En-ring explained

In mathematics, an

l{E} _{n} -algebra in a symmetric monoidal infinity category C consists of the following data:A(U) for any open subset U of Rⁿ homeomorphic to an n-disk.

• A multiplication map:

\mu: A(U _{1)} ⊗ … ⊗ A(U _{m)} \to A(V)

for any disjoint open disks

U _{j} contained in some open disk Vsubject 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

l{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

l{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 ^{n} _{*(\Omega} X; Λ ) defines an l{E} _{n} -algebra in the infinity category of chain complexes of Λ -modules.

See also

• Categorical ring
• Highly structured ring spectrum

References

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