View source for En-ring

{{DISPLAYTITLE:''E<sub>n</sub>''-ring}}
{{inline |date=May 2024}}
In [[mathematics]], an '''<math>\mathcal{E}_n</math>-algebra''' in a [[symmetric monoidal category|symmetric monoidal]] [[infinity category]] ''C'' consists of the following data:
*An [[object (category theory)|object]] <math>A(U)</math> for any [[open subset]] ''U'' of '''R'''<sup>n</sup> [[homeomorphic]] to an ''n''-disk.
*A multiplication map:
*:<math>\mu: A(U_1) \otimes \cdots \otimes A(U_m) \to A(V)</math>
:for any [[disjoint sets|disjoint]] open disks <math>U_j</math> contained in some open disk ''V''
subject to the requirements that the multiplication maps are compatible with composition, and that <math>\mu</math> is an equivalence if <math>m=1</math>.  An equivalent definition is that ''A'' is an [[algebra over an operad|algebra]] in ''C'' over the little ''n''-disks [[operad]].

== Examples ==
* An <math>\mathcal{E}_n</math>-algebra in [[vector space]]s over a [[field (mathematics)|field]] is a [[unital associative algebra]] if ''n'' = 1, and a unital [[commutative algebra|commutative associative algebra]] if ''n'' ≥ 2.{{fact|date=December 2018}}
* An <math>\mathcal{E}_n</math>-algebra in [[category (mathematics)|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>X \mapsto C_*(\Omega^n X; \Lambda)</math> defines an <math>\mathcal{E}_n</math>-algebra in the infinity category of [[chain complex]]es of <math>\Lambda</math>-[[module (mathematics)|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

== External links ==
*{{citation |title=En-algebra |url=http://ncatlab.org/nlab/show/En-algebra |website=ncatlab.org}}

[[Category:Higher category theory]]
[[Category:Homotopy theory]]


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