  In this article, we construct a $2$-category of Lagrangians in a fixed
shifted symplectic derived stack S. The objects and morphisms are all given by
Lagrangians living on various fiber products. A special case of this gives a
$2$-category of $n$-shifted symplectic derived stacks $Symp^n$. This is a
$2$-category version of Weinstein's symplectic category in the setting of
derived symplectic geometry. We introduce another $2$-category $Symp^{or}$ of
$0$-shifted symplectic derived stacks where the objects and morphisms in
$Symp^0$ are enhanced with orientation data. Using this, we define a partially
linearized $2$-category $LSymp$. Joyce and his collaborators defined a certain
perverse sheaf on any oriented $(-1)$-shifted symplectic derived stack. In
$LSymp$, the $2$-morphisms in $Symp^{or}$ are replaced by the hypercohomology
of the perverse sheaf assigned to the $(-1)$-shifted symplectic derived
Lagrangian intersections. To define the compositions in $LSymp$ we use a
conjecture by Joyce, that Lagrangians in $(-1)$-shifted symplectic stacks
define canonical elements in the hypercohomology of the perverse sheaf over the
Lagrangian. We refine and expand his conjecture and use it to construct $LSymp$
and a $2$-functor from $Symp^{or}$ to $LSymp$. We prove Joyce's conjecture in
the most general local model. Finally, we define a $2$-category of $d$-oriented
derived stacks and fillings. Taking mapping stacks into a $n$-shifted
symplectic stack defines a $2$-functor from this category to $Symp^{n-d}$.
