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# Homology, Homotopy and Applications

## Volume 5 (2003)

### Number 2

### Volume of a Workshop at Stanford University

### Discovering spaces

Pages: 1 – 17

DOI: http://dx.doi.org/10.4310/HHA.2003.v5.n2.a1

#### Author

#### 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 $|\stackrel{\rightarrow}{\pi}_1(\tilde{X},[x_0],x_1)|=1$ 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.

#### Keywords

covering spaces, abstract homotopy theory, dihomotopy theory

#### 2010 Mathematics Subject Classification

51H15, 54Exx, 57M10