Geometric Morphism

A geometric morphism $F:\mathcal{E}\to\mathcal{F}$ between to Topoi consists of a pair of Adjoint Functors $F^{*}\dashv F_{*}$ $F_{*}:\mathcal{E}\to\mathcal{F}$, $F^{*}:\mathcal{F}\to\mathcal{E}$ such that $F^{*}$ preserves Finite Limits.

By convention, the functor $F^{*}$ is called the "inverse image" and $F_{*}$ th "direct image".

Examples

• There is a unique geometric morphism from a Grothendieck Topos $\mathcal{E}$ to the Category of Sets: the inverse image sends a set $X$ to the Copower $\coprod_{x:X}\top$ of the Terminal Object of $\mathcal{E}$, and the direct image sends an object $X$ to it set of Global Elements ${{\top}\to{X}}$.

  Another way to see this connection is that the Category of Sets is Isomorphic to a Presheaf Category; namely, presheaves over the Terminal Category.