Dicovering Spaces

For a local po-space X and a base point x0 ∈ X, we define the universal dicovering space Π : X̃x0 → X. The image of Π is the future ↑ x0 of x0 in X and X̃x0 is a local po-space such that | π 1 (X̃, [x0], x1)| = 1 for the constant dipath [x0] ∈ Π−1(x0) and x1 ∈ X̃x0 . Moreover, dipaths and dihomotopies of dipaths (with a fixed starting point) in ↑ x0 lift uniquely to X̃x0 . The fibers Π −1(x) are discrete, but the cardinality is not constant. We define dicoverings P : X̂ → Xx0 and construct a map φ : X̃x0 → X̂ covering the identity map. Dipaths and dihomotopies in X̂ lift to X̃x0 , but we give an example where φ is not continuous.