# Terminal object in the category of sheaves?

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.

• The initial object is not the constant presheaf $0$ but rather the sheafification thereof. – Zhen Lin Apr 20 '16 at 5:48
• I actually wrote constant sheaf in my post. – ಠ_ಠ Apr 20 '16 at 9:27

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.
• Alternatively, I noticed that it follows from the sheafification adjunction, once we notice that $1_X$ is terminal in the category of presheaves on $X$. – ಠ_ಠ Apr 20 '16 at 23:23
• @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 $1_X$. – Qiaochu Yuan Apr 21 '16 at 2:47