Is there a notion of Hurewicz fibration with additional structure equivalent to UNIQUe path lifting

Question

Is there a notion of Hurewicz fibration with additional structure equivalent to UNIQUe path lifting? or even a UNIQUE homotopy lifting?cf parallel transport in diff geom

Answer 1

Yes. These are known as fibrations with unique path lifting. See for example

Spanier, Edwin H. Algebraic topology. Springer Science & Business Media, 1989.

Spanier defines this concept on p.68 and shows that it is a very useful generalization of the concept of a covering projection. They satisfy many properties of covering projections and behave better in some respect (for example, the composition of fibrations with unique path lifting is again a fibration with unique path lifting).

Note that a fibration with unique path lifting is automatically a fibration with unique homotopy lifting. In fact, let $p : E \to B$ be a fibration with unique path lifting and $h : X \times I \to B$ be a homotopy and $f : X \to E$ be a map such that $h(x,0) = p(f(x))$ for all $x \in X$. It has alift $H : X \times I \to E$ such that $H(x,0) = f(x)$ for all $x \in X$. We can regard the maps $H \mid_{\{x\} \times I}$ als lifts of the paths $h \mid_{\{x\} \times I}$ which are uniquely determined by their initial points $H(x,0) = f(x)$. This means that $H$ is the only lift of $h$.