# Logarithmically Regular Morphisms

## TL;DR

This paper provides a combinatorial framework for the stack of log schemes, establishes criteria for properties of log morphisms, and applies these to analyze logarithmic regularity and smoothness.

## Contribution

It introduces a concrete combinatorial presentation of the stack of log schemes and a criterion for when weak and strong properties of log morphisms coincide.

## Key findings

- Criteria for equivalence of weak and strong log morphism properties
- A chart criterion for logarithmic regularity and smoothness
- Main properties of logarithmic regularity derived

## Abstract

We consider the stack $\mathcal{L}og_X$ parametrizing log schemes over a log scheme $X$, and weak and strong properties of log morphisms via $\mathcal{L}og_X$, as defined by Olsson. We give a concrete combinatorial presentation of $\mathcal{L}og_X$, and prove a simple criterion of when weak and strong properties of log morphisms coincide. We then apply this result to the study of logarithmic regularity, derive its main properties, and give a chart criterion analogous to Kato's chart criterion of logarithmic smoothness.