En-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.

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