Moduli spaces of commutative ring spectra

2000

The category of modules over an S-algebra (A ∞ or E ∞ ring spectrum) has many of the good properties of the category of spectra. When the homotopy groups of the S-algebra in question form a sufficiently nice ring, it is possible to see the deviation of the category of modules over an S-algebra from the corresponding algebraic module category. In particular, many algebraic modules are realized as homotopy groups of topological modules over S-algebras. Examples studied include real and complex K-theory, both connective and periodic. Further, Bousfield localization by a smashing spectrum is shown to yield a category of modules over the localized sphere. For periodic K-theory, these methods yield an algebraic criterion to determine when a local spectrum is a module over the K-theory S-algebra, real or complex.

Proceedings of the Edinburgh Mathematical Society, 2004

We construct a Bousfield-Kan (unstable Adams) spectral sequence based on an arbitrary (and not necessarily connective) ring spectrum E with unit and which is related to the homotopy groups of a certain unstable E completion X ∧ E of a space X. For E an S-algebra this completion agrees with that of the first author and Thompson. We also establish in detail the Hopf algebra structure of the unstable cooperations (the coalgebraic module) E * (E * ) for an arbitrary Landweber exact spectrum E, extending work of the second author with Hopkins and with Turner and giving basis-free descriptions of the modules of primitives and indecomposables. Taken together, these results enable us to give a simple description of the E 2 -page of the E-theory Bousfield-Kan spectral sequence when E is any Landweber exact ring spectrum with unit. This extends work of the first author and others and gives a tractable unstable Adams spectral sequence based on a vn-periodic theory for all n.

2013

We develop homotopical algebraic geometry ([To-Ve 1, To-Ve 2]) in the special context where the base symmetric monoidal model category is that of spectra S, i.e. what might be called, after Waldhausen, brave new algebraic geometry. We discuss various model topologies on the model category of commutative algebras in S, and their associated theories of geometric S-stacks (a geometric S-stack being an analog of Artin notion of algebraic stack in Algebraic Geometry). Two examples of geometric S-stacks are given: a global moduli space of associative ring spectrum structures, and the stack of elliptic curves endowed with the sheaf of topological modular forms.

Publications Mathématiques de l'IHES, 1999

Advances in Mathematics

Spectral Mackey functors are homotopy-coherent versions of ordinary Mackey functors as defined by Dress. We show that they can be described as excisive functors on a suitable ∞-category, and we use this to show that universal examples of these objects are given by algebraic K-theory. More importantly, we introduce the unfurling of certain families of Waldhausen ∞-categories bound together with suitable adjoint pairs of functors; this construction completely solves the homotopy coherence problem that arises when one wishes to study the algebraic K-theory of such objects as spectral Mackey functors. Finally, we employ this technology to lay the foundations of equivariant stable homotopy theory for profinite groups; the lack of such foundations has been a serious impediment to progress on the conjectures of Gunnar Carlsson. We also study fully functorial versions of A-theory, upside-down A-theory, and the algebraic K-theory of derived stacks. Contents 0. Summary 1. Preliminaries on ∞-categories 2. The twisted arrow ∞-category 3. The effective Burnside ∞-category 4. Disjunctive ∞-categories 5. Disjunctive triples 6. Mackey functors 7. Mackey stabilization 8. Representable spectral Mackey functors and assembly morphisms 9. Mackey stabilization via algebraic K-theory 10. Waldhausen bicartesian fibrations 11. Unfurling 12. Horn filling in effective Burnside ∞-categories 13. The Burnside Waldhausen bicartesian fibration Example A. Coherent n-topoi and the Segal-tom Dieck splitting Example B. Equivariant spectra for a profinite group Example C. A-theory and upside-down A-theory of ∞-topoi Example D. Algebraic K-theory of derived stacks References 2 CLARK BARWICK

2003

Preprint, 1998

Proceedings of the American Mathematical Society, 1984

This paper presents the construction of a tower of spectra Yj with ¿.-invariants coming from the relations Sq'(^Sq2')Sq'(A'Sq2') = 0 in A/A Sq1. for 0 =sy «s 5 and A = Steenrod algebra mod 2, such that V5 has prescribed homotopy groups: f"(T5) = Z (integers) if n-2'+ '-2, and zero otherwise.

arXiv: Category Theory, 2018

In [Joyal] where the category $\Theta$ is first defined it is noted that the dimensional shift on $\Theta$ suggests an elegant presentation of the unreduced suspension on cellular sets. In this note we prove that the reduced suspension associated to that presentation is left Quillen with respect to the Cisinski model category structure presenting the $\left(\infty,1\right)$-category of pointed spaces and enjoys the correct universal property. More, we go on to describe how, in forthcoming work, inspired by the combinatorial spectra described in [Kan], this suspension functor entails a description of spectra which echoes the weaker form of the homotopy hypothesis, we describe the development of a presentation of spectra as locally finite weak $\mathbf{Z}$-groupoids.

Inventiones Mathematicae, 1968

The aim of this note is the study of the relations between the homotopy classes of maps of a spectrum A into a Kan spectrum B and the homotopy classes of maps of the components of the associated prespectrum A k into the spectrum B. The result is a little more general and consists in proving the validity for the spectra of MILNOR'S theorem, i.e. that for any countable family of subspectra A~ of A for which