In mathematics, an **-algebra** in a symmetric monoidal infinity category *C* consists of the following data:

- An object for any open subset
*U*of**R**^{n}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

- 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

## References

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

## External links