In point set topology, an open map is a continuous map that sends open sets to open sets. The notion of an open geometric morphism is a generalization of this notion from topology to topos theory.
From a logical perspective, a geometric morphism $f:\mathcal{F}\to\mathcal{E}$ is open iff it preserves the interpretation of first order logic. This contrasts with general geometric morphisms which are only bound to preserve geometric logic.
is called open if the following equivalent conditions hold
the localic reflection of $f$ over $\mathcal{E}$ is an overt internal locale in $\mathcal{E}$;
the inverse image $f^*$ is a Heyting functor, hence preserves first order logic.
A geometric morphism $f:\mathcal{F}\to\mathcal{E}$ is open iff the canonical map $\lambda:\Omega_\mathcal{E}\to f_\ast(\Omega_\mathcal{F})$ of poset objects in $\mathcal{E}$ has an internal left adjoint $\mu :f_\ast(\Omega_\mathcal{F})\to\Omega_\mathcal{E}$.
(cf. Mac Lane-Moerdijk (1994), p.502)
A geometric morphism $f:\mathcal{F}\to\mathcal{E}$ is open iff the pullback of any bounded geometric morphism with codomain $\mathcal{E}$ is skeletal iff the pullback of any localic geometric morphism with codomain $\mathcal{E}$ is skeletal.
This result appears as corollary 4.9 in Johnstone (2006).
Peter Johnstone, Open maps of toposes, Manuscripta Math. 31 no.1-3 (1980) pp.217-247. (gdz, doi)
Peter Johnstone, Sketches of an Elephant vol.II , Oxford UP 2002. (section C3.1, pp.606-625)
Peter Johnstone, Complemented sublocales and open maps , Annals of Pure and Applied Logic 137 (2006) pp.240–255.
André Joyal, Myles Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 309
(1984).
Saunders Mac Lane, Ieke Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994². (sections IX.6-8, pp.493ff; X.3, pp.535-538)
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