En-algebra (changes) in nLab

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Contents

Definition

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.

Examples

• universal enveloping E-n algebra

Properties

Relation to Poisson nn-algebras

The homology of an E nE_n-algebra for n≥2n \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	∞-algebra	grouplike version	in Top	generally
	A-∞ operad	A-∞ algebra	∞-group	A-∞ space, e.g. loop space	loop space object
	E-k operad	E-k algebra	k-monoidal ∞-group	iterated loop space	iterated loop space object
	E-∞ operad	E-∞ algebra	abelian ∞-group	E-∞ space, if grouplike: infinite loop space ≃\simeq ∞-space	infinite loop space object
				≃\simeq connective spectrum	≃\simeq connective spectrum object
stabilization				spectrum	spectrum object

• looping and delooping, stabilization hypothesis

• May recognition theorem

• monoidal groupoid, braided monoidal groupoid, symmetric monoidal groupoid

• 2-group, braided 2-group, symmetric 2-group

• k-tuply monoidal n-groupoid

• k-tuply monoidal (n,r)-category

• k-tuply groupal n-groupoid

• A-n algebra

• nonunital Ek-algebra

References

Section 5 of

• Jacob Lurie, Higher Algebra,

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 ℰ n\mathcal{E}_n-Rings (pdf)

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