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En ring : ウィキペディア英語版
En-ring

In mathematics, an \mathcal_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_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_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_n-algebra in the infinity category of chain complexes of \Lambda-modules.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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