What, exactly, is a vertical homotopy? As the question title suggests, what exactly is a vertical homotopy? Googling has failed to provide any results as so far as a clear definition goes...
 A: When I search for "vertical homotopy", results 1 and 3 are both this question. Result 4 is the book "Fibrewise Homotopy Theory" by Michael Charles Crabb and Ioan Mackenzie James. On page 21, it says

We say that two sections $s_0$ and $s_1$ of a fibrewise space $X \to B$ are vertically homotopic if they are homotopic through sections, that is, if there is a homotopy $s_t, 0 \le t \le 1$, where each map $s_t$ is a section.

A: I think that a "vertical homotopy" is the same thing than a "fiber homotopy" i.e. an homotopy in the category of maps over $B$ :
if $p : E \to B$ and $p':E' \to B$ are two maps over $B$, and $f_1,f_2:E \to E'$ are maps from $p$ to $p'$ (i.e. $B$-maps or $p'\circ f_i = p$), then a continuous map $H:E \times I \to E'$ such that $H(-,0) = f_0$ and $H(-,1) = f_1$ is a "fiber homotopy" between $f_0$ and $f_1$ if 
$\begin{equation}
\forall t, \  p' \circ H(-,t) = p.
\end{equation}$
And two section $s_1$ and $s_2$ of $p$ are vertically homotopic if the two $B$-morphisms $s_1 \circ p$ and $s_2 \circ p$ from $E$ to $E$ are fiber homotopic.
