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:
:

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.