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.{{fact|date=December 2018}}
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.

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

En-ring

Examples

See also

References

External links