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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.


  • 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


External links