On internally projective sheaves of groups

TL;DR

This paper investigates the properties of internally projective sheaves of abelian groups, providing a counterexample that demonstrates their instability under base change, thus clarifying the limitations of internal projectivity.

Contribution

It presents the first explicit example showing that internally projective sheaves of abelian groups are not stable under base change, highlighting a key difference from classical projectivity.

Findings

Internally projective sheaves are not stable under base change.

Counterexample derived from recent work on light condensed sets.

Internal projectivity is weaker than classical projectivity.

Abstract

A sheaf of modules on a site is said to be internally projective if sheaf hom with the module preserves epimorphism. In this note, we give an example showing that internally projective sheaves of abelian groups are not in general stable under base change to a slice. This shows that internal projectivity is weaker than projectivity in the internal logic of the topos, as expressed for example in terms of Shulman's stack semantics. The sheaf of groups that we use as a counterexample comes from recent work by Clausen and Scholze on light condensed sets.