ScalingStacks

Let us give two applications of the construction of §4.3. Let $({\mathcal{V}}_{1},E_{1},\tau_{1})$ and $({\mathcal{V}}_{2},E_{2},\tau_{2})$ be idempotent-complete strongly pretriangulated $2$-representations.

We view ${\mathcal{V}}_{1}\otimes{\mathcal{V}}_{2}$ as endowed with two strictly commuting actions of ${\mathcal{U}}$ given by $(E_{1}\otimes 1,\tau_{1}\otimes 1)$ and $(1\otimes E_{2},1\otimes\tau_{2})$: the isomorphism $\sigma:(1\otimes E_{2})\circ(E_{1}\otimes 1)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}(E_{1}\otimes 1)\circ(1\otimes E_{2})$ is the identity.

We define the tensor product $2$-representation

Given $(\Phi_{i},\varphi_{i}):{\mathcal{V}}_{i}\to{\mathcal{V}}^{\prime}_{i}$ a morphism of $2$-representations for $i\in\{1,2\}$, Proposition 4.3.9 provides a morphism of $2$-representations ${\mathcal{V}}_{1}{\hskip 1.9919pt\otimes\hskip-10.81218pt\bigcirc}{\mathcal{V}}_{2}\to{\mathcal{V}}^{\prime}_{1}{\hskip 1.9919pt\otimes\hskip-10.81218pt\bigcirc}{\mathcal{V}}^{\prime}_{2}$.

Given ${\mathcal{V}}_{1}$, ${\mathcal{V}}_{2}$ and ${\mathcal{V}}_{3}$ $2$-representations, Proposition 4.3.12 provides an isomorphism

that commutes with forgetful functors $\omega$.

Since the forgetful functors $\omega$ are faithful, we deduce that idempotent-complete strongly pretriangulated $2$-representations form a monoidal $2$-category.

Consider now $\operatorname{Hom}\nolimits({\mathcal{V}}_{1},{\mathcal{V}}_{2})$. It is endowed with two strictly commuting structures of $2$-representations: the first one is given by $((\Phi\mapsto\Phi\circ E_{1}),\Phi\tau_{1})$ and the second one by $((\Phi\mapsto E_{2}\circ\Phi),\tau_{2}\Phi)$. The isomorphism $\sigma$ is the identity.

We define the internal $\operatorname{Hom}\nolimits$ $2$-representation

The category ${\mathcal{H}}\hskip-5.69046pt{\mathcal{H}}{om}({\mathcal{V}}_{1},{\mathcal{V}}_{2})$ has objects pairs $(\Phi,\pi)$ where $\Phi:{\mathcal{V}}_{1}\to\overline{{\mathcal{V}}_{2}}^{i}$ is a differential functor and $\pi:E_{2}\Phi\to\Phi E_{1}$ is a closed natural transformation of functors such that

Note that $\operatorname{Hom}\nolimits_{\mathcal{U}}({\mathcal{V}}_{1},{\mathcal{V}}_{2})$ is the full subcategory of ${\mathcal{H}}\hskip-5.69046pt{\mathcal{H}}{om}({\mathcal{V}}_{1},{\mathcal{V}}_{2})$ with objects pairs $(\Phi,\pi)$ where $\Phi$ takes values in ${\mathcal{V}}_{2}$ and $\pi$ is invertible.

Given $(\Phi_{1},\varphi_{1}):{\mathcal{V}}^{\prime}_{1}\to{\mathcal{V}}_{1}$ and $(\Phi_{2},\varphi_{2}):{\mathcal{V}}_{2}\to{\mathcal{V}}^{\prime}_{2}$ two morphisms of $2$-representations, Proposition 4.3.9 provides a morphism of $2$-representations ${\mathcal{H}}\hskip-5.69046pt{\mathcal{H}}{om}({\mathcal{V}}_{1},{\mathcal{V}}_{2})\to{\mathcal{H}}\hskip-5.69046pt{\mathcal{H}}{om}({\mathcal{V}}^{\prime}_{1},{\mathcal{V}}^{\prime}_{2})$.