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Consider the preorder \( P = Q = \underline{3} \).

1) Let \( f , g \) be the monotone maps shown below:

Is it the case that \(f\) is left adjoint to \(g\)?

Check that for each \( 1 \le p, q \le 3 \), one has \( f(p) \le q \text{ iff } p \le g(q) \).

2) Let \( f , g \) be the monotone maps shown below:

Is it the case that \(f\) is left adjoint to \(g\)?

## Comments

My answer is yes for both pairs of adjoints.

Edit: And it is wrong, as Jonathan points below. My method of checking wasn't good enough.

`My answer is yes for both pairs of adjoints. Edit: And it is wrong, as Jonathan points below. My method of checking wasn't good enough.`

For the second part, we have \(2 \le g(1)\) but not \(f(2) \le 1\), since \(2 = g(1)\) and \(f(2) = 2\). So it can’t be adjoint.

`For the second part, we have \\(2 \le g(1)\\) but not \\(f(2) \le 1\\), since \\(2 = g(1)\\) and \\(f(2) = 2\\). So it can’t be adjoint.`

Consider

Remark 1.78where a claim is made about crossing arrows, if they are 'bending'. In the book it is not clear that the arrows are crossing, in the diagram above it is pretty clear that they cross.What does arrow crossing signify?

`Consider **Remark 1.78** where a claim is made about crossing arrows, if they are 'bending'. In the book it is not clear that the arrows are crossing, in the diagram above it is pretty clear that they cross. What does arrow crossing signify?`

I don't find the "arrow crossing" principle very convincing -- it isn't difficult to adjust the figure so that no arrows cross, even though there's no adjoint. And when the preorders are more complicated than simple total orders, I expect there to be cases where arrows must cross even though there is an adjunction.

It works here because the arrow crossing principle is presumably valid on figures drawn by the authors, else the authors would not provide said principle (or said figures).

EDIT:Perhaps it might work more generally if one adds the proviso that in-arrows and out-arrows aresortedin where they attach to the points. But I still suspect there will be edge cases.`I don't find the "arrow crossing" principle very convincing -- it isn't difficult to adjust the figure so that no arrows cross, even though there's no adjoint. And when the preorders are more complicated than simple total orders, I expect there to be cases where arrows must cross even though there is an adjunction. It works here because the arrow crossing principle is presumably valid on figures drawn by the authors, else the authors would not provide said principle (or said figures). **EDIT:** Perhaps it might work more generally if one adds the proviso that in-arrows and out-arrows are _sorted_ in where they attach to the points. But I still suspect there will be edge cases.`

I think the authors were trying to motivate

Proposition 1.86by their remark. As an alternative to the 'crossing' arrows idea is that of approaching from above [or as drawn from the right] by the right adjoint and from below [or as drawn from the left] by the left.`I think the authors were trying to motivate **Proposition 1.86** by their remark. As an alternative to the 'crossing' arrows idea is that of approaching from above [or as drawn from the right] by the right adjoint and from below [or as drawn from the left] by the left.`