En-ring

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:
$μ : A (U_{1}) \otimes \dots \otimes A (U_{m}) \to A (V)$

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.^{[citation needed]}
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 $X \mapsto C_{*} (Ω^{n} X; Λ)$ defines an $ℰ_{n}$ -algebra in the infinity category of chain complexes of $Λ$ -modules.

References

External links

"En-algebra", ncatlab.org

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En-ring

Examples

See also

References

External links