Geometric morphism

Summary

A geometric morphism $f \colon \mathcal{E} \to \mathcal{F}$ between toposes is a pair of adjoint functors $f^* \dashv f_*$, with $f_* \colon \mathcal{E} \to \mathcal{F}$, whose left adjoint $f^* \colon \mathcal{F} \to \mathcal{E}$ preserves finite limits. It is the notion of map between toposes that matches continuous maps between spaces. A continuous map $g \colon X \to Y$ gives a geometric morphism $\mathrm{Sh}(X) \to \mathrm{Sh}(Y)$, with $g_*$ the direct image and $g^*$ the inverse image of sheaves. Points of a topos, and the classifying toposes of geometric theories, are described through geometric morphisms.