En-ring

In mathematics, an [math]\displaystyle{ \mathcal{E}_n }[/math]-algebra in a symmetric monoidal infinity category C consists of the following data:

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

  [math]\displaystyle{ \mu: A(U_1) \otimes \cdots \otimes A(U_m) \to A(V) }[/math]

for any disjoint open disks [math]\displaystyle{ U_j }[/math] contained in some open disk V

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

Examples

• An [math]\displaystyle{ \mathcal{E}_n }[/math]-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 [math]\displaystyle{ \mathcal{E}_n }[/math]-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 [math]\displaystyle{ X \mapsto C_*(\Omega^n X; \Lambda) }[/math] defines an [math]\displaystyle{ \mathcal{E}_n }[/math]-algebra in the infinity category of chain complexes of [math]\displaystyle{ \Lambda }[/math]-modules.

See also

• Categorical ring

References

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

External links

• http://ncatlab.org/nlab/show/En-algebra