ScalingStacks

Let ${\mathcal{V}}$ be a differential pointed category. A bimodule $2$-represesentation on ${\mathcal{V}}$ is the data of a strict monoidal differential pointed functor from the $2$-category with one object given by ${\mathcal{U}}^{\bullet}$ to $\mathrm{Bimod}^{\bullet}$. Note that a bimodule $2$-representation on ${\mathcal{V}}$ gives rise to a bimodule $2$-representation on $k[{\mathcal{V}}]$.

A bimodule lax bi-$2$-representation is a lax differential pointed $2$-functor $\Upsilon:{\mathcal{U}}^{\bullet}\wedge{\mathcal{U}}^{\bullet}\to\mathrm{Bimod}^{\bullet}$. We say it is a bimodule lax bi-$2$-representation on $\Upsilon(\ast\wedge\ast)$.

A bimodule lax bi-$2$-representation on ${\mathcal{V}}$ is the same as the data of

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  $({\mathcal{V}},{\mathcal{V}})$-bimodules $E_{i,j}$ for $i,j\geq 0$

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  morphisms of differential pointed algebras $H_{i}\wedge H_{j}\to\operatorname{End}\nolimits(E_{i,j})$

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  morphisms $\mu_{(i,j),(i^{\prime},j^{\prime})}:E_{i,j}E_{i^{\prime},j^{\prime}}\to E_{i+i^{\prime},j+j^{\prime}}$ satisfying properties (1) and (2) of §4.2.1.

We define the differential pointed category $\Delta_{E}({\mathcal{V}})$ as the quotient of $T_{{\mathcal{V}}}(E_{0,1}E_{1,0})$ by the equivalence relation generated by $f\sim f^{\prime}$ if $(f,f^{\prime})$ is in the equalizer of a composition

Consider a differential pointed category ${\mathcal{V}}$ endowed with two bimodule $2$-representations $(F_{1},\tau_{1})$ and $(E_{2},\tau_{2})$ and a closed morphism $\lambda:F_{1}E_{2}\to E_{2}F_{1}$ such that the diagrams (4.2.1) commute.

We define the differential pointed category $\Delta^{\prime}_{\lambda}({\mathcal{V}})$ as the quotient of $T_{{\mathcal{V}}}(F_{1}E_{2})$ by the equivalence relation generated by

We define the differential pointed category $\Delta_{\lambda}({\mathcal{V}})$. We consider first the differential pointed category with same objects as ${\mathcal{V}}$ and pointed set of maps $v_{1}\to v_{2}$ given by $\bigvee_{i\geq 0}E_{2}^{i}F_{1}^{i}(v_{1},v_{2})$. The category $\Delta_{\lambda}({\mathcal{V}})$ is the quotient of that category by the equivalence relation generated by $(T_{r}\wedge 1)(f)\sim(1\wedge T_{r})(f)$ for $f\in E_{2}^{i}F_{1}^{i}$ and $1\leq r<i$.

Note that there is a canonical isomorphism of differential categories for $?\in\{\emptyset,\prime\}$