open geometric morphism
Could not include topos theory - contents
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 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.
A geometric morphism
is called open if the following equivalent conditions hold
A geometric morphism is open iff the canonical map of poset objects in has an internal left adjoint .
(cf. Mac Lane-Moerdijk (1994), p.502)
Peter Johnstone, Open maps of toposes, Manuscripta Math. 31 no.1-3 (1980) pp.217-247. (gdz)
Peter Johnstone, Sketches of an Elephant vol.II , Oxford UP 2002. (section C3.1, pp.606-625)
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)
Revised on August 2, 2015 09:13:50
by Thomas Holder