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I am sort of stuck with this problem and I have no idea if the answer is obvious and I am just too dumb to see it or if the question is really difficult.

What i would like to prove is a particular case of the following problem:

Let's take some objects $A$,$B$,$C$ in a category $\mathcal C$, and monomorphisms $f : A \to B$ and $g : A \to C$. Is it possible to prove that the morphisms from $B$ and $C$ to the coproduct of $B \oplus_A C$ are still monics? Or is there anything I can check on $\mathcal C$ in order to be sure that it is true?

If you want a closer look at my problem:

  • $\mathcal C$ is actually $\mathbf{Cat}$, the category of (small) categories. \item
  • $B$ and $C$ are two categories who share the same $0$-cells
  • $A$ is the set of those $0$-cells, seen as a category with only identity arrows
  • $f$ end $g$ are functors which are identities on the $0$-cells.

What I'd like to show is that there is an injection from the $1$-cells of $B$ into the $1$-cells of $B \oplus_A C$.

Maxime Lucas
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  • The general case: No. Look at $\mathsf{CRing}$ for example. – Martin Brandenburg Sep 19 '14 at 13:56
  • @MartinBrandenburg Could you please give an explicit counterexample? – Amitai Yuval Sep 19 '14 at 14:10
  • I'm going to guess: take rings with different characteristics. – Maxime Lucas Sep 19 '14 at 14:13
  • OK, but you need to embed $A$ in $B$ and in $C$... – Amitai Yuval Sep 19 '14 at 14:15
  • @skysurf3000 I think for your specific question the answer is yes, but you just have to prove it by hand. – Zhen Lin Sep 19 '14 at 14:23
  • True that. I'm not sure about the english terminology for this kind of things, but if you don't suppose that 1 is fixed by your morphisms, then you can embed Z/2Z in both Z/6Z and Z/4Z. I'm not very comfortable with rings though, so I'll let someone more qualified than me give a proper counter-example... – Maxime Lucas Sep 19 '14 at 14:26
  • @skysurf3000 A ring homomorphism, by definition, maps $1$ to $1$, so... – Amitai Yuval Sep 19 '14 at 14:28
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    @ZhenLin The truth is, my problem is about double-categories which share the same $0$ and $1$-cells, and at this point I have no idea of how to prove that by hand... For people who don't know what double categories are, they are not very far from $2$-categories, the point being that the exchange relation makes things complicated... – Maxime Lucas Sep 19 '14 at 14:31
  • @skysurf3000 I've never dealt with the category your concerning with, so I'll say nothing about it. However, in the categories of sets, topological spaces and abelian groups, it is easy to verify by hand that the answer is yes. I think it is also the case in CRing, even though Martin Brandenburg thinks it is not. – Amitai Yuval Sep 19 '14 at 14:54
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    @AmitaiYuval It is not true in $\mathbf{CRing}$. A counterexample is contained in my answer here. – Zhen Lin Sep 19 '14 at 15:35

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