En-ring

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 : 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 {\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

• 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 n X ; 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 {\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