En-ring
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In mathematics, an -algebra in a symmetric monoidal infinity category C consists of the following data:
- An object for any open subset U of Rn homeomorphic to an n-disk.
- A multiplication map:
- for any disjoint open disks contained in some open disk V
subject to the requirements that the multiplication maps are compatible with composition, and that is an equivalence if . An equivalent definition is that A is an algebra in C over the little n-disks operad.
Examples
[edit]- An -algebra in vector spaces over a field is a unital associative algebra if n = 1, and a unital commutative associative algebra if n ≥ 2.[citation needed]
- An -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 defines an -algebra in the infinity category of chain complexes of -modules.
See also
[edit]References
[edit]- http://www.math.harvard.edu/~lurie/282ynotes/LectureXXII-En.pdf
- http://www.math.harvard.edu/~lurie/282ynotes/LectureXXIII-Koszul.pdf
External links
[edit]- "En-algebra", ncatlab.org