# 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.

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 $\simeq$ Γ-spaceinfinite loop space object
$\simeq$ connective spectrum$\simeq$ connective spectrum object
stabilizationspectrumspectrum object

## References

Section 5 of

some summary of which is at Ek-Algebras.

Discussion of derived noncommutative geometry over formal duals of $E_n$-algebras is in

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

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

Revised on April 1, 2015 13:58:54 by Urs Schreiber (195.113.30.252)