Moduli stacks of Galois representations and the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$

We give a categorical formulation of the $p$-adic local Langlands correspondence for $\mathrm{GL}_2(\mathbb{Q}_p)$,as an embedding of the derived category of locally admissible representations into the category of Ind-coherent sheaves on the moduli stack of two-dimensional representations of $\mathr