Let $\mathcal{E}$ be a topos, and $\mathcal{U}$ an $\mathcal{E}$-universe (as discussed on this nLab page). Let $\mathcal{E}_\mathcal{U}$ be the full subcategory of $\mathcal{U}$-small objects in $\mathcal{E}$. Then I have three questions:
- Is $\mathcal{E}_\mathcal{U}$ a subtopos for any choice of $\mathcal{U}$?
- Given topoi $\mathcal{E}$ and $\mathcal{E'}$, we say that $\mathcal{E'}$ extends $\mathcal{E}$ when there exists some $\mathcal{E'}$-universe $\mathcal{U}$ such that $\mathcal{E}\simeq\mathcal{E'}_\mathcal{U}$. An extension is called trivial when $\mathcal{E}\simeq\mathcal{E'}$. My second question is, does every topos have a nontrivial extension?
- If 2 holds, then are the extensions of $\mathcal{E}$ totally-ordered by extension? Formally, fix a topos $\mathcal{E}$, and let $[\mathcal{E}]$ be the poset of topoi extending it, where $\mathcal{T}\leq\mathcal{T'}$ exactly when $\mathcal{T'}$ extends $\mathcal{T}$. That this is a poset is immediately obvious, but is it totally-ordered?
I interpret this as asking whether we can always extend a topos to a larger universe without losing the constructions and theorems we started with. It seems like it should be true, but I'm very new to topos theory and don't have a lot of trust for my intuitions. Thanks in advance for any help you might offer!