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Stream: theory: mathematics

Topic: Question about pullback *of* toposes.

finegeometer (May 10 2024 at 04:08):

Hi,

Suppose we have a pullback diagram of toposes, as shown at the following quiver link. (Just to be clear, this is a pullback in the 2-category $\mathbf{Topos}$. The bullets represent toposes, and the arrows geometric morphisms between them.)

https://q.uiver.app/#q=WzAsNCxbMCwwLCJcXGJ1bGxldCJdLFsxLDAsIlxcYnVsbGV0Il0sWzAsMSwiXFxidWxsZXQiXSxbMSwxLCJcXGJ1bGxldCJdLFswLDEsIlxcYWxwaGEiXSxbMCwyLCJcXGJldGEiLDJdLFsyLDMsImEiLDJdLFsxLDMsImIiXSxbMCwzLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=

There are two obvious natural transformations from the functor $b^* \circ a_*$ to the functor $\alpha_* \circ \beta^*$. Are these the same transformation? And are these transformations in fact isomorphisms?

I tried to disprove a specific example of the latter, but seemed to prove it instead. I have no idea what to do to attack the general case.

Morgan Rogers (he/him) (May 10 2024 at 09:08):

We have $b^*a_* \Rightarrow\alpha_*\alpha^*b^*a_* \cong \alpha_*\beta^*a^*a_* \Rightarrow\alpha_*\beta^*$ constructed from the unit of $\alpha$, the isomorphism $\alpha^*b^* \cong \beta^*a^*$ and the counit of $a$ and $b^*a_* \Rightarrow b^*a_*\beta_*\beta^* \cong b^*b_*\alpha_*\beta^* \Rightarrow\alpha_*\beta^*$ constructed from the unit of $\beta$, the isomorphism $a_*\beta_* \cong b_*\alpha_*$ and the counit of $b$. On the face of it these look like they could be distinct, but hiding here is the relationship between the isomorphism of left adjoints and the isomorphism of right adjoints! I can confirm for you that these are equal, but I will leave you to figure out the details, it's a good exercise.

As for whether this transposed morphism is an isomorphism, if it is, the square is said to satisfy the Beck-Chevalley condition. This comes up most often when $a$ or $b$ has the property that all such pullback squares satisfy the condition, which turns out to be equivalent to some quite natural properties of geometric morphisms (the properties of being open and being proper/stably closed are respectively defined by weaker versions of this condition); see C2.4.16 of Johnstone's Sketches of an Elephant for the definition and read Section C3 to understand the relationship with other properties.