nLab
A-infinity-algebra

An $A_{\mathrm{\infty }}$-algebra is an algebra for the cofibrant resolution of the operad $\mathrm{Ass}$ of associative algebras: it is an algebra which is associative only up to higher coherent homotopy.

Concretely, an $A_{\mathrm{\infty }}$-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

$$f_n:(A[1]{)}^{\otimes n}\to A[1]$$f_n : (A[1])^{\otimes n}\to A[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, $f_1\circ D_1=D_1\circ f_1$.

Remarks

• See also L-infinity-algebra, A-infinity-category.

References

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