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Let $\mathsf{C}$ denote one of the categories $\mathsf{Set}$, $\mathsf{Group}$, $\mathsf{Ring}$, or $\mathsf{Mod}_R$, or some other sensible category in which we would like sheaves to take their values. Let $\text{Sh}(X, \mathsf{C})$ denote the category of $\mathsf{C}$-valued sheaves on a space (or more generally a site) $X$.

The initial object in $\text{Sh}(X, \mathsf{C})$ is obviously the constant sheaf $0_X$ corresponding to the initial object $0$ in $\mathsf{C}$, since the constant sheaf functor is left adjoint to the global sections functor.

But what is the terminal object? I would conjecture that it should be $1_X$, where $1$ is the terminal object in $\mathsf{C}$, but I do not see how to prove it.

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  • $\begingroup$ The initial object is not the constant presheaf $0$ but rather the sheafification thereof. $\endgroup$ – Zhen Lin Apr 20 '16 at 5:48
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    $\begingroup$ I actually wrote constant sheaf in my post. $\endgroup$ – ಠ_ಠ Apr 20 '16 at 9:27
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The terminal presheaf is the constant presheaf with value the terminal object in $C$, since limits and colimits of presheaves are computed pointwise. The inclusion of sheaves into presheaves is a right adjoint, so preserves limits; hence the terminal presheaf is also the terminal sheaf.

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    $\begingroup$ This reads almost as if any right adjoint creates limits, one might mention the need for full faithfulness of the right adjoint to guarantee a terminal object exists at all. $\endgroup$ – Kevin Carlson Apr 20 '16 at 16:03
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    $\begingroup$ I definitely agree that right adjoints preserve limits, but you're taking a sheaf, noting its image under a right adjoint is a limit, and concluding it's a limit, yeah? That's asking for reflection of limits, or creation if you don't start from knowledge of the constant presheaf being a sheaf. And right adjoints certainly don't create or reflect limits in general. $\endgroup$ – Kevin Carlson Apr 20 '16 at 16:50
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    $\begingroup$ @Kevin: oops, you're right, I got confused. Fortunately the inclusion of sheaves into presheaves is also conservative, so reflects limits. Then we need to observe that the terminal presheaf is also a sheaf, as you say. $\endgroup$ – Qiaochu Yuan Apr 20 '16 at 20:42
  • $\begingroup$ Alternatively, I noticed that it follows from the sheafification adjunction, once we notice that $1_X$ is terminal in the category of presheaves on $X$. $\endgroup$ – ಠ_ಠ Apr 20 '16 at 23:23
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    $\begingroup$ @ಠ_ಠ: that's the point we've been discussing; the adjunction alone only gives you that if sheaves has a terminal object, then it must be $1_X$. $\endgroup$ – Qiaochu Yuan Apr 21 '16 at 2:47

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