representation, ∞-representation?
symmetric monoidal (∞,1)-category of spectra
An $E_n$-algebra is an ∞-algebra over the E-k operad?.
$E_1$-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.
The homology of an $E_2$-algebra in chain complexes is a Gerstenhaber algebra.
See E-∞ algebra.
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.
(∞,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 |
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)
Last revised on April 1, 2015 at 13:58:54. See the history of this page for a list of all contributions to it.