Exercise 9

Consider where is a poset considered as a category. We wish to characterize the subobject classifier of in this special case. A subset is a downset if implies . We use to denote the downset generated by , and to denote the set of downsets of . A sieve on is then precisely an element of . Thus, the subobject classifier can be defined with and the map that sends to if or itself otherwise.

The terminal presheaf is still , taking constant value . The monic will have components sending to (the maximal sieve). Take a monic , which we may consider to be an actual subfunctor in the sense that for all and is a restriction of for all arrows. The characteristic function has components with .

Consider as a category and consider . In this case, a downset of rational numbers corresponds to a one-sided Dedekind cut, thus each corresponds precisely to a real number via its supremum (see here for a discussion). However, we also consider a downset appearing in every ; we can represent this as an adjoined value . From this point of view, has a subobject classifer with .