nLab En-algebra




An E nE_n-algebra is an ∞-algebra over the E-k operad.

Special cases

E 1E_1-algebras

E 1E_1-algebras are often called A-∞ algebras. See also algebra in an (∞,1)-category.

An E 1E_1 algebra in the symmetric monoidal (∞,1)-category Spec of spectra is a ring spectrum.

E 2E_2-algebras

The homology of an E 2E_2-algebra in chain complexes is a Gerstenhaber algebra.

E E_\infty-algebra

See E-∞ algebra.



Relation to Poisson nn-algebras

The homology of an E nE_n-algebra for n2n \geq 2 is a Poisson n-algebra.

Moreover, in chain complexes over a field of characteristic 0 the E-n operad is formal, hence equivalent to its homology, and so in this context E nE_n-algebras are equivalent to Poisson n-algebras.

See there for more.

(∞,1)-operad∞-algebragrouplike versionin Topgenerally
A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operadE-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space \simeq ∞-spaceinfinite loop space object
\simeq connective spectrum\simeq connective spectrum object
stabilizationspectrumspectrum object


Section 5 of

some summary of which is at Ek-Algebras.

Discussion of derived noncommutative geometry over formal duals of E nE_n-algebras is in

  • John Francis, Derived algebraic geometry over n\mathcal{E}_n-Rings (pdf)

  • John Francis, The tangent complex and Hochschild cohomology of n\mathcal{E}_n-rings (pdf)

Last revised on June 25, 2022 at 19:04:33. See the history of this page for a list of all contributions to it.