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.

See also

*Categorical ring

References

*http://www.math.harvard.edu/~lurie/282ynotes/LectureXXII-En.pdf
*http://www.math.harvard.edu/~lurie/282ynotes/LectureXXIII-Koszul.pdf

External links

*http://ncatlab.org/nlab/show/En-algebra

Higher category theory
Homotopy theory

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