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?
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3Correct. Actually, you can conclude $p_A$ is a monomorphism if $f_B$ is. – Zhen Lin Apr 16 '15 at 20:45
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@ZhenLin: This is an answer, not a comment. – Martin Brandenburg Apr 17 '15 at 22:22
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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.
Zhen Lin
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