En-ring

In mathematics, an $$\mathcal{E}_n$$-algebra in a symmetric monoidal infinity category C consists of the following data:
 * An object $$A(U)$$ for any open subset U of Rn homeomorphic to an n-disk.
 * A multiplication map:
 * $$\mu: A(U_1) \otimes \cdots \otimes A(U_m) \to A(V)$$
 * for any disjoint open disks $$U_j$$ contained in some open disk V

subject 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 $$\mathcal{E}_n$$-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 $$\mathcal{E}_n$$-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 $$X \mapsto C_*(\Omega^n X; \Lambda)$$ defines an $$\mathcal{E}_n$$-algebra in the infinity category of chain complexes of $$\Lambda$$-modules.