# Monomorphisms vs pullbacks

Let $f:X\to Y$ be a morphism in a category. It is easy to see that if $f$ is a monomorphism then there exists a pullback $X \times_Y X$.

Here the question is whether the converse is true.

If two projections of $X\times_Y X$ equal then it is easy to check.

So my question is rewritten as whether the two projections equal always.

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Please clarify your question. Do you mean to ask, "if $X \times_Y X$ exists and the two projections $X \times_Y X \to X$ are equal, then $X \to Y$ is a monomorphism"? – Zhen Lin Oct 21 '12 at 12:11
No. It is easy. I want it without assumption of equal projection. – Tom Oct 21 '12 at 12:16
I want:If $X \times_Y X$ exists for morphism $f:X\to Y$ then $f$ is a monomorphism. – Tom Oct 21 '12 at 12:18
That's false in general. – Zhen Lin Oct 21 '12 at 12:28
No. He is right. I misunderstood what he means. – Tom Oct 21 '12 at 12:45

## 2 Answers

In Set, the pullback $X \times_Y X$ always exists and is given by $$\left\{ (x,x') \in X \times X \, \mid\, f(x) = f(x') \right\},$$ regardless of whether $f$ is a monomorphism or not.

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Thank you very much. – Tom Oct 21 '12 at 12:45

The correct statement is that $f$ is a monomorphism if and only if the pullback $X \times_Y X$ not only exists but is naturally isomorphic to $X$.

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"Naturally" here means that the two projections are equal and are isomorphisms. – Zhen Lin Oct 21 '12 at 22:37