En-ring

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

• An object $\displaystyle{ A(U) }$ for any open subset U of Rn homeomorphic to an n-disk.
• A multiplication map:
$\displaystyle{ \mu: A(U_1) \otimes \cdots \otimes A(U_m) \to A(V) }$
for any disjoint open disks $\displaystyle{ U_j }$ contained in some open disk V

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

Examples

• An $\displaystyle{ \mathcal{E}_n }$-algebra in vector spaces over a field is a unital associative algebra if n = 1, and a unital commutative associative algebra if n ≥ 2.[citation needed]
• An $\displaystyle{ \mathcal{E}_n }$-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 $\displaystyle{ X \mapsto C_*(\Omega^n X; \Lambda) }$ defines an $\displaystyle{ \mathcal{E}_n }$-algebra in the infinity category of chain complexes of $\displaystyle{ \Lambda }$-modules.