arXiv subjects
Publications and source records attributed to Sina Hazratpour.
Consider a locally cartesian closed category with an object I and a class of trivial fibrations, which admit sections and are stable under pushforward and retract as arrows. Define the fibrations to be those maps whose Leibniz exponential with the generic point of I defines a trivial fibration. Then the fibrations are also closed under pushforward.
Suppose an extension map $U\colon \mathbb{T}_1 \to \mathbb{T}_0$ in the 2-category $\mathfrak{Con}$ of contexts for arithmetic universes satisfies a Chevalley criterion for being an (op)fibration in $\mathfrak{Con}$. If $M$ is a model of $\mathbb{T}_0$ in an elementary topos $\mathcal{S}$ with nno, then the classifier $p\colon\mathcal{S}[\mathbb{T}_1/M]\to\mathcal{S}$ satisfies Johnstone's criterion for being an (op)fibration in the 2-category $\mathcal{E}\mathfrak{Top}$ of elementary toposes (with nno) and geometric morphisms. Along the way, we provide a convenient reformulation of Johnstone's criterion.