ScalingStacks

Let ${\mathcal{C}}$ be a category. We denote by ${\mathcal{C}}^{\operatorname{opp}\nolimits}$ the opposite category. We identify ${\mathcal{C}}$ with a full subcategory of $\operatorname{Hom}\nolimits({\mathcal{C}}^{\operatorname{opp}\nolimits},\mathrm{Sets})$ via the Yoneda embedding $c\mapsto\operatorname{Hom}\nolimits(-,c)$.

Given $(L,R)$ a pair of adjoint functors, we denote the unit of the adjunction by $\eta_{L,R}$ and the counit by $\varepsilon_{L,R}$.

When ${\mathcal{C}}$ is enriched in abelian groups, we denote by $\operatorname{add}\nolimits({\mathcal{C}})$ the smallest full subcategory of $\operatorname{Hom}\nolimits({\mathcal{C}}^{\operatorname{opp}\nolimits},\mathrm{Sets})$ containing ${\mathcal{C}}$ and closed under finite coproducts and isomorphisms.

Let ${\mathcal{X}}$ be a $2$-category. We denote by ${\mathcal{X}}^{\operatorname{opp}\nolimits}$ the $2$-category with same objects and ${{\mathcal{H}}om}(x,y)={{\mathcal{H}}om}(x,y)^{\operatorname{opp}\nolimits}$. We denote by ${\mathcal{X}}^{\mathrm{rev}}$ the $2$-category with the same objects and with ${{\mathcal{H}}om}(x,y)={{\mathcal{H}}om}(y,x)$ for $x$ and $y$ two objects of ${\mathcal{X}}$ (so that the composition of $1$-arrows is reversed).

Let ${\mathcal{C}}{at}$ be the $2$-category of categories. There is an equivalence ${\mathcal{C}}{at}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{C}}{at}^{\operatorname{opp}\nolimits}$ sending a category ${\mathcal{C}}$ to ${\mathcal{C}}^{\operatorname{opp}\nolimits}$.

Let ${\mathcal{C}}{at}^{r}$ (resp. ${\mathcal{C}}{at}^{l}$) be the $2$-full $2$-subcategory of ${\mathcal{C}}{at}$ with $1$-arrows those functors that admit a left (resp. right) adjoint. There is an equivalence of $2$-categories ${\mathcal{C}}{at}^{r}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}({\mathcal{C}}{at}^{l})^{\mathrm{rev}{\operatorname{opp}\nolimits}}$. It is the identity on objects and sends a functor to a left adjoint.