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Let $C$ be a category with a terminal object $1$. Is the unique arrow from an object into $1$ necessarily an epimorphism? If not, is it an epimorphism if $C$ is a topos?

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    $\begingroup$ What are the inconsistent definitions? $\endgroup$ Dec 5, 2013 at 0:05
  • $\begingroup$ I misread something. Sorry. $\endgroup$
    – sopot
    Dec 5, 2013 at 0:37
  • $\begingroup$ Arrow $e$ to a terminal object is constant. For each parallel pair $(f,g)$ of arrows you have $ef=eg$ (if they are defined). $\endgroup$
    – drhab
    Dec 6, 2013 at 16:47

2 Answers 2

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No to both. Let $\mathcal{C}$ be the topos $\mathbf{Set} \times \mathbf{Set}$ and consider the object $(\emptyset, 1)$; the terminal object in $\mathbf{Set} \times \mathbf{Set}$ is $(1, 1)$, but the unique morphism $(\emptyset, 1) \to (1, 1)$ is not an epimorphism. Note $(\emptyset, 1)$ is not an initial object either; if you allow that, then we have a counterexample even in $\mathbf{Set}$: the unique map $\emptyset \to 1$ is not a surjection.

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By duality, your question is equivalent to: Is every morphism from the initial object a monomorphism? No, for example in $\mathsf{CRing}$ with initial object $\mathbb{Z}$ this holds only for rings of characteristic $0$.

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  • $\begingroup$ Accordingly, the unique morphism of schemes $X \to \mathrm{Spec}(\mathbb{Z})$ doesn't have to be an epimorphism. $\endgroup$ Sep 22, 2014 at 13:39

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