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Stable homotopical algebra and Γspaces
, 1999
"... In this paper we advertise the category of Γspaces as a convenient framework for doing ‘algebra ’ over ‘rings ’ in stable homotopy theory. Γspaces were introduced by ..."
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In this paper we advertise the category of Γspaces as a convenient framework for doing ‘algebra ’ over ‘rings ’ in stable homotopy theory. Γspaces were introduced by
The realization space of a Πalgebra: a moduli problem in algebraic topology
 Topology
"... 2. Moduli spaces 7 3. Postnikov systems for spaces 10 4. Πalgebras and their modules 14 ..."
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2. Moduli spaces 7 3. Postnikov systems for spaces 10 4. Πalgebras and their modules 14
Homology and cohomology of E∞ ring spectra
 MATHEMATISCHE ZEITSCHRIFT
, 2005
"... Every homology or cohomology theory on a category of E∞ ring spectra is Topological André–Quillen homology or cohomology with appropriate coefficients. Analogous results hold more generally for categories of algebras over operads. ..."
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Every homology or cohomology theory on a category of E∞ ring spectra is Topological André–Quillen homology or cohomology with appropriate coefficients. Analogous results hold more generally for categories of algebras over operads.
Moduli problems for structured ring spectra
 DANIEL DUGGER AND BROOKE
, 2005
"... In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theore ..."
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In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theorem on the LubinTate or Morava spectra En; in particular, we showed how to prove that the moduli space of all E ∞ ring spectra X so that (En)∗X ∼ = (En)∗En as commutative (En) ∗ algebras had the homotopy type of BG, where G was an appropriate variant of the Morava stabilizer group. This is but one point of view on these results, and the reader should also consult [3], [38], and [41], among others. A point worth reiterating is that the moduli problems here begin with algebra: we have a homology theory E ∗ and a commutative ring A in E∗E comodules and we wish to discuss the homotopy type of the space T M(A) of all E∞ring spectra so that E∗X ∼ = A. We do not, a priori, assume that T M(A) is nonempty, or even that there is a spectrum X so that E∗X ∼ = A as comodules.
Co)homology theories for commutative Salgebras
"... The aim of this paper is to give an overview of some of the existing homology theories for commutative (S)algebras. We do not claim any originality; nor do we pretend to give a complete account. But the results in that field are widely spread in the literature, so for someone who does not actually ..."
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The aim of this paper is to give an overview of some of the existing homology theories for commutative (S)algebras. We do not claim any originality; nor do we pretend to give a complete account. But the results in that field are widely spread in the literature, so for someone who does not actually work in that subject, it can be difficult to trace all the relationships between the different homology theories. The theories we aim to compare are • topological AndréQuillen homology • Gamma homology • stable homotopy of Γmodules • stable homotopy of algebraic theories • the AndréQuillen cohomology groups which arise as obstruction groups in the GoerssHopkins approach As a comparison between stable homotopy of Γmodules and stable homotopy of algebraic theories is not explicitly given in the literature, we will give a proof of Theorem 2.1 which says that both homotopy theories are isomorphic
Every homotopy theory of simplicial algebras admits a proper model
 Topology Appl. 119
, 2002
"... Abstract. We show that any closed model category of simplicial algebras over an algebraic theory ..."
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Abstract. We show that any closed model category of simplicial algebras over an algebraic theory
Universal Toda brackets of ring spectra
, 2006
"... Abstract. We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of Rmodule spectra. It determines ..."
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Abstract. We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of Rmodule spectra. It determines for example all triple Toda brackets of R and the first obstruction to realizing a module over the homotopy groups of R by an Rmodule spectrum. For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. The real and complex Ktheory spectra serve as our main examples. 1.
M A Mandell, Homology and cohomology of E1 ring spectra
 Math. Z
"... Abstract. We show that every homology or cohomology theory on a category of E1 ring spectra is Topological Andre{Quillen homology or cohomology with appropriate coecients. We show that the cotangent complex of MU is MU ^ bu. ..."
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Abstract. We show that every homology or cohomology theory on a category of E1 ring spectra is Topological Andre{Quillen homology or cohomology with appropriate coecients. We show that the cotangent complex of MU is MU ^ bu.
Stable homotopical algebra and Gammaspaces
, 1999
"... 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. ..."
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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 EilenbergMacLane spectra counterparts. There are other settings for ring spectra, most notably the Smodules and Salgebras 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 Gammarings) 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 EilenbergMacLane 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 Gammarings. We give a quick proof (see Section 4) ...