On points of convergence lattices and sobriety for convergence spaces

We characterize the convergence spaces $(X,\xi)$ such that the space of points of $(\mathbb{P}X,\lim_{\xi})$ in the category of convergence lattices is $(X,\xi)$. On the way, we study variants of sobriety and of the axiom $T_{D}$ in convergence spaces. New phenomena appear when leaving the realm of topological spaces. We obtain new hindsight into the space of points of a convergence lattice and study a special quotient of it, which, in the case $L=(\mathbb{P}X,\lim_{\xi})$ for a topological space $(X,\xi)$, turns out to be homeomorphic to the sobrification of $X$.