En ring - Alchetron, The Free Social Encyclopedia

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

An object A ( U ) for any open subset U of Rⁿ homeomorphic to an n-disk.

A multiplication map:

for any disjoint open disks U j contained in some open disk V

subject to the requirements that the multiplication maps are compatible with composition, and that μ is an equivalence if m = 1 . An equivalent definition is that A is an algebra in C over the little n-disks operad.

Examples

An E n -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 E n -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 X ↦ C ∗ ( Ω n X ; Λ ) defines an E n -algebra in the infinity category of chain complexes of Λ -modules.

References

En-ring Wikipedia

(Text) CC BY-SA