Tensor product of a DGA and an $A_\infty$ algebra

Question

In general there seems no way to naturally define the tensor product of two $A_\infty$ algebras $A$ and $B$. But, if $(A, m^A_1,m^A_2)$ is only a DGA(differential graded algebra) and $(B, m^B_k, k\ge 1) $ is an $A_\infty$ algebra, then is there a natural way to get an $A_\infty$ algebra structure on the tensor product $A\otimes B$?

I guess this should be correct. But for the safety, I was wondering if there is a standard reference for this fact.

Moreover, if this is right, what I really want is an explicit formula of the $A_\infty$ algebra structure on $A\times B$ in terms of $m^A$ and $m^B$. Thank you!

Answer 1

In fact the tensor product of two $A_\infty$ algebras can be made into an $A_\infty$ algebra in an explicit way: there are two constructions, one by Saneblidze-Umble and one by Loday. See the paper https://arxiv.org/abs/0710.0572

(For cofibrancy reasons one also knows abstractly that there is such a tensor product, but this of course doesn't give a formula.)

Answer 2

To follow up on Dan's answer, a geometric construction of the Saneblidze--Umble diagonal is given by Masuda--Tonks--Thomas--Vallette in http://www.numdam.org/item/JEP_2021__8__121_0/, and an alternative description of the SU combinatorics is given here https://arxiv.org/abs/2308.12119 (see in particular Theorem 5.17).

A similar construction gives a tensor product of A-infinity morphisms https://jep.centre-mersenne.org/articles/10.5802/jep.221/. In fact, there are two distinct formulas for the tensor product of A-infinity morphisms (see Part III in https://arxiv.org/abs/2308.12119), whereas there is only one for the product of A-infinity algebras!