Composition and Product of fibrations

Question

Any help and hint on how to show that the composition and product of two fibrations are again fibrations. Thanks

Answer 1

Just verify the homotopy lifting property.

For example, if $p_1 : E \to F$ and $p_2 : F \to B$ are two fibrations, we wish to prove that the composition $p_2 \circ p_1 : E \to B$ is a fibration.

Given a map $f : X \times I \to B$, where $X$ is some other space and $I$ is the closed interval, and given a lift $\tilde f_0 : X \to E$ of $ f|_{X \times 0}$, we want to construct a lift $\tilde f : X \times I \to E$ of $f$ extending $\tilde f_0$.

First, $p_1 \circ \tilde f_0$ is certainly a lift of $f|_{X \times 0}$ to $F$. Since $p_2 : F \to B$ is a fibration, there exists a lift $f' : X \times I \to F$ of $f$ extending $p_1 \circ \tilde f_0$.

Next, $\tilde f_0$ is a lift of $p_1 \circ \tilde f_0 = f'|_{X \times 0}$ to $E$. Since $p_1 : E \to F$ is a fibration, there exists a lift $\tilde f : X \times I \to E$ of $f'$ extending $\tilde f_0$. Clearly, this lift $\tilde f$ is also a lift of $f$ itself. Thus, $p_2 \circ p_1 : E \to B$ obeys the homotopy lifting property, so it is a fibration, and we're done.