Dicovering spaces

Abstract

For a local po-space $X$ and a base point $x_0\in X$, we define the universal dicovering space $\Pi:\tilde{X}_{x_0}\to X$. The image of $\Pi$ is the future $\uparrow x_0$ of $x_0$ in $X$ and $\tilde{X}_{x_0}$ is a local po-space such that $|\vec{\pi}_1(\tilde{X},[x_0],x_1)|=1$ for the constant dipath $[x_0]\in\Pi^{-1}(x_0)$ and $x_1\in \tilde{X}_{x_0}$. Moreover, dipaths and dihomotopies of dipaths (with a fixed starting point) in $\uparrow x_0$ lift uniquely to $\tilde{X}_{x_0}$. The fibers $\Pi^{-1}(x)$ are discrete, but the cardinality is not constant. We define dicoverings $P:\hat{X}\to X_{x_0}$ and construct a map $\phi:\tilde{X}_{x_0}\to\hat{X}$ covering the identity map. Dipaths and dihomotopies in $\hat{X}$ lift to $\tilde{X}_{x_0}$, but we give an example where $\phi$ is not continuous.