Pullback stability?

Question

Suppose the following square is a pullback. $$\require{AMScd} \begin{CD} E\times _BA @>{\pi_2}>> A\\ @V{\pi_1}VV @VV{\alpha}V\\ E @>>{p}> B \end{CD}$$

The following is proposition 6.5.1 from Borceux & Janelidze's Galois Theories:

Proposition 6.5.1. Let $\mathcal M$ be a pullback stable class of arrows in $\mathsf{Top}$. Then $\alpha$ locally in $\mathcal M\implies \pi_1$ locally in $\mathcal M$.

Regardless of the definition of "locally in", isn't this exactly the definition of pullback stability for $\mathcal M$?

Answer 1

No, Proposition 6.5.1. is the statement that the class $\mathcal L$ of morphisms locally in $\mathcal M$ is pullback-stable. This is non-trivial even when $\mathcal M$ is pullback-stable because $\mathcal L$ contains different (strictly more) morphisms than $\mathcal M$ does.

The use of "locally in" is an instance of the red herring principle.

The mathematical red herring principle is the principle that in mathematics, a “red herring” need not, in general, be either red or a herring.

Frequently, in fact, it is conversely true that all herrings are red herrings. This often leads to mathematicians speaking of “non-red herrings,” and sometimes even to a redefinition of “herring” to include both the red and non-red versions.

In this particular case, "in $\mathcal M$" implies "locally in $\mathcal M$", but the converse is false.