#
Exponentiable Morphisms: Posets, Spaces, Locales, and Grothendieck Toposes

##
Susan Niefield

In this paper, we consider those morphisms $p : P\to B$ of posets for
which the induced geometric morphism of presheaf toposes is exponentiable
in the category of Grothendieck toposes. In particular, we show that a
necessary condition is that the induced map $p^{\downarrow} :
P^{\downarrow}\to B^{\downarrow}$ is exponentiable in the category of
topological spaces, where $P^{\downarrow}$ is the space whose points are
elements of $P$ and open sets are downward closed subsets of $P$. Along
the way, we show that $p^{\downarrow} : P^{\downarrow}\to
B^{\downarrow}$ is exponentiable if and only if $p : P\to B$ is
exponentiable in the category of posets and satisfies an additional
compactness condition. The criteria for exponentiability of morphisms of
posets is related to (but weaker than) the factorization-lifting property
for exponentiability of morphisms in the category of small categories
(considered independently by Giraud and Conduché).

Keywords: presheaf topos, poset, locale, exponentiable, factorization lifting, metastably
locally compact, discrete opfibration.

2000 MSC: 06D22, 06B35, 18B25, 18B30, 18D15, 54C35, 54D45.

*Theory and Applications of Categories*, Vol. 8, 2001, No. 2, pp 16-32.

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