36 problems

Wraith's conjecture. The theory of algebraically closed local rings should define the classifying theory of the fppf topos.

Let $A$ be a crisp set and $S$ a crisp discrete set, and let ${\mathord{\raisebox{0.5pt}{\text{\rm\esh}}}}$ denote the shape modality. The canonical map … is an equivalence. Contin…

Let a Grothendieck topos be a topos serving as a model of intuitionistic mathematics, and consider the $lambda-Pi$ calculus with universes and inductive objects described in the so…

Let $\bot$ denote the game-theoretic Katětov order restricted to ideals on $\omega$, and consider its equivalence classes under mutual comparability. Linearity conjecture. It is co…

Let $\mathcal{H}=\mathbf{Sh}(\mathbf{P},J_{\mathrm{can}})$ be the sheaf topos associated with the full subcategory $\mathbf{P}$ of $\omega\mathbf{CPO}$ consisting of retracts of Sc…

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.

Let $\mathsf{Q}$ be a commutative and divisible quantale, also known as a GL-monoid, and let $\mathsf{Q}\text{-}\mathbf{Set}$ be the category of $\mathsf{Q}$-sets and left adjoint…

Let $\mathcal{C}$ be a small extensive category, let $\mathrm{G}\mathcal{C}$ be its Gaeta topos, and let $X$ be an object of $\mathrm{G}\mathcal{C}$. Lawvere constructs a QD-subtop…

Internal monotone-operator conjecture. The paper's axioms are sound when interpreted in the internally monotone operators $j: \Omega \to \Omega$ of any elementary topos…

A topos is locally finitely presentable when it is locally finitely presentable as a category, and it has enough points when its points jointly detect distinct morphisms. Dimitri's…

A topos with enough points is a topos whose points form a conservative family of geometric morphisms to the category of sets. A topos is locally finitely presentable when it is loc…

Existence conjecture. There exists an elementary topos containing an $LL$-algebra that is not free on its subdcpo of positive elements.

Shulman's spatial type theory conjecture. The split-context spatial type theory should be interpretable in any local topos.

Characterization conjecture. Right Kan injectives with respect to flat embeddings of coherent topoi should be precisely coreflections of coherent topoi. Equivalently, such a result…

Lamarche conjecture. For any category $C$, the category

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

The discussion concerns geometric constructions that can be implemented uniformly in elementary topoi with a natural numbers object. The proposed constructions include finite limit…

Let $n$ and $m$ be natural numbers, and let $f_1,\ldots,f_m$ be homogeneous polynomials in $n+1$ variables, with a finite set of variables recording their coefficients. In the intu…

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…

Forster–Lewicki–Vidrine's conjecture. Every model of $T$ has a subcategory which is a topos.

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

Funk and Steinberg construct a universal covering topos for each inverse semigroup, with a concrete interpretation of the associated fundamental group. A comprehension scheme is a…

Let $\mathcal{T}$ be a topos. Its diagonal and the diagonal of its diagonal are geometric morphisms associated with the corresponding iterated fiber products. A topos is absolutely…