nLab En-algebra

Context

Higher algebra

higher algebra

universal algebra

Contents

Definition

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

Special cases

$E_1$-algebras

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

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

$E_2$-algebras

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

See E-∞ algebra.

Properties

Relation to Poisson $n$-algebras

The homology of an $E_n$-algebra for $n \geq 2$ 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 $E_n$-algebras are equivalent to Poisson n-algebras.

See there for more.

E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space $\simeq$ Γ-spaceinfinite loop space object
$\simeq$ connective spectrum$\simeq$ connective spectrum object