E_n-ring

In mathematics, an ${\mathcal {E}}_{n}$ -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:
$\mu :A(U_{1})\otimes \cdots \otimes A(U_{m})\to A(V)$

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 $\mu$ 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 ${\mathcal {E}}_{n}$ -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 ${\mathcal {E}}_{n}$ -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 $X\mapsto C_{*}(\Omega ^{n}X;\Lambda )$ defines an ${\mathcal {E}}_{n}$ -algebra in the infinity category of chain complexes of $\Lambda$ -modules.

References

External links

http://ncatlab.org/nlab/show/En-algebra

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En-ring

Examples

See also

References

External links

E_n-ring