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 Jun 2018

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Remark1.73.IfPandQare total orders andf:P!Qand1:Q!Pare drawn witharrows bending as in Exercise 1.72, we believe thatfis left adjoint to1iff the arrows donot cross. But we have not proved this, mainly because it is difficult to state precisely,and the total order case is not particularly general

The preservation of meets and joins, and hence whether a monotone map sustainsgenerative effects, is tightly related to the concept of a Galois connection, or moregenerally an adjunction.

Galois connections between posets were first considered by Évariste Galois—whodidn’t call them by that name—in the context of a connection he found between “fieldextensions” and “automorphism groups”. We will not discuss this further,
