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Suppose $\mathcal{C}$ is a category such that for every $c \in \mathrm{Ob}(\mathcal{C})$, the slice category $\mathcal{C}/c$ is equivalent to a small category. I need to show that the category of presheaves $[\mathcal{C}^\mathrm{op}, \mathbf{Set}]$ is an elementary topos.

I understand the standard argument used when $\mathcal{C}$ is a small category — for example, to construct the exponential $G^F$ of two presheaves, we apply the Yoneda lemma and see that we are forced to set $G^F (c) = \mathrm{Hom}(\mathbf{y}c \times F, G)$, where $\mathbf{y}c = \mathrm{Hom}(-, c) : \mathcal{C}^\mathrm{op} \to \mathbf{Set}$ is the contravariant hom-functor. The main obstruction, then, to using this argument is showing that $\mathrm{Hom}(\mathbf{y}c \times F, G)$ is indeed a set under these weaker assumptions. Well, actually, I'd first have to show that $\mathbf{y}c$ is actually a set-valued functor... but isn't this the same as showing that $\mathcal{C}$ is locally small? It's intuitively plausible that $\mathcal{C}/c$ being equivalent to a small category implies $\mathcal{C}$ itself is locally small, but I imagine, from the phrasing of the problem, that it's not the case.

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  • $\begingroup$ Have you by any chance managed to solve this problem? $\endgroup$ – Arrow Oct 6 '15 at 7:53
  • $\begingroup$ We must add the hypothesis that $\mathcal{C}$ is locally small. $\endgroup$ – Zhen Lin Oct 6 '15 at 8:20
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We must add the hypothesis that $\mathcal{C}$ is locally small.

Indeed, consider the case where $\mathcal{C}$ is a groupoid with a unique object $c$. Then $\mathcal{C}_{/ c}$ is equivalent to the terminal groupoid – in particular, it is always essentially small. But $\mathcal{C}$ may not be locally small.

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