Gepner, David An introduction to higher categorical algebra. (English) Zbl 1476.55008 Miller, Haynes (ed.), Handbook of homotopy theory. Boca Raton, FL: CRC Press. CRC Press/Chapman Hall Handb. Math. Ser., 487-548 (2020). The Handbook of homotopy theory has been reviewed in [Zbl 1468.55001].Summary: This chapter deals with deformations of commutative algebras. Higher algebra is the study of algebraic structures that arise in the setting of higher category theory. Higher algebra generalizes ordinary algebra, or algebra in the setting of ordinary category theory. Ordinary algebra is set based, meaning that it is carried out in the language of ordinary categories. Higher categorical algebra is truly homotopical and not just homological in nature, meaning that many of its most important objects simply do not exist within the world of chain complexes or derived categories. The portion of the theory that can be formulated in these terms is differential graded algebra, the abstract study of which employs the language of differential graded categories. The chapter briefly reviews the theory of localization of ring spectra. An associative ring spectrum is an algebra object of the monoidal \(\infty\)-category of spectra. A commutative ring spectrum is a commutative algebra object of the symmetric monoidal \(\infty\)-category of spectra.For the entire collection see [Zbl 1468.55001]. Cited in 1 Review MSC: 55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology 13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) 13D09 Derived categories and commutative rings 16E45 Differential graded algebras and applications (associative algebraic aspects) 18C10 Theories (e.g., algebraic theories), structure, and semantics 18C35 Accessible and locally presentable categories 55U35 Abstract and axiomatic homotopy theory in algebraic topology 55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology Keywords:commutative algebras; deformations; higher category theory; differential graded algebra; localization of ring spectra Citations:Zbl 1468.55001 PDFBibTeX XMLCite \textit{D. Gepner}, in: Handbook of homotopy theory. Boca Raton, FL: CRC Press. 487--548 (2020; Zbl 1476.55008) Full Text: DOI arXiv