Let \(p\colon\tilde X\to X\) be a covering space. A deck transformation of \(p\) is a homeomorphism \(\psi\colon\tilde X\to\tilde X\) with \(p\circ\psi=p\). The deck transformations form a group \(\Deck(\tilde X\to X)\) under composition.
In the slice category \(\mathsf{Top}_{/X}\), whose objects are continuous maps to \(X\) and whose morphisms are maps over \(X\), the group \(\Deck(\tilde X\to X)\) is the automorphism group \(\Aut_{\mathsf{Top}_{/X}}(p)\) of the object \(p\colon\tilde X\to X\). A deck transformation permutes each fiber \(\inverseof{p}(x)\). If \(\tilde X\) is connected, a deck transformation is determined by the image of one point, so a deck transformation with a fixed point is the identity [@Hat02].