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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.
@article{TAC_2014_29_a19,
author = {Matias Menni},
title = {Continuous cohesion over sets},
journal = {Theory and applications of categories},
pages = {542--568},
publisher = {mathdoc},
volume = {29},
year = {2014},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2014_29_a19/}
}
Matias Menni. Continuous cohesion over sets. Theory and applications of categories, Tome 29 (2014), pp. 542-568. http://geodesic.mathdoc.fr/item/TAC_2014_29_a19/