Geometric Morphism

Let $\mathcal{E}$ and $\mathcal{F}$ be topoi (either elementary or Grothendieck).

Definition. A geometric morphism $f: \mathcal{E} \to \mathcal{F}$ is an adjunction $f^* \dashv f_*$ where:

• $f_*: \mathcal{E} \to \mathcal{F}$ is the direct image functor,
• $f^*: \mathcal{F} \to \mathcal{E}$ is the inverse image functor,

and $f^*$ is left exact (i.e., preserves finite limits).

The direction convention is: $f$ points in the direction of $f_*$, so that $f^*$ goes “backward,” mirroring the situation for continuous maps of topological spaces.

Proposition. The inverse image functor $f^*$ preserves the terminal object, binary products, and equalizers. Since every finite limit is built from these, left exactness is equivalent to preserving these three classes of limits.

Proposition. Since $f^*$ is a left adjoint, it preserves all colimits. Hence $f^*$ is an exact functor: it preserves all finite limits and all colimits.

Proposition. A geometric morphism $f: \mathcal{E} \to \mathcal{F}$ is an equivalence if and only if $f^*$ is an equivalence of categories (equivalently, if and only if both $f^*$ and $f_*$ are).

Proof sketch. If $f^*$ is an equivalence, its quasi-inverse serves as both a left and right adjoint to $f^*$, so $f_*$ is also an equivalence.

Proposition (Geometric morphisms from sheaf topoi). Let $X$ and $Y$ be sober topological spaces. There is a bijection between geometric morphisms $\mathrm{Sh}(X) \to \mathrm{Sh}(Y)$ and continuous maps $X \to Y$.

Proof sketch. A continuous map $\varphi: X \to Y$ induces $\varphi_*(F)(V) = F(\varphi^{-1}(V))$, and $\varphi^*$ is obtained by sheafifying the inverse image presheaf. Left exactness of $\varphi^*$ follows from exactness of the stalk functors and exactness of sheafification. Conversely, any geometric morphism between spatial topoi determines a continuous map on the underlying sober spaces via the correspondence between points of the topos and points of the space.

Examples.

• Global sections: For any Grothendieck topos $\mathcal{E}$, the unique geometric morphism $\gamma: \mathcal{E} \to \mathbf{Set}$ has direct image $\gamma_* = \Gamma = \mathrm{Hom}(1, -)$ (global sections) and inverse image $\gamma^* = \Delta$ (constant sheaf functor).
• Inclusions of subtopoi: A Lawvere-Tierney topology $j$ on $\mathcal{E}$ determines a geometric morphism $\mathrm{Sh}_j(\mathcal{E}) \hookrightarrow \mathcal{E}$ whose direct image is the inclusion and whose inverse image is $j$-sheafification. This is a geometric embedding: $f_*$ is full and faithful.
• Higher setting: For ($\infty$,1)-topoi, geometric morphisms are adjunctions $f^* \dashv f_*$ where $f^*$ preserves finite $\infty$-limits. The site-based correspondence generalizes accordingly.