En-ring

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In mathematics, an ${\displaystyle {\mathcal {E}}_{n}}$-algebra in a symmetric monoidal infinity category C consists of the following data:

• An object ${\displaystyle A(U)}$ for any open subset U of Rⁿ homeomorphic to an n-disk.
• A multiplication map:

  ${\displaystyle \mu :A(U_{1})\otimes \cdots \otimes A(U_{m})\to A(V)}$

for any disjoint open disks ${\displaystyle U_{j}}$ contained in some open disk V

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

Examples

• An ${\displaystyle {\mathcal {E}}_{n}}$-algebra in vector spaces over a field is a unital associative algebra if n = 1, and a unital commutative associative algebra if n ≥ 2.^{[citation needed]}
• An ${\displaystyle {\mathcal {E}}_{n}}$-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 ${\displaystyle X\mapsto C_{*}(\Omega ^{n}X;\Lambda )}$ defines an ${\displaystyle {\mathcal {E}}_{n}}$-algebra in the infinity category of chain complexes of ${\displaystyle \Lambda }$-modules.

See also

• Categorical ring
• Highly structured ring spectrum

References

• J. Lurie, http://www.math.harvard.edu/~lurie/282ynotes/LectureXXII-En.pdf
• J. Lurie, http://www.math.harvard.edu/~lurie/282ynotes/LectureXXIII-Koszul.pdf

External links

• "En-algebra", ncatlab.org

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