Exercise 2.50

To answer 1., the objects of \(\cat{X}\) are simply the elements of the preorder, i.e. \(Ob(\cat{X}) = P\). For every pair of objects \((x, y)\) we need an element of \(\BB = \{false, true\}\). As before, simply take \(true\) if \(x \leq y\) and \(false\) otherwise to define the hom-object \(\cat{X}(x, y)\).

To go backwards as in Theorem 2.49, let \(P := Ob(\cat{X})\). For the \(\leq\) relation, let’s declare \(x \leq y\) iff \(\cat{X}(x, y) = true\). We’ve clearly gotten back to where we started.

To answer 2., let \(X := Ob(\cat{X})\). For the \(\leq\) relation, let’s declare \(x \leq y\) iff \(\cat{X}(x, y) = true\).

To go back to a \(\Bool\)-category, the objects of \(\cat{X}\) are simply the elements of the preorder, i.e. \(Ob(\cat{X}) = X\). For every pair of objects \((x, y)\) we need an element of \(\BB = \{false, true\}\). As before, simply take \(true\) if \(x \leq y\) and \(false\) otherwise to define the hom-object \(\cat{X}(x, y)\). We’ve clearly gotten back to where we started.