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

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

• A multiplication map: μ μ : A ( U 1 ) ⊗ ⊗ ⋯ ⋯ ⊗ ⊗ A ( U m ) → → A ( V ) {\displaystyle \mu :A(U_{1})\otimes \cdots \otimes A(U_{m})\to A(V)}

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

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

• Examples

• See also

• References

• External links

• An E n {\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.

• An E n {\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 X ↦ ↦ C ∗ ∗ ( Ω Ω n X ; Λ Λ ) {\displaystyle X\mapsto C_{*}(\Omega ^{n}X;\Lambda )} defines an E n {\displaystyle {\mathcal {E}}_{n}} -algebra in the infinity category of chain complexes of Λ Λ {\displaystyle \Lambda } -modules.

• Categorical ring

• Highly structured ring spectrum

• http://www.math.harvard.edu/~lurie/282ynotes/LectureXXII-En.pdf

• http://www.math.harvard.edu/~lurie/282ynotes/LectureXXIII-Koszul.pdf

• "En-algebra", ncatlab.org