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I'm not familiar with the subject and i don't know if my question make sense.

In Homological mirror symmetry and torus fibrations (http://arxiv.org/abs/math/0011041) a structure of a non-unital $A_{\infty}$-algebra on a $\mathbb{Z}$-graded vector space $B$ is defined by a codifferential $d$ of degree $1$ on the cofree tensor coalgebra $T_{+}(B[1]):=\oplus_{n\geq 1}(B[1])^{\otimes n}$. This differential graded coalgebra is non-unital. The coderivation is uniquely determined by $$ m_{n}\: :\: B^{\otimes n}\to B,\quad n\geq 0. $$ Assume $B=\oplus_{n\geq 0}B^{n}$ is a differential graded algebra, then it carries a non-unital $A_{\infty}$-algebra structure where $m_{1}$ is equal to the differential of $B$, $m_{2}$ is the multiplication and the higher $m_{n}$ are all equal to $0$. Now assume that $B$ has a unit $1\in B^{0}$. It turns out that $B$ is a non-unital $A_{\infty}$-algebra equipped with a strict unit (in the sense of $A_{\infty}$ algebras)!

Is there somewhere a notion of unital $A_{\infty}$-algebra? I guess that the reason for the name non-unital is the fact that $T_{+}(B[1])$ is non-unital as a coalgebra.

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  • $\begingroup$ A nice reference (but in french) is the chapter 3 of the Phd thesis of Kenji Lefevre-Hasegawa: "Sur les $A_{\infty}$-categories". $\endgroup$
    – Ilias A.
    Aug 17, 2015 at 19:15
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    $\begingroup$ Your question makes me think of operads. From that POV, a unital $A_\infty$ algebras is an algebra over a certain operad $O$, which has only the base field $k$ in degree 1. $B$'s structure is given by $O \to End(B)$, the endomorphism operad of $B$, or equivalently by maps $O(n)\otimes B^{\otimes n}\to B$. I think it's a good idea to work out what this implies about the elements of $B^1$. Anyway, up to quasi-isomorphism I don't think there's a difference, since both types of $A_\infty$ are supposed to rectify to dga structure. $\endgroup$ Aug 18, 2015 at 12:03
  • $\begingroup$ @DavidWhite Maybe I misunderstood your notation but it seems to me that there is a confusion. An operad with $\mathcal{O}(1)=k$ is a unital operad in the sense that this operad is a monoid with unit. If this operad has the property that $\mathcal{O}(0)=k$ (it is called reduced opera) then $\mathcal{O}$-algebras are unital. $\endgroup$
    – Ilias A.
    Aug 18, 2015 at 18:21
  • $\begingroup$ @AmraniIlias: Wow, that is an unfortunate state of affairs. So really the OP is asking about $A_\infty$ as a reduced operad. That's fine too, and the remark about rectification still holds, but it's just annoying that the terminology doesn't line up. $\endgroup$ Aug 18, 2015 at 22:23

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