I'm stuck on the following problem.

In a category with a zero object; if an epimorphism is a zero morphism, then its target is a zero object.


Note that in a cat. with zero object, any initial object is the zero object (because all initial objects are isomorphic).

Thus, it is enough to show that the codomain $y$ of the said zero epimorphism $x \stackrel{0}\twoheadrightarrow y$ is initial. So given $z$, pick any two morphisms $\alpha, \beta: y \rightarrow z$ (also show that there is at least one such morphism). Now show that $\alpha$ is necessarily equal to $\beta$.

  • $\begingroup$ Thank you for your answer. The part of that I am not able to prove is the uniqueness of the morphism from y to z. To prove this, I must use that $x \stackrel{0}\twoheadrightarrow y$ is an epimorphism. Yet, in order to do so, I must first show that for $\alpha \circ 0 = \beta \circ 0$. This is the part where I'm stuck. $\endgroup$ – Jules Pitcho Apr 12 '18 at 18:30
  • $\begingroup$ @JulesPitcho Well, for any two pairs of objects $x', y'$, the zero map $x'\rightarrow y'$ is the unique map factorizing through the zero object. Do $\alpha \circ 0, \beta \circ 0$ factorize through the zero object? $\endgroup$ – Pavel Čoupek Apr 12 '18 at 18:42
  • $\begingroup$ It's called a zero map because composing with it acts like multiplying by zero: absorbs it, trivializes it, etc.: making the composition a zero map. $\endgroup$ – Berci Apr 12 '18 at 23:43

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