symmetric monoidal (∞,1)-category of spectra
Let here be the category of chain complexes . Notice that often in the literature this choice of is regarded as default and silently assumed.
An -algebra in chain complexes is concretely the following data.
A chain -algebra is the structure of a degree 1 coderivation
Coderivations on free coalgebras are entirely determined by their “value on cogenerators”, which allows one to decompose as a sum:
with each specified entirely by its action
which is a map of degree (or can be alternatively understood as a map of degree ).
is the differential with ;
is the product in the algebra;
is the associator which measures the failure of to be associative;
is the pentagonator (or so) which measures the failure of to satisfy the pentagon identity;
and so on.
One can also allow , in which case one talks about weak -algebras, which are less understood.
There is a resolution of the operad of associative algebras (as operad on the category of chain complexes) which is called the -operad; the algebras over the -operad are precisely the -algebras.
A morphism of -algebras is a collection of maps
of degree satisfying
For example, .
(Kadeishvili (1980), Merkulov (1999))
If is a dg-algebra, regarded as a strictly associative -algebra, its chain cohomology , regarded as a chain complex with trivial differentials, naturally carries the structure of an -algebra, unique up to isomorphism, and is weakly equivalent to as an -algebra.
More details are at Kadeishvili's theorem.
This theorem provides a large supply of examples of -algebras: there is a natural -algebra structure on all cohomologies such as
|(∞,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 Γ-space||infinite loop space object|
|connective spectrum||connective spectrum object|
|dimension||classical field theory||Lagrangian BV quantum field theory||factorization algebra of observables|
|general||P-n algebra||BD-n algebra?||E-n algebra|
|Poisson 0-algebra||BD-0 algebra? = BD algebra||E-0 algebra? = pointed space|
|P-1 algebra = Poisson algebra||BD-1 algebra?||E-1 algebra? = A-∞ algebra|
A survey of -algebras in chain complexes is in
Classical articles on -algebra in topological spaces are
A brief survey is in section 1.8 of