4.2 Non-unital A ∞ -algebras and A ∞ -categories [03R0]

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4.2 Non-unital $A_{\infty}$ -algebras and $A_{\infty}$ -categories

Let $A=\oplus_{i\in Z}A^{i}$ be a ${\bf Z}$ -graded module over a ${\bf Z}$ -graded commutative associative algebra $k$ . As usual, we will denote by $A[n]$ the graded $k$ -module such that $(A[n])^{i}=A^{i+n}$ for all $i$ .

Definition 3

A structure of non-unital $A_{\infty}$ -algebra on $A$ is given by a codifferential $d$ of degree $+1$ on the cofree tensor coalgebra $T_{+}(A[1])=\oplus_{n\geq 1}(A[1])^{\otimes n}$ .

The codifferential $d$ is by definition a coderivation, such that $d^{2}=0$ . It is uniquely determined by its “Taylor coefficients” $m_{n}:A^{\otimes n}\to A[2-n],n\geq 1$ . The condition $d^{2}=0$ can be rewritten as a sequence of quadratic equations

\sum_{i+j=n+1}\sum_{0\leq l\leq i}\epsilon(l,j)m_{i}(a_{0},...,a_{l-1},m_{j}(a_{l},...,a_{l+j-1}),a_{l+j},...,a_{n})=0

where $a_{m}\in A$ , and $\epsilon(l,j)=(-1)^{j\sum_{0\leq s\leq l-1}deg(a_{s})}$ . In particular, $m_{1}^{2}=0$ .

Definition 4

A morphism of non-unital $A_{\infty}$ -algebras ( $A_{\infty}$ -morphism for short) $(V,d_{V})\to(W,d_{W})$ is a morphism of tensor coalgebras $T_{+}(V[1])\to T_{+}(W[1])$ of degree zero, which commutes with the codifferentials.

A morphism $f$ of non-unital $A_{\infty}$ -algebras is determined by its “Taylor coefficients” $f_{n}:V^{\otimes n}\to W[1-n],n\geq 1$ satisfying the system of equations

$\sum_{1\leq l_{1}<...,<l_{i}=n}\pm m_{i}^{W}(f_{l_{1}}(a_{1},...,a_{l_{1}}),$
$f_{l_{2}-l_{1}}(a_{l_{1}+1},...,a_{l_{2}}),...,f_{n-l_{i-1}}(a_{n-l_{i-1}+1},...,a_{n}))=$

$\sum_{s+r=n+1}\sum_{1\leq j\leq s}\pm f_{s}(a_{1},...,a_{j-1},m_{r}^{V}(a_{j},...,a_{j+r-1}),a_{j+r},...,a_{n}).$

We leave to the reader as an exercise to write down the formulas for the signs in terms of degrees of $a_{i}$ and $f_{i}$ .

Definition 5

A non-unital $A_{\infty}$ -category ${\cal C}$ over $k$ is given by the following data:

1) A class of objects $Ob({\cal C})$ .

2) For any two objects $X_{1}$ and $X_{2}$ a ${{\bf Z}}$ -graded $k$ -module of morphisms $Hom(X_{1},X_{2})$ .

3) For any sequence of objects $X_{0},...,X_{n}$ , $n\geq 1$ , a morphism of $k$ -modules (called a composition map) $m_{n}:\otimes_{0\leq i\leq n-1}Hom(X_{i},X_{i+1})\to Hom(X_{0},X_{n})[2-n]$ .

It is required that for any sequence of objects $X_{0},...,X_{N}$ , $N\geq 0$ the graded $k$ -module $A=A(X_{0},...,X_{N}):=\oplus_{i,j}Hom(X_{i},X_{j})$ , equipped with the direct sum of the compositions $m_{n},n\geq 1$ , is a non-unital $A_{\infty}$ -algebra.

The class of objects $Ob({\cal C})$ will be often denoted by ${\cal C}$ . We hope it will not lead to a confusion.

Remark 8

A non-unital $A_{\infty}$ -algebra $A$ can be considered as a non-unital $A_{\infty}$ -category with one object $X$ such that $Hom(X,X)=A$ .

Definition 6

A functor $F:{\cal C}_{1}\to{\cal C}_{2}$ between non-unital $A_{\infty}$ -categories is given by the following data:

1) A map of classes of objects $\phi:{\cal C}_{1}\to{\cal C}_{2}$ .

2) For any finite sequence of objects $X_{0},...,X_{n}$ , $n\geq 0$ , a morphism of graded $k$ -modules $f_{n}:\otimes_{0\leq i\leq n-1}Hom_{{\cal C}_{1}}(X_{i},X_{i+1})\to Hom_{{\cal C}_{2}}(\phi(X_{0}),\phi(X_{n}))[1-n].$

The following condition holds for any $X_{1},...,X_{N}\in{\cal C}_{1}$ : the sequence $f_{n},n\geq 1$ defines an $A_{\infty}$ -morphism

\oplus_{i,j}Hom_{{\cal C}_{1}}(X_{i},X_{j})\to\oplus_{i,j}Hom_{{\cal C}_{2}}(\phi(X_{i}),\phi(X_{j})).

Remark 9

Let ${\cal C}$ be a non-unital $A_{\infty}$ -category. Let us replace spaces of morphisms by their cohomology with respect to $m_{1}$ . In other words, we define $Hom_{H({\cal C})}(X,Y):=\{Ker\,m_{1}\}/\{Im\,m_{1}\}$ , where $m_{1}:Hom_{\cal C}(X,Y)\to Hom_{\cal C}(X,Y)[1]$ is the composition map. Then $H({\cal C})=({\cal C},Hom_{H({\cal C})}(\cdot,\cdot))$ gives rise to a “non-unital” category structure with the class of objects ${\cal C}$ and composition of morphisms induced by $m_{2}$ . We write “non-unital” because there are no identity morphisms $id_{X}\in Hom_{H({\cal C})}(X,X)$ .