A Derived Legendrian Category for Shifted Contact Stacks

We construct the derived Legendrian category ${\mathcal{F}}_c(X)$ for an $n$-shifted contact derived Artin stack $X$ and the $(\mathrm{\infty },2)$-category $Le{g}_n$ of Legendrian correspondences in the context of derived algebraic geometry, with several applications to moduli theory. In brief, the objects of the category ${\mathcal{F}}_c(X)$ are Legendrian morphisms; the morphism spaces and composition operations are defined using equivariant descent. We also establish that ${\mathcal{F}}_c(X)$ embeds into an $(\mathrm{\infty },2)$-category of spans defined by the AKSZ construction. We further evaluate topological cobordisms as Lagrangian correspondences to define derived Legendrian surgery.