  Given a category fibered in groupoids over schemes with a log structure, one
produces a category fibered in groupoids over log schemes. We classify the
groupoid fibrations over log schemes that arise in this manner in terms of a
categorical notion of "minimal" objects. The classification is actually a
purely category-theoretic result about groupoid fibrations over fibered
categories, though most of the known applications occur in the setting of log
geometry, where our categorical framework encompasses many notions of
"minimality" previously extant in the literature.
