Abstract not found

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

The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model category structure. We show that the associated stable homotopy theory is completely determined by a ring spectrum functorially associated with the algebraic theory. For several familiar algebraic theories we can identify the parameterizing ring spectrum; for other theories we obtain new examples of ring spectra. For the theory of commutative algebras we obtain a ring spectrum which is related to AndreH}Quillen homology via certain spectral sequences. We show that the (co-)homology of an algebraic theory is isomorphic to the topological Hochschild (co-)homology of the parameterizing ring spectrum. # 2000 Elsevier Science Ltd. All rights reserved. MSC: 55U35; 18C10 Keywords: Algebraic theories; Ring spectra; AndreH}Quillen homology; #-spaces The original motivation for this paper came from the attempt to generalize a rational result about the homotopy theory of commutative rings. For...

combinatorial models for spaces ~ E X, where X is a connected based space. The first such result was given by James [26], who showed that ~EX lent to the James construction MX. is equiva-The unpublished preprint form of Dyer and gashof's paper [25] gave an approximation to QX = lira ~nEnx, and Milgram [41] gave a cellular model for ~2nEnx for all finite n. Starting from Boardman and Vogt's spaces ~ of j-tuples of little n-cubes n,j [5], Dold and Thorn's treatment of the infinite symmetric product NX in terms of quasifibrations [24], and the category theorists ' comparison between finitary alge-braic theories and monads (as for example in Beck [4]), I gave a new approxima-

Abstract. Given an algebraic theory T, a homotopy T-algebra is a simplicial set where all equations from T hold up to homotopy. All homotopy T-algebras form a homotopy variety. We will give a characterization of homotopy varieties analogous to the characterization of varieties. We will also study homotopy models of limit theories which leads to homotopy locally presentable categories. These were recently considered by Simpson, Lurie, Toën and Vezzosi. 1.