En-ring

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 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. 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 Categorical ring Highly structured ring spectrum

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

External links