nLab open geometric morphism

Idea

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.

Definition

A geometric morphism

$$f := (f^* \dashv f_*) : \mathcal{F} \to \mathcal{E}$$

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.

Examples

• locally connected geometric morphisms are open. (cf. Johnstone (2002), p.650)

Properties

Proposition

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)

Proposition

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).

Related entries

• open morphism
• open subtopos
• proper geometric morphism

References

• 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)