symmetric monoidal (∞,1)-category of spectra
An -algebra is a monoid internal to a homotopical category such that the associativity law holds not as an equation, but only up to higher coherent homotopy.
An -algebra is an algebra over an operad over an A-∞ operad.
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
on the reduced tensor coalgebra (with the standard noncocommutative coproduct, see differential graded Hopf algebra) over a graded vector space such that
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 ).
Then:
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
etc.
An -algebra in Top is also called an A-∞ space .
Every loop space is canonically an A-∞ space. (See there for details.)
Every -space is weakly homotopy equivalent to a topological monoid.
This is a classical result by (Stasheff 1963, BoardmanVogt). It follows also as a special case of the more general result on rectification in a model structure on algebras over an operad (see there).
See ring spectrum and algebra spectrum.
-algebra, A-∞-category
L-∞ algebra, .
algebraic deformation quantization
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
Jim Stasheff, Homotopy associativity of H-spaces I, Trans. Amer. Math. Soc. 108 2 (1963) 275-292 [doi:10.2307/1993608]
Jim Stasheff, Homotopy associativity of H-spaces II 108 2 (1963) 293-312 [doi:10.2307/1993609, doi:10.1090/S0002-9947-1963-0158400-5]
Michael Boardman and Rainer Vogt, Homotopy invariant algebraic structures on topological spaces , Lect. Notes Math. 347 (1973).
A brief survey is in section 1.8 of
The 1986 thesis of Alain Prouté explores the possibility of obtaining analogues of minimal models for algebras. It was published in TAC much later.
Last revised on July 5, 2022 at 12:03:56. See the history of this page for a list of all contributions to it.