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Exercise 82 - Chapter 1

edited June 2018 in Exercises

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

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

monotone maps

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:

map 2

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

Comments

  • 1.
    edited May 2018

    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.

    Comment Source: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.
  • 2.
    edited May 2018

    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.

    Comment Source: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.
  • 3.
    edited May 2018

    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?

    Comment Source: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?
  • 4.
    edited May 2018

    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.

    Comment Source: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.
  • 5.
    edited May 2018

    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.

    Comment Source: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.
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