Logarithmically regular morphisms

We consider the stack [Formula: see text] parametrizing log schemes over a log scheme X, and weak and strong properties of log morphisms via [Formula: see text] , as defined by Olsson. We give a concrete combinatorial presentation of [Formula: see text] , 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.