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5

I really like this question, I've been trying to sort out some of these ideas for a little while. I don't know the answer to your questions about conilpotence and twisting morphisms vs twisted arrows. I do have reason to believe that twisted arrows between A and C are the same as the twisted arrows from A to conil(C) but I don't know how to prove that. I ...

1

Yes. The map $G_{A'} \to F_f \circ G_A$ is called the mate of $T_f : F_A \to F_{A'} \circ F_f$. It's the composite $$G_{A'} \to G_{A'} F_A G_A \to G_{A'} F_{A'} F_f G_A \to F_f G_A$$ of $T_f$ with the unit of the adjunction $G_A \dashv F_A$ and the counit of the adjunction $G_{A'}\dashv F_{A'}$.

4

Daniel Schäppi's answer made me realize that I actually can say something about this. I'll keep Daniel's notation. Something more than what Daniel said is true, and it holds in a more general context: Given that $\mathcal{A}$ has finite limits, $F$ is left exact iff $\mathrm{Lan}_Y YF$ is. This is shown by Kelly (mimic Daniel's argument and use Kelly's Thm ...

6

This is true, even if the ring extension is not necessarily flat. It follows from the fact that $F$ is right exact. First some generalities: given an additive category $\mathcal{A}$ I will write $\mathcal{PA}$ for the category of additive presheaves (that is, additive functors ${\mathcal{A}}^{\mathrm{op}} \rightarrow \mathrm{Ab}$). The question when ...

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