This entry collects links to $n$Lab-entries related to Higher Algebra: algebra inside monoidal (infinity,1)-categories .

Idea

Recall that ordinary algebra concerns itself with monoids internal to monoidal categories:

• a monoid internal to Set is just an ordinary monoid;

• a monoid internal to Ab, the category of abelian groups, is a ring;

• a monoid internal to Vect is an ordinary algebra: a vector space equipped with a linear binary associative product with unit;

• a monoid in a category of chain complexes is a differential graded algebra;

etc.

Higher algebra is similarly the study of monoids internal to higher categories.

A central motivating example example for or special case of the study of higher algebra was

• monoids internal to the stable (infinity,1)-category of spectra – called commutative ring spectra

The “higher algebra” embodied by commutative ring spectra has been called brave new algebra by F. Waldhausen.

More recently Jacob Lurie argued that the natural ambient formalism for “brave new algebra” is that of (symmetric) monoidal (infinity,1)-categories:

We should emphasize that the theory of A-infinity rings is not new. There are various definitions available in the the literature; see for example EKMM97. We have chosen to present the subject using the language of (infinity,1)-categories, which we feel is the natural home for these ideas. #

Related entries

The following links are ordered following the sections of the articles

• Jacob Lurie

  • Noncommutative algebra (pdf)

  • Commutative algebra (pdf)

See also

• EKMM97 Elmendorf, Kriz, Mandell, May, Rings, modules and algebras in stable homotopy theory, Mathematical surveys and monographs 47, American Mathematical Society, 1997

Monoidal $(\mathrm{\infty }\mathrm{,1})$-Categories

• monoidal category

  • monoid

  • algebra

• monoidal (infinity,1)-category

• algebra in an (infinity,1)-category

Modules

• module

Monads and the Barr-Beck theorem

• monad

The monoidal structure on stable homotopy theory

• stable homotopy theory

  • stable (infinity,1)-category of spectra

  • stable homotopy category

• smash product of spectra

  • symmetric monoidal smash product of spectra

Symmetric monoidal $(\mathrm{\infty }\mathrm{,1})$-categories and commutative algebra

• symmetric monoidal (infinity,1)-category

• commutative algebra in an (infinity,1)-category

• examples

  • stable (infinity,1)-category of spectra

  • symmetric monoidal (infinity,1)-category of presentable (infinity,1)-categories

Commutative ring spectra

• stable (infinity,1)-category of spectra

  • spectrum

  • spectrum object

• associative ring spectrum

  • A-infinity ring

• commutative ring spectrum

  • E-infinity ring

Symmetric monoidal model categories

• model category