symmetric monoidal (∞,1)-category of spectra
An $A_\infty$-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 $A_\infty$-algebra is an algebra over an operad over an A-∞ operad.
Let here $\mathcal{E}$ be the category of chain complexes $\mathcal{Ch}_\bullet$. Notice that often in the literature this choice of $\mathcal{E}$ is regarded as default and silently assumed.
An $A_\infty$-algebra in chain complexes is concretely the following data.
A chain $A_\infty$-algebra is the structure of a degree 1 coderivation
on the reduced tensor coalgebra $T^c V = \oplus_{n\geq 1} V^{\otimes n}$ (with the standard noncocommutative coproduct, see differential graded Hopf algebra) over a graded vector space $V$ such that
Coderivations on free coalgebras are entirely determined by their “value on cogenerators”, which allows to decompose
with each $D_k$ specified entirely by its action
which is a map of degree $2-k$ (or can be alternatively understood as a map $D_k : (V[1])^{\otimes k}\to V[1]$ of degree $1$).
Then:
$D_1 : V\to V$ is the differential with $D_1^2 = 0$;
$D_2 : V^{\otimes 2} \to V$ is the product in the algebra;
$D_3 : V^{\otimes 3} \to V$ is the associator which measures the failure of $D_2$ to be associative;
$D_4 : V^{\otimes 4} \to V$ is the pentagonator (or so) which measures the failure of $D_3$ to satisfy the pentagon identity;
and so on.
One can also allow $D_0$, in which case one talks about weak $A_\infty$-algebras, which are less understood.
There is a resolution of the operad $\mathrm{Ass}$ of associative algebras (as operad on the category of chain complexes) which is called the $A_\infty$-operad; the algebras over the $A_\infty$-operad are precisely the $A_\infty$-algebras.
A morphism of $A_\infty$-algebras $f : A\to B$ is a collection $\lbrace f_n\rbrace_{n\geq 1}$ of maps
of degree $0$ satisfying
For example, $f_1\circ D_1 = D_1\circ f_1$.
(Kadeishvili (1980), Merkulov (1999))
If $A$ is a dg-algebra, regarded as a strictly associative $A_\infty$-algebra, its chain cohomology $H^\bullet(A)$, regarded as a chain complex with trivial differentials, naturally carries the structure of an $A_\infty$-algebra, unique up to isomorphism, and is weakly equivalent to $A$ as an $A_\infty$-algebra.
More details are at Kadeishvili's theorem.
This theorem provides a large supply of examples of $A_\infty$-algebras: there is a natural $A_\infty$-algebra structure on all cohomologies such as
etc.
An $A_\infty$-algebra in Top is also called an A-∞ space .
Every loop space is canonically an A-∞ space. (See there for details.)
Every $A_\infty$-space is weakly homotopy equivalent to a topological monoid.
This is a classical result by (Stasheff, 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.
$A_\infty$-algebra, A-∞-category
L-∞ algebra, .
algebraic deformation quantization
dimension | classical field theory | Lagrangian BV quantum field theory | factorization algebra of observables |
---|---|---|---|
general $n$ | P-n algebra | BD-n algebra? | E-n algebra |
$n = 0$ | Poisson 0-algebra | BD-0 algebra? = BD algebra | E-0 algebra? = pointed space |
$n = 1$ | P-1 algebra = Poisson algebra | BD-1 algebra? | E-1 algebra? = A-∞ algebra |
A survey of $A_\infty$-algebras in chain complexes is in
Classical articles on $A_\infty$-algebra in topological spaces are
A brief survey is in section 1.8 of
Martin Markl, Steve Shnider, James D. Stasheff, Operads in algebra, topology and physics, Math. Surveys and Monographs 96, Amer. Math. Soc. 2002.