finite total dimension, and also impose strict unitality properties. Bimodules form an $A_\infty$-category $(A,\tilde{A})^{\mathit{mod}}$ (just like in the case of modules, this has vanishing higher order compositions, hence could be considered as a dg category).

The standard example is the diagonal $A$-bimodule, $P(X,Y) = \mathit{hom}_A(X,Y)$ with $\mu^{s|1|t}_P = \mu^{s+1+t}_A$ (this also indirectly illustrates our sign conventions). We usually just write $P = A$. Another example is the one which appeared in : if $G: A \rightarrow \tilde{A}$ is an $A_\infty$-functor, one can define an $A$-bimodule $P(X,Y) = \mathit{hom}_{\tilde{A}}(GX,GY)$, with structure maps similar to the diagonal one but plugging in multiple copies of $G$. We again denote this by $P = \tilde{A}$, but always make sure to mention that the pullback to an $A$-bimodule is intended.

One traditional use of bimodules is as “kernels” defining “convolution functors” between categories of modules. The tensor product of an $A$-module $M$ and an $(A,\tilde{A})$-bimodule $P$ is an $\tilde{A}$-module $\tilde{M} = M \otimes_A P$, given by a bar construction $$\label{eq:ainfty-tensor}
\tilde{M}(\tilde{X}) = \bigoplus M(X_r) \otimes \mathit{hom}_A(X_{r-1},X_r)[1] \otimes \cdots \mathit{hom}_A(X_0,X_1)[1] \otimes P(\tilde{X},X_0),$$ where the sum is over all $r \geq 0$ and objects $(X_0,\dots,X_r)$. The induced differential is $$\begin{aligned}
& \mu^1_{\tilde{M}}(m \otimes a_r \otimes \cdots \otimes 