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Let $k$ be a commutative ring and $C$ a $ \mathbb{Z}$ graded $k$ module. A $B_\infty$ structure on $C$ is the datum of a differential and a multiplication on $ BC:= \bigoplus^\infty_{i =0} C[1]^{\otimes i}$, the graded tensor coalgebra, such that $(BC, d, \mu, \delta, \epsilon, \eta)$ is a differential graded bialgebra. I haven't been able to find an explicit definition of morphism between two $B_\infty$ algebras. I would suspect that it would be a graded morphism of $k$ modules such that the induced map on the $BC$ respects the differential graded bialgebra structure. Is this the right idea?

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  • $\begingroup$ There are many undefined terms in your question. $\endgroup$ May 28, 2014 at 21:42
  • $\begingroup$ Yes, I think you are right that it will be given by a map of graded modules such that the induced map on bar coalgebras is a map of dg-bialgebras. In his paper "Derived Invariance of Higher Structures on the Hochschild Complex" Bernhard Keller mentions (just after defining them) that $B_{\infty}$-structure can be equivalently described as a family of maps $m_{k}: C^{\otimes k} \rightarrow C$ (ie. an $A_{\infty}$-structure) together with a family of maps $m_{k, l}: C^{\otimes k+l} \rightarrow C$. $\endgroup$ May 29, 2014 at 11:16
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    $\begingroup$ This is done by more-or-less explicit formulas, since the bar coalgebra is just a direct sum of products (of suspensions) of $C$, so any map defined on it will be in fact a family of maps on tensor products of (suspensions of) $C$. Thus, a map of graded modules $C_{1} \rightarrow C_{2}$ will commute with $B_{\infty}$-strutures (the way you defined it) if and only if it commutes with these families $m_{k}, m_{k, l}$. Moreover, to say that the maps defined on $BC_{i}$ give the structure of a dg-bialgebra could be in principle translated into a set of equations on these families of maps. $\endgroup$ May 29, 2014 at 11:21
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    $\begingroup$ The fact that you can do this translation is implicit in Keller's paper when he writes that "$B_{\infty}$-algebras can be considered as the algebras over a certain asymmetric operad", one can imagine that this will be something like a free operad generated by these maps $m_{k}, m_{k, l}$ subject to some complicated relations. However, a map of algebras over this operad will be exactly a map of graded modules commuting with all these maps (since they generate the whole operad). $\endgroup$ May 29, 2014 at 11:26
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    $\begingroup$ You are actually correct I think: See this paper on pg. 1562. win.ua.ac.be/~wlowen/pdf%20papers/COM%20144.6%20Lowen.pdf :) $\endgroup$
    – Anette
    May 29, 2014 at 11:40

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