8 problems

Cauchy-completeness separation conjecture. 1. There exists an elementary topos in which $\mathbb{R}_C\neq\mathbb{R}_E$; equivalently, there exists an elementary topos in which…

Choice-separation conjecture. 1. There exists an elementary topos in which $\mathrm{AC}_{\mathbb{N}\text{-bounded}}$ holds but $\mathrm{AC}_{\mathbb{N}\mathbb{N}}$ does not. 2. The…

Descent conjecture. The descent object $\mathcal{X}/\mathcal{G}$ of $\mathcal{G}$-equivariant sheaves on $\mathcal{X}$ is also preorderlike.

Artin-gluing conjecture. The theory $\mathbb{T}+\mathbb{E}/\phi$ is classified by the topos

Let $\mathcal{E}$ be a Grothendieck topos, and let $\omega \leq \lambda \leq \kappa$. Relative Scott adjunction conjecture. There is a $2$-adjunction … Moreover, if $\mathcal{E}$ i…

Counterpart conjecture. Some of these local operators have counterparts in the toposes

Let $cmathcal{V}(cmathcal{H})$ be the context category of the Hilbert space $cmathcal{H}$, let $cmathcal{V}_f(cmathcal{H})$ be the same category with the group action forgotten, le…

Let $cmathcal{V}(cmathcal{H})$ be the poset of contexts of the Hilbert space $cmathcal{H}$, let $cmathcal{V}_f(cmathcal{H})$ denote the corresponding context category with the grou…