In recent years the theory of structured ring spectra (formerly known as A # - and E # -ring spectra) has been signicantly simplified by the discovery of categories of spectra with strictly associative and commutative smash products. Now a ring spectrum can simply be dened as a monoid with respect to the smash product in one of these new categories of spectra. In order to make use of all of the standard tools from homotopy theory, it is important to have a Quillen model category structure [##] available here. In this paper we provide a general method for lifting model structures to categories of rings, algebras, and modules. This includes, but is not limited to, each of the new theories of ring spectra. One model for structured ring spectra is given by the S-algebras of [##]. This example has the special feature that every object is brant, which makes it easier to fo...

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...

Abstract. We define a sequence of purely algebraic invariants – namely, classes in the Quillen cohomology of the Π-algebra π∗X – for distinguishing between different homotopy types of spaces. Another sequence of such cohomology classes allows one to decide whether a given abstract Π-algebra can be realized as the homotopy Π-algebra of a space. 1.

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 =

In [HS] and [F1] Halperin, Stasheff, and Félix showed how an inductively-defined sequence of elements in the cohomology of a graded commutative algebra over the rationals can be used to distinguish among the homotopy types of all possible realizations, thus providing a collection

this paper we advertise the category of #-spaces as a convenient framework for doing `algebra' over `rings' in stable homotopy theory. #-spaces were introduced by Segal [Se] who showed that they give rise to a homotopy category equivalent to the usual homotopy category of connective (i.e. (-1)-connected) spectra. Bousfield and Friedlander [BF] later provided model category structures for #-spaces. The study of `rings, modules and algebras' based on #-spaces became possible when Lydakis [Ly] introduced a symmetric monoidal smash product with good homotopical properties. Here we develop model category structures for modules and algebras, set up (derived) smash products and associated spectral sequences and compare simplicial modules and algebras to their Eilenberg--MacLane spectra counterparts. There are other settings for ring spectra, most notably the S-modules and S-algebras of [EKMM] and the symmetric spectra of [HSS], each of these with its own advantages and disadvantages. We believe that one advantage of the #- space approach is its simplicity. The definitions of the stable equivalences, the smash product and the `rings' (which we call Gamma-rings) are given on a few pages. Another feature is that #-spaces nicely reflect the idea that spectra are a homotopical generalization of abelian groups, that the smash product generalizes the tensor product and that algebras over the sphere spectrum generalize classical rings. There is an Eilenberg--MacLane functor H which embeds the category of simplicial abelian groups as a full subcategory of the category of #-spaces. The embedding has a left adjoint, left inverse which on cofibrant objects models spectrum homology. Similarly, simplicial rings embed fully faithfully into Gamma-rings. We give a quick proof (see Section 4) ...

Abstract. Given a suitable functor T: C → D between model categories, we define a long exact sequence relating the homotopy groups of any X ∈ C with those of TX, and use this to describe an obstruction theory for lifting an object G ∈ D to C. Examples include finding spaces with given homology or homotopy groups. A number of fundamental problems in algebraic topology can be described as measuring the extent to which a given functor T: C → D between model categories induces an equivalence of homotopy categories: more specifically, which objects (or maps) from D are in the image of T, and in how many different ways. For example:

: The homology of a homotopy inverse limit can be studied by a spectral sequence with has E 2 term the derived functors of limit in the category of coalgebras. These derived functors can be computed using the theory of Dieudonn'e modules if one has a diagram of connected abelian Hopf algebras. One of the standard problems in homotopy theory is to calculate the homology of a given type of inverse limit. For example, one might want to know the homology of the inverse limit of a tower of fibrations, or of the pull-back of a fibration, or of the homotopy fixed point set of a group action, or even of an infinite product of spaces. This paper presents a systematic method for dealing with this problem and works out a series of examples. It simplifies the foundational questions present when dealing with inverse limits to work with simplicial sets rather than topological spaces. So this paper is written simplicially; that is, a space is a simplicial set. As usual this doesn't affect the homotop...

Abstract. In this paper we construct internal cohomomorphism objects in various categories of operads (ordinary, cyclic, modular, properads...) and algebras over operads. We argue that they provide an approach to symmetry and moduli objects in non-commutative geometries based upon these “ring–like ” structures. We give also a unified axiomatic treatment of operads as functors on labeled graphs. Finally, we extend internal cohomomorphism constructions to more general categorical contexts. 0.1. Internal 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