  Inspired by Zermelo's quasi-categoricity result characterizing the models of
second-order Zermelo-Fraenkel set theory $\text{ZFC}_2$, we investigate when
those models are fully categorical, characterized by the addition to
$\text{ZFC}_2$ either of a first-order sentence, a first-order theory, a
second-order sentence or a second-order theory. Thus we mount an analysis of
the categorical large cardinals. This mathematical analysis leads naturally to
philosophical issues concerning structuralism and realism, including especially
the tension between categoricity and reflection. Ultimately we identify grounds
for the preference of noncategoricity in one's foundations.
