nLab
A-infinity-algebra

Idea

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

Realizations

In chain complexes

Let here $ℰ$ be the category of chain complexes ${\mathrm{𝒞𝒽}}_{•}$. Notice that often in the literature this choice of $ℰ$ is regarded as default and silently assumed.

An $A_{\mathrm{\infty }}$-algebra in chain complexes is concretely the following data.

A chain $A_{\mathrm{\infty }}$-algebra is the structure of a degree 1 coderivation

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

on the reduced tensor coalgebra? $T^{c}V={\oplus }_{n\ge 1}{V}^{\otimes n}$ over a graded vector space $V$ (the coproduct is the unshuffle product) such that

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

Coderivations on free coalgebras are entirely determined by their “value on cogenerators”, which allows to decompose

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

with each $D_k$ specified entirely by its action

$$D_k:V^{\otimes k}\to V.$$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 $A_{\mathrm{\infty }}$-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 $A_{\mathrm{\infty }}$-operad; the algebras over the $A_{\mathrm{\infty }}$-operad are precisely the $A_{\mathrm{\infty }}$-algebras.

A morphism of $A_{\mathrm{\infty }}$-algebras $f:A\to B$ is a collection $\{f_n{\}}_{n\ge 1}$ of maps

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

of degree $0$ satisfying

$$\sum _{0\le i+j\le n}{f}_{i+j+1}\circ (1^{\otimes i}\otimes D_{n-i-j}\otimes 1^{\otimes j})=\sum _{i_1+\dots +i_r=n}{D}_r\circ (f_{i_1}\otimes \dots f_{i_r}).$$\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 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 $A_{\mathrm{\infty }}$-algebra, its chain cohomology $H^{•}(A)$, regarded as a chain complex with trivial differentials, naturally carries the structure of an $A_{\mathrm{\infty }}$-algebra, unique up to isomorphism, and is weakly equivalent to $A$ as an $A_{\mathrm{\infty }}$-algebra.

Remark

This theorem provides a large supply of examples of $A_{\mathrm{\infty }}$-algebras: there is a natural $A_{\mathrm{\infty }}$-algebra structure on all cohomologies such as

• de Rham cohomology

• Hochschild cohomology

etc.

In Topological space

An $A_{\mathrm{\infty }}$-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 $A_{\mathrm{\infty }}$-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).

In spectra

See ring spectrum and algebra spectrum.

Related concepts

• $A_{\mathrm{\infty }}$-algebra, A-∞-category

• 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 e.g. infinite loop space $\simeq$ Γ-space	infinite loop space object
				$\simeq$ connective spectrum	$\simeq$ connective spectrum object
stabilization				spectrum	spectrum object

• looping and delooping, stabilization hypothesis

References

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

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

Classical articles on $A_{\mathrm{\infty }}$-algebra in topological spaces are

• Jim Stasheff, Homotopy associativity of H-spaces I , Trans. Amer. Math. Soc. 108 (1963), p. 275-292.

• 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.