nLab A-infinity-algebra

Idea

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.

Definition

Definition

An $A_\infty$-algebra is an algebra over an operad over an A-∞ operad.

Realizations

In chain complexes

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

$$D : T^c V \to T^c V$$

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

$$D^2 = 0 \,.$$

Coderivations on free coalgebras are entirely determined by their “value on cogenerators”, which allows one to decompose $D$ as a sum:

$$D = D_1 + D_2 + D_3 + \cdots$$

with each $D_k$ specified entirely by its action

$$D_k : V^{\otimes k} \to V \,.$$

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

$$f_n : (A[1])^{\otimes n}\to B[1]$$

of degree $0$ satisfying

$$\sum_{0\leq i+j\leq n} f_{i+j+1}\circ(1^{\otimes i}\otimes D_{n-i-j}\otimes 1^{\otimes j}) = \sum_{i_1+\ldots+i_r=n} D_r\circ (f_{i_1}\otimes\ldots \otimes f_{i_r}).$$

For example, $f_1\circ D_1 = D_1\circ f_1$.

Rectification

Theorem

(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.

Remark

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

• de Rham cohomology

• Hochschild cohomology

etc.

In Topological space

An $A_\infty$-algebra in Top is also called an A-∞ space .

Examples

Every loop space is canonically an A-∞ space. (See there for details.)

Rectification

Theorem

Every $A_\infty$-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).

In spectra

See ring spectrum and algebra spectrum.

Related concepts

• $A_\infty$-algebra, A-∞-category

  • augmented A-∞ algebra

  • curved A-∞ algebra

• An-space

• A-n algebra

• E-∞ algebra

• L-∞ algebra, .

	(∞,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

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

References

A survey of $A_\infty$-algebras in chain complexes is in

• Bernhard Keller, A brief introduction to $A_\infty$-algebras (pdf)

Classical articles on $A_\infty$-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

• Martin Markl, Steve Shnider, James D. Stasheff, Operads in algebra, topology and physics, Math. Surveys and Monographs 96, Amer. Math. Soc. 2002.

The 1986 thesis of Alain Prouté explores the possibility of obtaining analogues of minimal models for $A_\infty$ algebras. It was published in TAC much later.

• Alain Prouté, Algèbres différentielles fortement homotopiquement associatives ($A_\infty$-algèbres), thesis, available as Reprints in Theory and Applications of Categories, No. 21, 2011, pp. 1–99