  This paper focuses on RCD(0,2)-spaces, which can be thought of as possibly
non-smooth metric measure spaces with non-negative Ricci curvature and
dimension less than 2. First, we establish a list of the compact topological
spaces admitting an RCD(0,2)-structure. Then, we describe the associated moduli
space of RCD(0,2)-structures for each of them. In particular, we show that all
these moduli spaces are contractible.
