Logarithmic stacks and minimality

TL;DR

This paper develops a categorical framework to classify groupoid fibrations over log schemes, unifying various notions of minimality in log geometry through a purely category-theoretic approach.

Contribution

It introduces a categorical classification of fibrations over log schemes based on minimal objects, connecting existing concepts in log geometry.

Findings

Classification of fibrations via minimal objects

Unification of minimality notions in log geometry

Purely category-theoretic framework for log structures

Abstract

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.