Continuous cohesion over sets

Summary

Summary: A pre-cohesive geometric morphism $p:\cal E \rightarrow \cal S$ satisfies $Continuity$ if the canonical $p_! (X^{p^* S}) \rightarrow (p_! X)^S$ is an iso for every $X$ in $\cal E$ and $S$ in $\cal S$. We show that if $\cal S = Set$ and $\cal E$ is a presheaf topos then, $p$ satisfies Continuity if and only if it is a quality type. Our proof of this characterization rests on a related result showing that Continuity and Sufficient Cohesion are incompatible for presheaf toposes. This incompatibility raises the question whether Continuity and Sufficient Cohesion are ever compatible for Grothendieck toposes. We show that the answer is positive by building some examples.

18B25, 03G30

topos