Derived Algebraic Geometry II: Noncommutative Algebra [article]

Jacob Lurie
2007 arXiv   pre-print
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
more » ... elop an infinity-categorical theory of monads, and prove a version of the Barr-Beck theorem.
arXiv:math/0702299v5 fatcat:6tdcjvtqtjctvlqxahu7kvrtcu