I am reading the book "Sheaves in Geometry and Logic" by MacLane and Moerdijk.
Early in the book, the authors show that $Sets^{\mathscr{C}^{op}}$ where $\mathscr{C}$ is a small category is an elementary topos.
My Question: Is it true, more generally, that $\mathscr{E}^{\mathscr{C}^{op}}$ where $\mathscr{E}$ is a topos, itself is a topos?
I am scratching my head and to me, at least so far, it seems that if $\mathscr{E}$ is Boolean then yes. If this is true: is being Boolean a necessary condition?
Thanks in advance for your help.
Best Answer
Being boolean is neither necessary nor sufficient. (What would the subobject classifier of $\textbf{FinSet}^\omega$, where $\omega$ is the preorder category $\{ 0 \to 1 \to 2 \to \cdots \}$ be?) What is sufficient is that the category be internalisable into the topos, because we have the following:
Theorem. If $\mathcal{E}$ is a topos and $\mathcal{C}$ is an internal category in $\mathcal{E}$, then the category $\textbf{Psh} (\mathcal{C})$ of internal presheaves on $\mathcal{C}$ in $\mathcal{E}$ is also a topos.
In the first place, what does it mean for an ordinary category $\mathcal{C}$ to be internalisable into a topos $\mathcal{E}$? There are a few ways you could define it, but the point is that we want the (ordinary) category of functors $\mathcal{C}^\textrm{op} \to \mathcal{E}$ to be equivalent to the (ordinary) category of internal presheaves on some internal category in $\mathcal{E}$. For example, $\omega^\textrm{op}$ is not internalisable in $\textbf{FinSet}$ – not surprising, considering it has infinitely many non-isomorphic objects! But usually there is no problem, because:
Theorem. Let $\kappa$ be an infinite cardinal. If $\mathcal{C}$ is equivalent to a category with $< \kappa$ morphisms, and $\mathcal{E}$ is a topos in which $\coprod_{i \in I} 1$ exists for all sets $I$ of cardinality $< \kappa$, then $\mathcal{C}$ is internalisable in $\mathcal{E}$. In particular, if $\mathcal{C}$ is essentially small and $\mathcal{E}$ is a cocomplete topos, then $\mathcal{C}$ is internalisable in $\mathcal{E}$.
That said, internalisability is not necessary for $\mathcal{E}^{\mathcal{C}^\textrm{op}}$ to be a topos either. For example, $\textbf{FinSet}^{\omega^\textrm{op}}$ is a topos even though $\omega$ is not internalisable in $\textbf{FinSet}$. Similarly, $\textbf{Set}^{\textbf{ON}^\textrm{op}}$ is a topos, where $\textbf{ON}$ is the preorder category of all small ordinals.