ScalingStacks

Let us recall some aspects of Douglas-Manolescu’s theory [DouMa].

Note that Douglas and Manolescu work in the differential graded setting, and we translate their constructions to the differential setting.

Their nil-Coxeter $2$-algebra [DouMa, §2.2] can be viewed as the same data as our monoidal category ${\mathcal{U}}$ (cf [DouMa, Remark 2.4]). A bottom-algebra module [DouMa, §2.4] for the nil-Coxeter $2$-algebra is the same data as a lax bimodule $2$-representation on a differential algebra $A$, where a lax bimodule $2$-representation on $A$ is defined to be a lax $2$-functor $\Upsilon:{\mathcal{U}}\to\mathrm{Bimod}$ with $\Upsilon(1)$ the differential category with one object whose endomorphism ring is $A$. They also consider top-algebra modules, where ${\mathcal{U}}$ above is replaced by ${\mathcal{U}}^{\operatorname{opp}\nolimits}$. Using the isomorphism ${\mathcal{U}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{U}}^{\operatorname{opp}\nolimits}$ (§4.1.1), a top-algebra module can be viewed as a bottom-algebra module, hence as a lax bimodule $2$-representation.

Douglas and Manolescu define a tensor product of a top algebra-module and a bottom algebra-module [DouMa, Definition 2.11]. This corresponds to our construction of a differential algebra $A$ as a tensor product ${\hskip 1.9919pt\otimes\hskip-10.81218pt\bigcirc}$. Note that they do not endow this tensor product with any algebra-module structure.