open geometric morphism
Cohomology and homotopy
In higher category theory
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)
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)
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)
Revised on August 7, 2015 04:59:52
by Thomas Holder