Abstract. In this paper we construct a small E ∞ chain operad S which acts naturally on the normalized cochains S ∗ X of a topological space. We also construct, for each n, a suboperad Sn which is quasi-isomorphic to the normalized singular chains of the little n-cubes operad. The case n = 2 leads to a substantial simplification of our earlier proof of Deligne’s Hochschild cohomology conjecture. 1. Introduction. This paper has two goals. The first (see Theorem 2.15 and Remark 2.16(a)) is to construct a small E ∞ chain operad S which acts naturally on the normalized cochains S∗X of a topological space X. This is of interest in view of a theorem of Mandell [15, page 44] which states that if O is any E ∞ chain operad over Fp (the algebraic closure of the field with

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this paper we will discuss the combination of two classical notions of category theory, both treated extensively in [CWM]. One of these is the notion of a monad or triple on a category, which goes back to Godement [G] and was rst developed by Eilenberg, Moore, Beck and others. The other is that of a monoidal category or tensor category, which originates with Benabou [Be] and with Mac Lane's famous coherence theorem [MacL], and which pervades much of present day mathematics. For a monad S on a tensor category, there is a natural additional structure that one can impose, namely that of a comparison map S(X

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We give a universal construction of families of Hopf P-algebras for any Hopf operad P. As a special case, we recover the Connes-Kreimer Hopf algebra of rooted trees.

Abstract. We give a new construction of the algebraic K-theory of small permutative categories that preserves multiplicative structure, and therefore allows us to give a unified treatment of rings, modules, and algebras in both the input and output. This requires us to define multiplicative structure on the category of small permutative categories. The framework we use is the concept of multicategory, a generalization of symmetric monoidal category that precisely captures the multiplicative structure we have present at all stages of the construction.

Abstract. In a previous paper, Auslander-Reiten triangles and quivers were introduced into algebraic topology. This paper shows that over a Poincaré duality space, each component of the Auslander-Reiten quiver is isomorphic to ZA∞. In [5], the concepts of Auslander-Reiten triangles and Auslander-Reiten quivers from the representation theory of Artin algebras were introduced into algebraic topology. The main theorem was that Auslander-Reiten triangles exist precisely

We study the splitting of the Goodwillie towers of functors in various settings. In particular, we produce splitting criteria for functors F: A → MA from a pointed category with coproducts to A-modules in terms of differentials of F. Here A is a commutative S-algebra. We specialize to the case when A is the category of a-algebras for an operad a and F is the forgetful functor, and derive milder splitting conditions in terms of the derivative of F. In addition, we describe how triples induce operads, and prove that, roughly speaking, a triple T is naturally equivalent to the product of its Goodwillie layers if and only if it is an algebra over its induced operad. Key words: spectra with additional structure, Goodwillie Calculus, algebras over operads

Abstract. Differential Graded Algebras can be studied through their Differential Graded modules. Among these, the compact ones attract particular attention. This paper proves that over a suitable chain Differential Graded Algebra R, each compact Differential Graded module M satisfies amp M ≥ ampR, where amp denotes amplitude which is defined in a straightforward way in terms of the homology of a DG module. In other words, the homology of each compact DG module M is at least as long as the homology of R itself. Conversely, DG modules with shorter homology than R are not compact, and so in general, there exist DG modules with finitely generated homology which are not compact. Hence, in contrast to ring theory, it makes no sense to define finite global dimension of DGAs by the condition that each DG module with finitely generated homology must be compact.

Abstract. The concept of Koszul differential graded algebra (Koszul DG algebra) is introduced. Koszul DG algebras exist extensively, and have nice properties similar to the classic Koszul algebras. A DG version of the Koszul duality is proved. When the Koszul DG algebra A is AS-regular, the Ext-algebra E of A is Frobenius. In this case, similar to the classical BGG correspondence, there is