pushout with zero upper map

Question

If \begin{array}{ccC} A &\xrightarrow {0} & B\\ \downarrow & &\downarrow \\ C & \xrightarrow{g} & D \end{array} is a pushout. is $g$ zero map?

I guess $g$ is rather the cokernel of $A\to C$. – Berci Feb 26 '20 at 12:03 — Berci, Feb 26 '20 at 12:03
What category is this in? – Daniel Donnelly Mar 31 '20 at 02:14 — Daniel Donnelly, Mar 31 '20 at 02:14

score 1 · Accepted Answer · answered May 19 '23 at 10:31

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No, for example \begin{array}{ccC} 0 & \to & B\\ \downarrow & &\downarrow \\ C & \xrightarrow{g} & B \oplus C \end{array} is a pushout with $g$ the "inclusion", in say vector spaces.

answered May 19 '23 at 10:31

ronno

11,390

1

Indeed your diagram is a correct pushout in any category with binary coproducts where $B\oplus C$ stands for $B\sqcup C$ and $0$ stands for an initial object – FShrike May 19 '23 at 10:35
@FShrike Is it clear in that generality that $g \ne 0$? – ronno Feb 26 '24 at 22:28
Not assuming the category has a zero object, i.e. only assuming $0$ is initial and perhaps not terminal, that would imply that if there exists an arrow $B\to X$ there exists at most one arrow $C\to X$, and existence of an arrow $C\to 0$ would entail $C\to0\to X$ surely always exists and therefore that $B\oplus C\cong B$. This isn't enough to show $C=0$ though, I don't think. Hmm. But if you live in a category where the binary coproduct inclusions are monomorphisms then $g$ is zero iff. $C$ is isomorphic to $0$, we can say that – FShrike Feb 26 '24 at 23:56

pushout with zero upper map

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