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I apologise for the vagueness of this question - it's just something I was idly wondering about.

Let $(\mathcal{C},J)$ be a site with $\mathcal{C}$ small. The axioms for the Grothendieck topology $J$ imply that $J$ is in fact a presheaf on $\mathcal{C}$, with some extra properties. However, for each object $U\in\mathcal{C}$, every covering sieve $R\in J(U)$ can itself be identified with a subpresheaf $\hat{R}\in [\mathcal{C}^\text{op},\text{Set}]$ of the representable presheaf $h_U = \mathcal{C}(-,U)$ given by $\hat{R}(V) = \left\{f\in R: \operatorname{domain}(f) = V\right\}$. So, roughly, $J$ is a presheaf on $\mathcal{C}$ with the sections on each object themselves presheaves on $\mathcal{C}$.

I guess that the hypothesis that $\mathcal{C}$ is small might be the saviour here, but does this lead to any set-theoretical problems or require shifting universes in going from $\hat{R}$ to $J$? I hardly know anything about set theory, but it seems as though $J$ exists "at a higher level" than the presheaves $\hat{R}$ as it somehow contains them as sections.

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  • $\begingroup$ It is not necessary to increase the universe level if $\mathcal{C}$ is small. In fact, $J$ is literally a presheaf – a subpresheaf of the presheaf of all sieves. $\endgroup$ – Zhen Lin Dec 21 '15 at 2:00

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