Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.