# Question on Fibration

Can anyone please help me with question?

Question: Let $p: E \rightarrow B$ be a fibration. Suppose there is a commuting diagram $$\begin{equation*}\begin{array}{ccl} (\mathbb{I}\times\{0\}\bigcup\{0,1\} \times \mathbb{I}) \times X&\stackrel{H}{\longrightarrow}&E\\ \!\!\!\!\!\!\!\!{\scriptstyle}\downarrow&&\downarrow{p}\\ \!\!\!\!\mathbb{I} \times \mathbb{I} \times X&\stackrel{\phi}{\longrightarrow}&B \end{array}\qquad\qquad\end{equation*}$$

Show that $\phi$ has a lift $\phi^\star$ to $E$ extending $H$. Notation $\mathbb{I}= [0,1]$.

Thanks.

## 1 Answer

$f:\mathbb{I}\times \{0\}\times X\rightarrow \mathbb{I}\times \mathbb{I}\times X$ and $g:\mathbb{I}\times \{0,1\}\times X\rightarrow \mathbb{I}\times \mathbb{I}\times X$ are weak cofibrations, if the diagram commutes, the following diagrams commute

$$\begin{equation*}\begin{array}{ccl} (\{0,1\} \times \mathbb{I}) \times X&\stackrel{H}{\longrightarrow}&E\\ \!\!\!\!\!\!\!\!{\scriptstyle}\downarrow&&\downarrow{p}\\ \!\!\!\!\mathbb{I} \times \mathbb{I} \times X&\stackrel{\phi}{\longrightarrow}&B \end{array}\qquad\qquad\end{equation*}$$

$$\begin{equation*}\begin{array}{ccl} (\mathbb{I}\times\{0\} ) \times X&\stackrel{H}{\longrightarrow}&E\\ \!\!\!\!\!\!\!\!{\scriptstyle}\downarrow&&\downarrow{p}\\ \!\!\!\!\mathbb{I} \times \mathbb{I} \times X&\stackrel{\phi}{\longrightarrow}&B \end{array}\qquad\qquad\end{equation*}$$

Since in a Quillen model, the class of maps which has the left lifting property in respect of the class of fibrations is the class of weak cofibrations, you can find a lift $\phi_1$ and $\phi_2$ of the two previous diagrams which will enable you to construct $\phi^*$.

• Thanks for you answer. @Tsemo could you please show me how to find a $phi_1$. I should be about to figure it out from there. This topic is just newly taught in class and I am struggling a bit with it. – Jaynot Mar 15 '17 at 18:24
• The existence of $\phi_1$ is true in in a general model category in particular in Top. – Tsemo Aristide Mar 15 '17 at 18:26