contractible nonsymmetric operads. The combinatorics of these higher homotopies is well understood and is extremely useful. For n ≥ 2 the theory of symmetric operads encapsulated the corresponding higher homotopies, yet hid the combinatorics and it has remain a mystery for almost 40 years. However, the recent developments in many fields ranging from algebraic topology and algebraic geometry to mathematical physics and category theory show that this combinatorics in higher dimensions will be even more important than the one dimensional case. In this paper we are going to show that there exists a conceptual way to make these combinatorics explicit using the so called higher nonsymmetric n-operads.

Schwänzl and Vogt in [2] can be seen as a weaker structure than a symmetric or even braided monoidal category. In this paper we show that it is still sufficient to permit a good definition of (n-fold) operads in a k-fold monoidal category which generalizes the definition

Abstract. We describe a new sequence of polytopes which characterize A ∞ maps from a topological monoid to an A ∞ space. Therefore each of these polytopes is a quotient of the corresponding multiplihedron. Our sequence of polytopes is demonstrated not to be combinatorially equivalent to the associahedra, as was previously assumed in both topological and categorical literature. They are given the new collective name composihedra. We point out how these polytopes are used to parameterize compositions in the formulation of the theories of enriched bicategories and pseudomonoids in a monoidal bicategory. We also present a simple algorithm for determining the extremal points in Euclidean space whose convex hull is the nth polytope in the sequence of

Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads [1] to this task. We present a general construction of a tensor product on the category of n-globular sets from any normalised (n + 1)-operad A, in such a way that the algebras for A may be recaptured as enriched categories for the induced tensor product. This is an important step in reconciling the globular and simplicial approaches to higher category theory, because in the simplicial approaches one proceeds inductively following the idea that a weak (n + 1)category is something like a category enriched in weak n-categories. In this paper we reveal how such an intuition may be formulated in terms of globular operads.