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I'm working through Category theory for the sciences (by David Spivak) and I'm a bit stuck in section 2 - my problem is with Proposition 2.7.5.5 and Exercise 2.7.5.6, I hope someone can help :)

The proof goes a bit fast for me, there are a few things I don't get:

  • Why do we start by defining $f' \circ n = f' \circ m$ ? This is what we want to prove, right?

  • How is the universal property of pullback used?

With the figure in the book, the way I understand it, I would write the proof as follows, but I'm not confident that this proof fully holds (I'm not entirely comfortable in the notation and rules of these systems yet):

  1. We take an arbitrary $m$ and $n$ as illustrated.

  2. We define $q := g' \circ m$ and $r := g' \circ n$.

  3. Since $f$ is monomorphic, $q = r \Rightarrow g' \circ m = g' \circ n \Rightarrow m = n$.

  4. Considering we started with arbitrary $m$ and $n$ the function at that position in the graph is unique, implying $f'$ is monomorphic.

Am I cutting corners here?

And as a side question, since we know that now for an arbitrary $B$, $m$, $n$ we can prove that $m = n$, does this imply that not only $f$ and $f'$ are monomorphic, but also $g$ and $g'$?

It would be great if someone a bit more experienced in this subject could tell me if my reasoning makes sense - the fact that the text suggests that the universal property of pullback is needed, while I don't see how, makes me a bit insecure about the problem.

Kind regards, Sam

Sam van Herwaarden
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  • You need to check your understanding of the definition of monomorphism. To show that $f'$ is a monomorphism, you start by assuming $f' \circ n = f' \circ m$; what you need to prove is that $n = m$. – Zhen Lin May 30 '15 at 16:43
  • Thank you! I think I get it now: $f \circ m = f \circ n$ does NOT necessarily imply $m = n$, this is what the whole concept of monomorphism is about (I got that before but somehow lost track of it). So, first, we prove that $g \circ p = f \circ r = f \circ q$. Having proved this, we're allowed to use the universal property of pullbacks due to the structure of the diagram. Conclusion: $B \rightarrow X \times_{Y} A$ is unique. This proves that $m = n$. – Sam van Herwaarden May 30 '15 at 17:48
  • https://math.stackexchange.com/questions/1980887/how-to-prove-that-pullback-preserves-monomorphisms – Arnaud D. Apr 09 '19 at 21:01

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