#
Continuous cohesion over sets

##
Matias Menni

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.

Keywords:
topos, Axiomatic Cohesion

2010 MSC:
18B25, 03G30

*Theory and Applications of Categories,*
Vol. 29, 2014,
No. 20, pp 542-568.

Published 2014-09-17.

http://www.tac.mta.ca/tac/volumes/29/20/29-20.pdf

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