In mathematics, an E n {\displaystyle {\mathcal {E}}_{n}} -algebra in a symmetric monoidal infinity category C consists of the following data:
An object A ( U ) {\displaystyle A(U)} for any open subset U of Rn homeomorphic to an n-disk. A multiplication map:
μ : A ( U 1 ) ⊗ ⋯ ⊗ A ( U m ) → A ( V ) {\displaystyle \mu :A(U_{1})\otimes \cdots \otimes A(U_{m})\to A(V)}
for any disjoint open disks U j {\displaystyle U_{j}} contained in some open disk V subject to the requirements that the multiplication maps are compatible with composition, and that μ {\displaystyle \mu } is an equivalence if m = 1 {\displaystyle m=1} . An equivalent definition is that A is an algebra in C over the little n-disks operad.
Examples An E n {\displaystyle {\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. An E n {\displaystyle {\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 ↦ C ∗ ( Ω n X ; Λ ) {\displaystyle X\mapsto C_{*}(\Omega ^{n}X;\Lambda )} defines an E n {\displaystyle {\mathcal {E}}_{n}} -algebra in the infinity category of chain complexes of Λ {\displaystyle \Lambda } -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 "En-algebra", ncatlab.org