# 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.