0
$\begingroup$

Suppose you are in the category of sets or more generally in a topos (i.e. sheaf topos) and $f:A\rightarrow C$, $g:B\rightarrow C$ are two morphisms.

There is a canonical map $u:D\to C$ from $D$ (defined as the pushout of the diagram $A\leftarrow A\times_C B\rightarrow B$ consisting of the two projections) into $C$.

Presumably $u$ doesn't have to be a monomorphism in general, however I can't think of a counterexample. In my situation, it is supplementary given that the two projections $pr_1:A\times_C B\rightarrow A$ and $pr_2:A\times_C B\rightarrow B$ and $f$ are monomorphisms each. Does $u$ have to be a monomorphism then?

  • 0
    Just because it's category theory doesn't mean it can be dualised! (The reason why dualising works in abelian categories is because the opposite of an abelian category is an abelian category. But this is not true for a topos.) In this case, your dual question also has a negative answer. A counterexample can be found in $\textbf{Set}$.2012-05-14

1 Answers 1

1
  1. Let $C = 1$, let $A$ be an object such that $A \to 1$ is not a monomorphism, and let $B = 0$. Then, $A \times_C B = 0$, but $D = A$, so $D \to C$ is not a monomorphism. (For this to work we only need to know that $0$ is a strict initial object.)

  2. Let us consider the topos of sheaves on the discrete space $\{ a, b \}$. Let $C = 1$, let $A$ be the subsheaf of $C$ such that $A_a = 1$ and $A_b = 0$, and let $B$ be a sheaf such that $B_a = 1$ and $B_b = 2$. Then, $f : A \to C$ is monic, $g : B \to C$ is epic but not monic, and both $p_1 : A \times_C B \to A$ and $p_2 : A \times_C B \to B$ are monic. But $D_a = 1$ and $D_b = 2$, so $D \to C$ is not monic.

    Morally, what's happening here is that your hypotheses only guarantee that the restriction of $B$ (considered as a sheaf over $C$) to $A$ is monic, so you have no control over what $B$ looks like over the complement of $A$ in $C$. Nonetheless, this plays a role in the construction of $D$ and so influences whether $D \to C$ is monic or not.