We give a definition of weak n-categories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘S-operads’, and given such an operad O, we denote its set of operations by elt(O). Then for any S-operad O there is an elt(O)-operad O + whose algebras are S-operads over O. Letting I be the initial operad with a one-element set of types, and defining I 0+ = I, I (i+1)+ = (I i+) +, we call the operations of I (n−1)+ the ‘n-dimensional opetopes’. Opetopes form a category, and presheaves on this category are called ‘opetopic sets’. A weak n-category is defined as an opetopic set with certain properties, in a manner reminiscent of Street’s simplicial approach to weak ω-categories. In a similar manner, starting from an arbitrary operad O instead of I, we define ‘n-coherent O-algebras’, which are n times categorified analogs of algebras of O. Examples include ‘monoidal n-categories’, ‘stable n-categories’, ‘virtual n-functors ’ and ‘representable n-prestacks’. We also describe how n-coherent O-algebra objects may be defined in any (n + 1)-coherent O-algebra. 1

There are many different types of algebra: associative, associative and commutative, Lie, Poisson, etc., etc. Each comes with an appropriate notion of a module. As is becoming more and more important in a variety of fields, it is often necessary to deal with algebras and modules of these sorts “up to homotopy”. I shall give a very partial overview, concentrating on algebra, but saying a little about the original use of operads in topology. The development of abstract frameworks in which to study such algebras has a long history. As this conference attests, it now seems to be widely accepted that, for many purposes, the most convenient setting is that given by operads and their actions. While the notion was first written up in a purely topological framework [19], it was thoroughly understood by 1971 [12] that the basic definitions apply equally well in any underlying symmetric monoidal ( = tensor) category. The definitions and ideas had many precursors. I will indicate those that I was aware of at the time. • Algebraists such as Kaplansky, Herstein, and Jacobson systematically studied

Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X; G); we will single out three of them for discussion here. First of all, we have the singular cohomology groups H n sing (X; G), which are defined to be cohomology of a chain complex of G-valued singular cochains on X. An alternative is to regard H n (•, G) as a representable functor on the homotopy category

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Let X be a topological space and G an abelian group. There are many different definitions for the cohomology group H n (X, G); we will single out three of them for discussion here. First of all, one has the singular cohomology H n sing(X, G), which is defined as the cohomology of a complex of G-valued singular cochains. Alternatively, one may regard H n (•, G) as a representable functor on the homotopy category of topological spaces, and thereby define H n rep(X, G) to be the set of homotopy classes of maps from X into an Eilenberg-MacLane space K(G, n). A third possibility is to use the sheaf cohomology H n sheaf (X, G) of X with coefficients in the constant sheaf G on X. If X is a sufficiently nice space (for example, a CW complex), then all three of these definitions agree. In general, however, all three give different answers. The singular cohomology of X is constructed using continuous maps from simplices ∆k into X. If there are not many maps into X (for example if every path in X is constant), then we cannot expect H n sing (X, G) to tell us very much about X. Similarly, the cohomology group H n rep(X, G) is defined using maps from X into a simplicial complex, which (ultimately) relies on the existence of continuous real-valued functions on X. If X does not admit many real-valued functions, we should not expect H n rep (X, G) to be a useful invariant. However, the sheaf cohomology of X seems to be a good invariant for arbitrary spaces: it has excellent formal properties in general and sometimes yields

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

Abstract. With motivation from algebraic topology, algebraic geometry, and string theory, we study various topics in differential homological algebra. The work is divided into five largely independent parts: I Definitions and examples of operads and their actions II Partial algebraic structures and conversion theorems III Derived categories from a topological point of view IV Rational derived categories and mixed Tate motives V Derived categories of modules over E ∞ algebras In differential algebra, operads are systems of parameter chain complexes for multiplication on various types of differential graded algebras “up to homotopy”, for example commutative algebras, n-Lie algebras, n-braid algebras, etc. Our primary focus is the development of the concomitant theory of modules up to homotopy and the study of both classical derived categories of modules over DGA’s and derived categories of modules up to homotopy over DGA’s up to homotopy. Examples of such derived categories provide the appropriate setting for one approach to mixed Tate motives in algebraic geometry, both rational and integral.

Abstract We show that the topological Hochschild homology THH(R) of an En-ring spectrum R is an En−1-ring spectrum. The proof is based on the fact that the tensor product of the operad Ass for monoid structures and the the little n-cubes operad Cn is an En+1-operad, a result which is of independent interest. In 1993 Deligne asked whether the Hochschild cochain complex of an associative ring has a canonical action by the singular chains of the little 2-cubes operad. Affirmative answers for differential graded algebras in characteristic 0 have been found by Kontsevich and Soibelman [11], Tamarkin [15] and [16], and Voronov [18]. A more general proof, which also applies to associative ring spectra is due to McClure and Smith [14]. In [10] Kontsevich extended Deligne’s question: Does the Hochschild cochain complex of an En differential graded algebra carry a canonical En+1-structure? We consider the dual problem: Given a ring R with additional structure, how much structure does the topological Hochschild homology THH(R) of R inherit from R? The close connection of THH with algebraic K-theory and with structural questions in the category of spectra make multiplicative structures on THH desirable. In his early work on topological Hochschild homology of functors with smash product Bökstedt proved that THH of a commutative such functor is a commutative ring spectrum (unpublished). The discovery of associative, commutative and unital smash product functors of spectra simplified the definition of THH and the proof of the corresponding result for E∞-ring spectra considerably (e.g. see [13]). 1 In this paper we morally prove Theorem A: For n ≥ 2, if R is an En-ring spectrum then THH(R) is an En−1-ring spectrum. The same result has been obtained independently by Basterra and Mandell using different techniques [2]. Why “morally”? To define THH(R) we need R to be a strictly associative spectrum. In general, En-structures do not have a strictly associative substructure. So we have to replace R by an equivalent strictly associative ring spectrum Y, whose multiplication extends to an En-structure. Then the statement makes sense for Y. Here is a more precise reformulation of

Abstract. We apply the Dwyer-Kan theory of homotopy function complexes in model categories to the study of mapping spaces in quasi-categories. Using this, together with our work on rigidification from [DS1], we give a streamlined proof of the Quillen equivalence between quasi-categories and simplicial categories. Some useful material about relative mapping spaces in quasi-categories is developed along the way. Contents

There are a number of theorems to the effect that spaces of the form Q. n 11 n X split stably into wedges of simpler spaces when X is connected (by which we mean pathwise connected). The proofs generally proceed by exploitation of combinatorially manageable approximations for Q. n *L n X. The earliest theorem of this kind is due to Milnor(18), who exploited the semisimplicial version of the James construction on X to split ZQ ~LX into V 2>X lq \ where X [q Q>1) denotes the g-fold smash product of X with itself. More recently, Kahn(iO) proved that the suspension spectrum of QX = lim £2 7l £ n X splits as the wedge of the suspension spectra of the extended powers ~* DqX = E-Lt AxqXM, where Sg is the symmetric group on q letters, E"Lq is a contractible space on which Sg acts freely, and Y+ denotes the union of a space Y and a disjoint basepoint. He exploited Barratt's semisimplicial approximation to QX, and a proof of this splitting has also been given by Barratt and Eccles(i). A bit later, Snaith(22) proved that the suspension spectrum of Q m 2 n X for 1 ^ n