Let $\mathcal E$ and $\mathcal F$ be two toposes. A geometric morphism $f\colon \mathcal E\to\mathcal F$ consists of an adjunction $f^\ast\dashv f_\ast$.

Let $f,g\colon \mathcal E\to\mathcal F$ be two geometric morphisms.

Questions: What is a 2-morphism from $f$ to $g$ in the 2-category of toposes?

Is it a natural transformation from $f^\ast$ to $g^\ast$ or from $f_\ast$ to $g_\ast$?

Can the notion of a 2-morphism from $f$ to $g$ be defined based on direct image and the inverse image (separately)?

Is a natural transformation from $f^\ast$ to $g^\ast$ the same as a natural transformation from $f_\ast$ to $g_\ast$?


1 Answer 1


A two-morphism $f \to g$ is called a geometric transformation. By definition, it is a natural transformation $f^* \Rightarrow g^*$. As mentioned in the nLab article, this is the same thing as a natural transformation $g_* \Rightarrow f_*$.

The convention that it is a natural transformation $f^* \Rightarrow g^*$ (rather than a natural transformation $f_* \Rightarrow g_*$) makes sense if you look at Diaconescu's Theorem. For a presheaf topos $\mathbf{PSh}(\mathcal{C})$ there is an equivalence of categories $$\mathbf{Geom}(\mathcal{E},\mathbf{PSh}(\mathcal{C})) \simeq \mathbf{Flat}(\mathcal{C},\mathcal{E})$$ where $\mathbf{Flat}(\mathcal{C},\mathcal{E})$ denotes the category of flat functors $\mathcal{C} \to \mathcal{E}$ and natural transformations between them. So the direction of the geometric transformation is the same as the direction of the natural transformation between the corresponding flat functors.


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