On internally projective sheaves of groups

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.