Monomorphisms in Categories of Log Schemes

In the present paper, we study category-theoretic properties ofmonomorphisms in categories of log schemes. This study allows one to give a purely categorytheoretic reconstruction of the log scheme that gave rise to the category under consideration. We also obtain analogous results for categories of schemes of locally finite type over the ring of rational integers that are equipped with “archimedean structures”. Such reconstructions were discussed in two previous papers by the author, but these reconstructions contained some errors, which were pointed out to the author by C. Nakayama and Y. Hoshi. These errors revolve around certain elementary combinatorial aspects of fan decompositions of two-dimensional rational polyhedral cones — i.e., of the sort that occur in the classical theory of toric varieties — and may be repaired by applying the theory developed in the present paper.