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A model for the homotopy theory of homotopy theory
We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, “functors between two homotopy theories form a homotopy theory”, or more…
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Spaces of algebra structures and cohomology of operads
The aim of this paper is two-fold. First, we compare two notions of a "space" of algebra structures over an operad A: 1. the classification space, which is the nerve of the category of weak…
Simplicial structures on model categories and functors
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…
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A cartesian presentation of weak n–categories
We propose a notion of weak (n+k,n)-category, which we call (n+k,n)-Theta-spaces. The (n+k,n)-Theta-spaces are precisely the fibrant objects of a certain model category structure on the category of…
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An ∞‐categorical approach to R‐line bundles, R‐module Thom spectra, and twisted R‐homology
We develop a generalization of the theory of Thom spectra using the language of ∞‐categories. This treatment exposes the conceptual underpinnings of the Thom spectrum functor: we use a new model of…
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Notes on the Hopkins-Miller theorem
We give an exposition of the proof of a theorem of Hopkins and Miller, that the spectra En admit an action of the Morava stabilizer group.
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Units of ring spectra and Thom spectra
We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. We recall (from May, Quinn, and Ray) that a commutative ring spectrum A has a spectrum of units…
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The units of a ring spectrum and a logarithmic cohomology operation
Recall that if R is a commutative ring, then the set R× ⊂ R of invertible elements of R is naturally an abelian group under multiplication. This construction is a functor from commutative rings to…
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The congruence criterion for power operations in Morava $E$-theory
We prove a congruence criterion for the algebraic theory of power operations in Morava E-theory, analogous to Wilkerson's congruence criterion for torsion free lambda-rings. In addition, we provide a…
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Topological Modular Forms of Level 3
We describe and compute the homotopy of spectra of topological modular forms of level 3. We give some computations related to the "building complex" associated to level 3 structures at the prime 2.…
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