An internal description of constructible objects in an $\infty$-topos
Li He
TL;DR
The paper provides an internal description of $P$-constructible objects in an $\infty$-topos by identifying them with objects locally constant internal to $\mathrm{Fun}(P,\mathrm{An})$ when $P$ is a noetherian poset. It introduces the $\mathscr{B}$-locally constant notion for a fixed morphism $\pi^*:\mathscr{B} \to \mathscr{X}$ and proves the central equivalence $\mathrm{LC}_{\mathrm{Fun}(P,\mathrm{An})}(\mathscr{X}) \simeq \mathrm{Cons}_P(\mathscr{X})$. The result connects to exodromy by showing that constructible sheaves can be realized as internal locally constant objects, with the equivalence established via a combination of recollement, étale morphisms, and noetherian descent. Overall, the work provides a robust internal framework for stratified, constructible phenomena in $\infty$-topoi and clarifies how exodromy can be understood through internal constancy.
Abstract
We give an internal description of constructible objects in an $\infty$-topos. More precisely, $P$-consctructible objects are locally constant objects internal to Fun($P$,An), for any noetherian poset $P$.