What does it mean for pullbacks to preserve monomorphisms?

Question

If two arrows $f_A : A \to C$, $f_B : B \to C$ are monomorphisms, then their pullback arrows $p_A : P \to A$, $p_B : P \to B$ are monomorhisms too. Is that what is meant by pullbacks preserving monomorphisms?

Answer 1

There are two possible meanings:

• For any pullback square as below, $$\require{AMScd} \begin{CD} X' @>>> X \\ @VVV @VVV \\ Y' @>>> Y \end{CD}$$ if $X \to Y$ is a monomorphism, then $X' \to Y'$ is also a monomorphism.

• For any commutative diagram of the form below, $$\begin{CD} X' @>>> X \\ @VVV @VVV \\ Y' @>>> Y \\ @VVV @VVV \\ S' @>>> S \end{CD}$$ where the lower square and outer rectangle are pullback diagrams, if $X \to Y$ is a monomorphism, then $X' \to Y'$ is also a monomorphism.

Of course, both are true. In fact, the two are inter-derivable: the first one is a special case of the second (by taking $Y' \to S'$ and $Y \to S$ to be the identity), and you can deduce the second one from the first by using the pullback pasting lemma.