The goal of the paper is to set the foundations and prove some topological results about moduli spaces of non-smooth metric measure structures with non-negative Ricci curvature in a synthetic sense (via optimal transport) on a compact topological space; more precisely, we study moduli spaces of RCD(0, N )structures. First, we relate the convergence of RCD(0, N )-structures on a space to the associated lifts' equivariant convergence on the universal cover. Then we construct the Albanese and soul maps, which reflect how structures on the universal cover split, and we prove their continuity. Finally, we construct examples of moduli spaces of RCD(0, N )-structures that have non-trivial rational homotopy groups.CONTENTS 2.1. Covering space theory of RCD(0,N)-spaces 2.2. Splittings and topological invariants 2.3. Moduli spaces and their topology 2.4. Maps between moduli spaces 3. Proof of the main results 3.1. Proof of Theorem A 3.2. Proof of Theorem B 3.3. Proof of Theorem C Appendix References

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. CONTENTS1. Introduction 1.1. Basic definitions 1.2. Main results 1.3. Sketch of the proofs Organization of the paper Acknowledgements Preliminaries 2. Equivariance 3. Albanese variety and soul 4. Essential dimension and topological obstructions Computations of Moduli spaces 5. Helpful results 5.1. The case of compact flat manifolds 5.2. The case of surfaces 6. The 1-dimensional case 6.1. The circle S 1 6.2. The closed unit interval I 7. The 2-dimensional case 7.1. The 2-Torus T 2 and the Klein bottle K 2 7.2. The Möbius band M 2 and the cylinder S 1 × I 7.3. The 2-sphere S 2 , the projective plane RP 2 , and the closed disc D References

The goal of the paper is to set the foundations and prove some topological results about moduli spaces of non-smooth metric measure structures with non-negative Ricci curvature in a synthetic sense (via optimal transport) on a compact topological space; more precisely, we study moduli spaces of $${{\,\mathrm{\textrm{RCD}}\,}}(0,N)$$ RCD ( 0 , N ) -structures. First, we relate the convergence of $${{\,\mathrm{\textrm{RCD}}\,}}(0,N)$$ RCD ( 0 , N ) -structures on a space to the associated lifts’ equivariant convergence on the universal cover. Then we construct the Albanese and soul maps, which reflect how structures on the universal cover split, and we prove their continuity. Finally, we construct examples of moduli spaces of $${{\,\mathrm{\textrm{RCD}}\,}}(0,N)$$ RCD ( 0 , N ) -structures that have non-trivial rational homotopy groups.