nLab A-infinity-algebra

1. 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.

2. Definition

Definition 2.1. An $A_\infty$-algebra is an algebra over an operad over an A-∞ operad.

3. 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 3.1. (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.

Remark 3.2. 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 3.3. 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.

4. 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

5. References

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

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

• Bernhard Keller, Introduction to A-infinity algebras and modules, Homology, Homotopy and Applications 3 1 (2001) 1-35 [doi:10.4310/hha.2001.v3.n1.a1, arXiv:math/9910179]

• Bernhard Keller, A-infinity algebras, modules and functor categories, Contemp. Math. 406 (2006) 67–93 [doi:10.1090/conm/406, arXiv:math/0510508]

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, Springer Berlin, Heidelberg 1973. [doi:10.1007/BFb0068547]

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. [doi:10.1090/surv/096]

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