Summary: We address the problem of building valid representations of non-manifold \(d\)-dimensional objects through an approach based on decomposing a non-manifold \(d\)-dimensional object into an assembly of more regular components. We first define a standard decomposition of \(d\)-dimensional non-manifold objects described by abstract simplicial complexes. This decomposition splits a non-manifold object into components that belong to a well-understood class of objects, that we call initial quasi-manifold. Initial quasi-manifolds cannot be decomposed without cutting them along manifold faces. They form a decidable superset of \(d\)-manifolds for \(d\geq 3\), and coincide with manifolds for \(d \leq 2\). We then present an algorithm that computes the standard decomposition of a general non-manifold complex. This decomposition is unique, and removes all singularities which can be removed without cutting the complex along its manifold faces.