What is the relationship between the path-loop space fibration and path induction?

Question

I have an intuition that I can't quite put into words that the path-loop space fibration $\Omega X \rightarrow PX \rightarrow X$ and the based path induction axiom are related, but I don't know enough homotopy theory to formalise it. Can someone formally elaborate this relationship?

Answer 1

Based path induction is essentially the statement that, for any $x_0 : X$, the type $\sum_{x_1 : X} (x_0 = x_1)$ is contractible. See Theorem 5.8.4 in the Homotopy type theory book. The type $(x_0 = x_0)$ does not come into it.