We study differential complexes of Kolmogorov-Alexander-Spanier type on metric measure spaces associated with unbounded non-local operators, such as operators of fractional Laplacian type. We define Hilbert complexes, observe invariance properties and obtain self-adjoint non-local analogues of Hodge Laplacians. For d-regular measures and operators of fractional Laplacian type we provide results on removable sets in terms of Hausdorff measures. We prove a Mayer-Vietoris principle and a Poincaré lemma and verify that in the compact Riemannian manifold case the deRham cohomology can be recovered. Contents1. Introduction 1 2. Preliminaries 4 3. Complexes of elementary functions 5 4. Kernels and measures 9 5. Non-local Hilbert complexes 13 5.1. Regularity and density 13 5.2. Closed extensions 14 5.3. Remarks on invariance 17 5.4. Non-local Hodge Laplacians 19 5.5. Approximation results 21 5.6. Removable sets 23 6. Covers and cohomology 27 6.1. Partitions of unity 27 6.2. Mayer-Vietoris sequences 28 6.3. Poincaré lemma 31 6.4. Recovering deRham cohomology 36 7. Some basic examples 37 References 39