1.1. Simplicial sets 5 1.2. Symmetric spectra 6

The main theorem of [4] say that there is a proper closed simplicial model category

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this paper) are well known: They are the FSPs, which were introduced in 1985 by Bokstedt [B]. There are interesting constructions with FSPs, e.g., topological cyclic homology, which can be considerably simplified and conceptualized using the smash product of simplicial functors [LS]. Note however that it is not clear that an E1-FSP has a commutative model, although the analogous statement is true for S-modules and symmetric spectra. This has, e.g., the disadvantage, that it is not clear how to give a model category structure to (or even how to define) MU-algebras using simplicial functors (although this can be done for modules over any any FSP, and algebras over any commutative FSP, by similar methods as in [Sch]). Returning to examining the special features of simplicial functors, in constrast to

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. We review Quillen's concept of a model category as the proper setting for defining derived functors in non-abelian settings, explain how one can transport a model structure from one category to another by mean of adjoint functors (under suitable assumptions), and define such structures for categories of cosimplicial coalgebras. 1. Introduction Model categories, first introduced by Quillen in [Q1], have proved useful in a number of areas -- most notably in his treatment of rational homotopy in [Q2], and in defining homology and other derived functors in non-abelian categories (see [Q3]; also [BoF, BlS, DwHK, DwK, DwS, Goe, ScV]). From a homotopy theorist's point of view, one interesting example of such non-abelian derived functors is the E 2 -term of the mod p unstable Adams spectral sequence of Bousfield and Kan. They identify this E 2 -term as a sort of Ext in the category CA of unstable coalgebras over the mod p Steenrod algebra (see x7.4). The original purpose of this note w...

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...

this paper, we study certain categories of "diagram ring spectra" that are isomorphic to corresponding categories of "functors with smash product (FSP's)". The notion of an FSP was introduced by Bokstedt [2], and his use of FSP's to define topological Hochschild homology established their convenience and importance in stable homotopy theory. Versions of FSP's had been defined earlier: in different language, what we call FSP's in the category of symmetric spectra were defined by Gunnarson [7], and what we call FSP's in the category of orthogonal spectra were defined by the second author and others [18, 17] in the early 1970's. As we shall see, FSP's are defined in terms of "external smash products". It is a crucial insight of Jeff Smith that external smash products can be internalized. We show that each category of generalized FSP's is isomorphic to the category of monoids in an associated symmetric monoidal category of diagram spectra. Such monoids are what we mean by diagram ring spectra. Smith introduced the category of symmetric spectra, showed that it is symmetric monoidal, and observed that its externally defined FSP's and internally defined monoids give isomorphic categories. We study a general form of the construction. In fact, the relevant categorical framework was already in place by 1970, in work of Day [5]. Hovey, Shipley, and Smith [8] have studied the category of symmetric spectra and its homotopy theory, and the papers [21] and [23] go further with the homotopical study of its ring spectra. There is a coordinate-free analogue of the category of symmetric spectra, which we call the category of orthogonal spectra; it was introduced by May [17, x5] (who called its objects I -prespectra). The names come from the fact that actions by symmetric groups and by or...

this paper we shall provide a homotopy-theoretic computation of the derivative,

Introduction \Gamma-spaces were introduced by Segal [S], who proved that they are combinatorial models for connective spectra (see also [A], [BF]). Based on Kan-Thurston theorem we show that any \Gamma-space is stably weak equivalent to a discrete \Gamma-group. By a well-known theorem of Dold-Kan the Moore normalization establishes the equivalence between the category of simplicial abelian groups and the category of chain complexes (see [DP]). mimicking the construction of normalization of simplicial groups, we give a similar construction for \Gamma-groups. This construction is based on the notion of cross-effects of functors [BP], which is a generalizatin of the classical definition of Eilenberg and Mac Lane [EM] to the non-abelian setup. Finally a Dold-Kan type theorem for the category of \Gamma-groups is proved. In abelian case our theorem claims that the category of abelian \Gamma-groups is equivalent to the category of functors Ab\Omega , where\Om