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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 ...


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 $G_{A'}\dashv F_{A'}$.


Daniel Sch├Ąppi'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 ...


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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