En-ring

In mathematics, an -algebra in a symmetric monoidal infinity category C consists of the following data:

  • An object for any open subset U of Rn homeomorphic to an n-disk.
  • A multiplication map:
for any disjoint open disks contained in some open disk V

subject to the requirements that the multiplication maps are compatible with composition, and that is an equivalence if . An equivalent definition is that A is an algebra in C over the little n-disks operad.

Examples

  • An -algebra in vector spaces over a field is a unital associative algebra if n=1, and a unital commutative associative algebra if n≥2.
  • An -algebra in categories is a monoidal category if n=1, a braided monoidal category if n=2, and a symmetric monoidal category if n≥3.
  • If Λ is a commutative ring, then defines an -algebra in the infinity category of chain complexes of -modules.

See also

References


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