Derived Algebraic Geometry II: Noncommutative Algebra

Details Derived Algebraic Geometry II: Noncommutative Algebra Derived Algebraic Geometry II: Noncommutative Algebra

2007-02-11

arxiv.org/abs/math/0702299v5

Mathematics

Category Theory

170 pages; corrected some erroneous claims about the category of symmetric spectra, other minor changes. 9/19/07: minor modifi

Scientific paper

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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