nLab En-algebra

Context

Higher algebra

higher algebra

universal algebra

Contents

Definition

An ${E}_{n}$-algebra is an ∞-algebra over the E-k operad.

Examples

${E}_{1}$-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 BV-algebra.

See E-∞ algebra.

Examples

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