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Derived Algebraic Geometry II: Noncommutative Algebra

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arxiv math/0702299 v5 pith:M6U56AQV submitted 2007-02-11 math.CT math.AT

Jacob Lurie This is my paper · ORCID
classification math.CTmath.AT
keywords theorymonoidalspectracategorydevelopinfinityinfinity-categoricalprove
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In this paper, we present an infinity-categorical version of the theory of monoidal categories. We show that the infinity category of spectra admits an essentially unique monoidal structure (such that the tensor product preserves colimits in each variable), and thereby recover the classical smash-product operation on spectra. We develop a general theory of algebras in a monoidal infinity category, which we use to (re)prove some basic results in the theory of associative ring spectra. We also develop an infinity-categorical theory of monads, and prove a version of the Barr-Beck theorem.

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