2. Moduli spaces 7 3. Postnikov systems for spaces 10 4. Π-algebras and their modules 14

We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or "continuous," functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence.

Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# -algebra A in E#E-comodules, is there an E# -ring spectrum X with E#X = A as comodule algebras? We will formulate this as a moduli problem, and give a way -- suggested by work of Dwyer, Kan, and Stover -- of dissecting the resulting moduli space as a tower with layers governed by appropriate Andre-Quillen cohomology groups. A special case is A = E#E itself. The final section applies this to discuss the Lubin-Tate or Morava spectra En .

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 Hopkins-Miller theorem on the Lubin-Tate 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 non-empty, or even that there is a spectrum X so that E∗X ∼ = A as comodules.

We outline an obstruction theory for deciding when a homotopy commutative and associative ring spectrum is actually an E∞ ring spectrum. The obstruction groups are André-Quillen cohomology groups of an algebra over an E∞ operad. The same cohomology theory is part of a spectral sequence for computing the homotopy type of mapping spaces between E∞ ring spectrum. The obstruction theory arises out of techniques of Dwyer, Kan, and Stover, and the main application here is to prove an analog of a theorem of Haynes Miller and the second author: the Lubin-Tate spectra En are E∞ and the space of E∞ self-maps has weakly contractible components.

Abstract. We show how a certain type of CW simplicial resolutions of spaces by wedges of spheres may be constructed, and how such resolutions yield an obstruction theory for a given space to be a loop space. 1.

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

Let C be a closed model category. Following work of Dwyer, Kan, and Stover, we develop a technique for building simplicial resolutions of objects in C which are free, or homotopically free, in a precise sense. Specifically, we specify in advance a set A of small, co brant, co-H objects in C and the simplicial resolutions X will have the property that at each level n, the object Xn is homotopy equivalent to a coproduct of objects of A. These Stover-type resolutions are modeled on the free resolutions developed by Quillen, and are meant to be a replacement for projective resolutions in this context. In the end we develop some computational tools for spectra and structured ring spectra.

We examine the foundations of simplicial algebras in spectra over a simplicial operad. We are led to simplicial operads and simplicial algebras over simplicial operads because certain operads which are notoriously hard to work with { mainly the E1 operad { can be simplicially resolved by simpler pieces. Our main goals are to a build spectral sequence for computing spaces of maps between structured ring spectra, and to develop a Dwyer-Kan-Stover style obstruction theory for deciding when a spectrum actually can be a structured ring spectrum. In this paper we work out some of the foundations of the homotopy theory of simplicial ring spectra over a simplicial operad. This is not a gratuitous act of generalization. Simplicial objects in any category are a standard mechanism for building the resolutions necessary for computations; this is how simplicial spectra arise. The simplicial operads arise as an answer to an immediate practical problem. If T is an E1 operad over the linear isometrie...

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